Sharp Hardy–Littlewood–Sobolev inequalities on quaternionic Heisenberg groups

Sharp Hardy–Littlewood–Sobolev inequalities on quaternionic Heisenberg groups

Linear Algebra Applications Nonlinear Analysis and 130 its (2016) 361–395 466 (2015) 102–116 Contents lists at ScienceDirect Contents lists available...

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Linear Algebra Applications Nonlinear Analysis and 130 its (2016) 361–395 466 (2015) 102–116

Contents lists at ScienceDirect Contents lists available at available ScienceDirect

Linear Algebra and its Applications Nonlinear Analysis www.elsevier.com/locate/laa www.elsevier.com/locate/na

Inverse eigenvalue problem of Jacobi matrix Sharp Hardy–Littlewood–Sobolev inequalities on quaternionic with mixed data Heisenberg groups 1 Ying Wei Michael Christ a , Heping Liu b , An Zhang a,b,∗ a b

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA Nanjing 210016, PR China School of Mathematical Sciences, Peking University, Beijing, 100871, China

highlights

a r t i c l e

i n f o

a b s t r a c t

Article history: In this paper, the inverse eigenvalue problem of reconstructing Received 16 January 2014 a Jacobi matrix from its eigenvalues, its leading principal Accepted 20 September 2014 • We get sharp Hardy–Littlewood–Sobolev inequalities on Quaternionic Heisenberg groups. submatrix and part of the eigenvalues of its submatrix AvailableLog-Sobolev online 22 October 2014 are given. • Dual Sobolev and endpoint inequalities is considered. The necessary and sufficient conditions for Submitted by Y. Wei

• The method is symmetrization-free and center of the group is the highexistence dimensional. and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical MSC: examples are given. abstract a r t i c l e i n f15A18 o © 2014 Published by Elsevier Inc. 15A57 Article history: Keywords: Received 27 January 2015 Jacobi matrix Accepted 16 October 2015 Eigenvalue Communicated by S. Carl Inverse problem Submatrix MSC: 26D10 35A23 35R03 42B37 53C17 Keywords: Extremizers Hardy–Littlewood–Sobolev inequalities Quaternionic Heisenberg group Conformal symmetry

1. Introduction 1

In this paper, we get several sharp Hardy–Littlewood–Sobolev-type inequalities on quaternionic Heisenberg groups, using the symmetrization-free method of Frank and Lieb, who considered the analogues on the Heisenberg group. First, we give the sharp Hardy–Littlewood–Sobolev inequality on the quaternionic Heisenberg group and its equivalent on the sphere, for singular exponent of partial range λ ≥ 4. The extremal function, as we guess, is “almost” uniquely constant function on the sphere. Then their dual form, a sharp conformally-invariant Sobolev-type inequality involving a (fractional) intertwining operator, and the right endpoint case, a Log-Sobolev-type inequality, are also obtained. Higher dimensional center brings extra difficulty. The conformal symmetry of the inequalities, zero center-mass technique and estimates involving meticulous computation of eigenvalues of singular kernels play a critical role in the argument. © 2015 Elsevier Ltd. All rights reserved.

E-mail address: [email protected] Tel.: +86 13914485239.

Sharp constants and extremal functions for important inequalities, especially Sobolev-type, have been http://dx.doi.org/10.1016/j.laa.2014.09.031 studied since many 0024-3795/© years ago. 2014 It has been abygeneral Published Elsevier hot Inc. topic in analysis, geometry, probability, PDE and ∗ Corresponding author. E-mail addresses: [email protected] (M. Christ), [email protected] (H. Liu), [email protected] (A. Zhang).

http://dx.doi.org/10.1016/j.na.2015.10.018 0362-546X/© 2015 Elsevier Ltd. All rights reserved.

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quantum field theory. They play an important role because they almost always contain or reveal profound geometric and probabilistic information of the underlying space, manifold or group. Vast beautiful literature has been done on this subject. Despite rich profound results in the framework of Riemannian geometry, the problem in the sub-Riemannian world is more interesting, but far away from being absolutely understood while some conclusive results have been obtained recently. Among so many geometric inequalities, Sobolev-type inequalities exceptionally attract attention of analysts. In this paper, we will discuss some Hardy–Littlewood–Sobolev-type (HLS) inequalities. We use notations “∼” and “.” respectively for “=” and “<” through this paper, modulo a constant we do not care in the present context. s The classical HLS inequality on Rn states that for 0 < s < n, the Riesz potential denoted by (−∆)− 2 , a convolution operator with kernel |x|s−n (or called fractional integral operator), is a linear operator of strongtype (p, q) for particular exponents, i.e., it is bounded from Lebesgue spaces Lp to Lq . This inequality can be viewed as the weak Young inequality, considering the fact that function |x|−λ is indeed contained in weak n Lebesgue space Lwλ , although not in any Lp . For symmetry, we can write it in a sesquilinear form      f (x)g(y)   dxdy   . ∥f ∥p ∥g∥q  Rn ×Rn |x − y|λ  with 0 < λ < n and 1 < p, q < ∞ satisfying p1 + 1q + nλ = 2. It was found by Hardy, Littlewood and Sobolev almost a century ago [26,27,43]. The existence of extremal functions, called extremizers, was first proved by Lieb in [39], combining the Riesz rearrangement inequality, refined Fatou lemma and compactness argument. Another proof of existence was given by Lions in [40], using the famous concentration compactness principle. By exploiting the intrinsic conformal symmetries and using symmetric decreasing rearrangement, Lieb [39] 2n−λ 2n that function (1 + |x|2 )− 2 is the “almost unique” also proved for symmetric exponents p = q = 2n−λ extremizer, ignoring constant multiples, translations and dilations, and the sharp constant, the infima for the inequality to hold, was also given. Actually, there exists for the inequality a large conformal symmetry (invariance) group, consisting of not only translations, dilations, rotations, but also an interesting inversion f (x) → |x|−(2n−λ) f (x|x|−2 ). This large symmetry group is the main difficulty for the existence proof of extremizers, which tells non-uniqueness and easily vanishes the weak limit of any extremizing sequence. However, it also gives some convenience to find some special extremizer and a unified competing symmetry method for the existence and calculation of extremizers was given by Carlen and Loss in [7]. They constructed a special strong limit using alternatively the conformal action and rearrangement to any positive Lp function, which ingeniously balanced between the “bad” and “good” roles of the symmetry group. Other symmetric rearrangement-free methods of finding the sharp constant and extremizers can be found in the work of Frank and Lieb [17,18]. The first reference used inversion-positivity to get a result for partial exponent λ ≥ n − 2, while the second one demonstrate on Rn the method used for the Heisenberg group in a seminal paper [19]. The analogous HLS inequality on the Heisenberg group Hn parameterized by Cn × R originates from Folland and Stein [16], which states that      f (u)g(v)   dudv   . ∥f ∥p ∥g∥q  Hn ×Hn |u−1 v|λ  λ with 0 < λ < Q and 1 < p, q < ∞ satisfying p1 + 1q + Q = 2, where u−1 v, du, Q, | · | are respectively the left translation, Haar measure, homogeneous dimension and norm associated to dilation δ : u = (z, t) → δu = (δz, δ 2 t). For the sharp problem, existence of extremizers for general exponents was recently proved by Han [24] using Lions’ concentration compactness lemmas. For calculation of extremizers for symmetric exponents p = q, first important progress was due to Jerison and Lee [32] for λ = Q − 2, whose extremizer was proved to be “uniquely”  − Q+2 4 f ∼ g ∼ (1 + |z|2 )2 + |t|2 ,

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modulo a conformal symmetry group and they used the extremizers to study Cauchy–Riemann Yamabe problem about solutions to related Euler–Lagrange equation. Actually the authors gave a dual sharp Sobolev embedding inequality of order 2, and general Sobolev-embedding states that ∥f ∥p∗ . ∥Xf ∥p , with 1 < p < Q pQ and p∗ = Q−p , where X is the horizontal gradient. Recently, Branson, Fontana and Morpurgo gave in [5] the endpoint case λ → 0, a sharp Log-HLS inequality, which states that for any two normalized nonnegative functions f, g ∈ L log L ∩ L log(1 + |u|2 ),     |S2n+1 | 2 f log f + g log g + 2 log 2 |S2n+1 | , f (u) log −1 2 g(v)dudv ≤ |u v| Q Hn Hn Hn ×Hn with almost unique extremizer  − Q f ∼ g ∼ (1 + |z|2 )2 + |t|2 2 , from which it is also conjectured that function 

(1 + |z|2 )2 + |t|2

− 2Q−λ 4

(1.1)

be the almost unique extremizer for general λ. Inspired by the former two special cases, Frank and Lieb [19] recently killed this conjecture completely, by proving that a cleverly chosen extremizer is just the function above (1.1). It is easy to note from the explicit formula (1.1) that this extremizer is similar to, but very different from that on Rn . The level set of the Euclidean extremizer is just the Euclidean sphere, however, because of the different degrees of the coordinates in (1.1), the level set of this extremizer is neither that of the homogeneous norm, nor the isoperimetric surface (w.r.t the perimetric measure [6]). So, the symmetric rearrangement method for the Euclidean case will not work again for the Heisenberg case. However, Frank and Lieb cutely choose a nice direction for the second variation around a special extremizer, which is fortunately proved to be nonnegative1 with equality reached only by the right function. This has similar flavor to an idea of Chang and Yang [8], originating from [28]. They also used an alternative ingenious compactness argument to prove the existence of extremizers. A related doubly-weighted version of the HLS (Stein–Weiss inequalities) on the Heisenberg group was recently obtained by Han–Lu–Zhu [25] where, however, the sharp constants still remain open. The asymptotic behavior for the possible extremizers was also considered there. Some other recent, important related results for the so-called Moser–Trudinger inequalities, an exponential class embedding at the endpoint, are Cohn–Lu [10,11], Lam–Lu [38]. Now, as conjectured in [19], we want to extend the results onto more general 2-step nilpotent Lie groups of Heisenberg type (H-type) introduced by Kaplan [34]. We first consider a subclass—groups of Iwasawa type (Itype) [13],2 the nilpotent part of the Iwasawa decomposition of a semisimple Lie group of rank one, which can be viewed as the isometry group of associated noncompact symmetric space.3 It is known from the Clifford algebra characterization that there are only four cases for groups of I-type, including Euclidean spaces, classical Heisenberg groups, quaternionic Heisenberg groups and a 15-dimensional octonionic Heisenberg group. This paper is focused on the quaternionic Heisenberg group G with 3-dimensional center, parameterized by its Lie algebra Hn ×Im H (H is the quaternions4 ) and we get a parallel result to that on the Heisenberg group. 2Q Accurately, we proved that, for Q > λ ≥ 4 and symmetric exponent p = 2Q−λ , extremizer exists for the sharp HLS  f (u)g(v) . ∥f ∥p ∥g∥p , (1.2) −1 v|λ G×G |u 1 The second variation around an extremizer should be non-positive. 2 This work gave a sufficient and necessary J -condition for a H-type group to be of I-type, which allows to study rank one 2 non-compact symmetric spaces avoiding classification and the semisimple Lie group theory. 3 Harmonic extension of the I-type group. 4 It is easy to distinguish from the notation Hn for the n order Heisenberg group.

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and is almost uniquely  − 2Q−λ 4 , f ∼ g ∼ (1 + |q|2 )2 + |w|2

(1.3)

with group element u = (q, w) and Q, | · | is the homogeneous dimension and norm. Already known results by us are for λ = Q − 2 in a series of papers [21,29,30]. The authors considered a dual sharp embedding inequality ∥f ∥ 2Q . ∥Xf ∥2 when studying the sub-Riemannian Yamabe problem, where X is the horizontal Q−2

gradient. The authors in [21] proved in the unifying I-type framework that extremizer satisfying a partial symmetry for above embedding inequality is almost uniquely  − Q−2 4 , (1 + |q|2 )2 + |w|2 which is also used there to give all solutions with additional symmetry to the Yamabe equation. [29] gave all extremizers on the seven dimensional sphere, while [30] extended the result to all dimensions. Both the latter two papers got rid of the symmetric assumption. So, our result also extends [30]. The conformal symmetry group is again the core in our analysis. We use the strategy of Frank and Lieb in our proof with extra difficulty from higher dimension of center. The existence was proved in a dual form as the distance power kernel is positive definite, combining a refined HLS, Brezis–Lieb lemma and a Rellich–Kondrachov-type lemma. As before on Rn and Hn , there is a large conformal symmetry group for (1.2), which makes it very easy for the weak limit of an extremizing sequence to vanish. However, it is proved that left translations and dilations are the only ways of losing compactness, which means, through them we can recover a strong limit by pulling the extremizing sequence back and the refined HLS is a beautiful tool to realize this idea. An alternative method for general non-symmetric exponents is Lions’ concentration compactness argument. In finding the explicit extremizers, boundary extension of the Cayley transform, which is an isomorphism between two models of related hyperbolic space, Siegel upper domain D and unit ball B, is a basic tool to shift the inequality from the group onto the sphere. First, it is easier to deal on the sphere. Second, it gives a more clear view about how to choose the special extremizer that we finally proved to be constant, which just corresponds to the function (1.3). The critical step to filter the extremizer is utilizing the method from Herch, Chang and Yang to prove a purported inverse second-variation (ISV) inequality, which reaches equality only by constant functions. Actually, it is a natural idea to break the huge conformal symmetry group by restricting extremizers to functions satisfying certain zero center-mass condition as in above references. Fortunately, by checking the eigenvalues of the quadratic forms in the ISV inequality, any extremizer with zero center-mass property can only be constant function and finally, by turning back, we figure out all extremizers. However, a bit pity is that the special extremizer we choose is proved to be constant function only for partial exponent λ ≥ 4. Note the special case λ = Q − 2 = 4n + 4 ≥ 4 is included. For λ < 4, it seems that the chosen second variation and zero center-mass condition is not enough to filter the extremizers to be constant function. So, maybe more information of the conformal symmetry group should be excavated, while Euler–Lagrange equation can also do some favor. Several examples were computed, which guide us to a positive answer. We anticipate that constant should still be the “unique” extremizer for λ in this gap, which will be studied in our forthcoming paper. However, we remark here a very weak “local” extreme that the second variation around 1 is strictly negative. The dual form of the sharp HLS, a sharp conformally-invariant Sobolev-type (CIS) inequality involving intertwining operators associated to the complementary series representation of Lorentz group Sp(n, 1), and the right endpoint case, a sharp Log-Sobolev-type (LogS) inequality, are also obtained in this note. In [9], we worked out a similar result about the sharp HLS on the octonionic Heisenberg group, which together with [39,19] and this paper, give a (partial) answer to the conjecture on I-type groups. Related stability on remainder terms was obtained in [41]. For general H-type or Siegel-type (high rank) groups, the higher dimensional center and lack of good symmetry structure as I-type both give more involved difficulty. This problem and a multilinear HLS will be considered in our forthcoming works.

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Arrangement of this paper: In Section 2, we will give some notations, definitions and basic formulas. In Section 3, we give our main theorems. In Section 4, we proved the existence of extremizers for HLS. In Section 5, we proved an almost uniqueness and give characterization of all extremizers. Section 5.1 breaks the conformal symmetry by adding the zero center-mass condition. Then we compute the second variation and reduce the problem to a claim about a quadratic ISV inequality in Section 5.2. The last but critical step is to prove the claim using Funk–Hecke formula in Sections 5.4 and 5.5. The sharp constants and explicit formulas of extremizers were proved in Section 5.3 and Appendix B. 2. Preliminary Quaternionic Heisenberg group. The quaternions H = {q = a + bi + cj + dk | a, b, c, d ∈ R; i2 = j 2 = k 2 = ijk = −1}, endowed with multiplication qq ′ = (aa′ − bb′ − cc′ − dd′ ) + (ab′ + a′ b + cd′ − dc′ )i + (ac′ + a′ c + db′ − bd′ )j + (ad′ + a′ d + bc′ − cb′ )k, form a real division algebra. The real and imaginary parts, conjugate and norm of q are Re q = a,

Im q = bi + cj + dk,

q¯ = a − bi − cj − dk,

|q|2 = a2 + b2 + c2 + d2

satisfying qq ′ = q ′ q¯,

|q|2 = q q¯,

|qq ′ | = |q| |q ′ |.

The quaternions H ≃ C2 as q = (a + bi) + (c + di)j, which can be represented by 2-order complex matrix   a + bi c + di Aq = , −(c − di) a − bi preserving the multiplications, i.e. Aqq′ = Aq Aq′ . So, if we use notation q = (ζ 1 , ζ 2 )(ζ 1 , ζ 2 ∈ C) for q = ζ 1 + ζ 2 j and take q ′ = (η 1 , η 2 ), then q¯ = (ζ 1 , −ζ 2 ) and qq ′ = (ζ 1 η 1 − ζ 2 η 2 , ζ 1 η 2 + ζ 2 η 1 ), qq ′ = (ζ 1 η 1 + ζ 2 η 2 , ζ 2 η 1 − ζ 1 η 2 ). n For q = (q1 , . . . , qn ) ∈ Hn , we use scalar product q · q ′ = j=1 qj qj′ , then quaternionic unitary group, also called compact symplectic group, Sp(n) = {A ∈ Mn (H) : AT A = AAT = I} preserves this product by right action on Hn . The quaternionic Heisenberg group is a 2-step nilpotent Lie group identified with its Lie algebra G = Hn × ImH with group multiplication law5 uu′ = (q, w)(q ′ , w′ ) = (q + q ′ , w + w′ + 2Imq · q ′ ), where u = (q, w), u′ = (q ′ , w′ ) are two group elements. Here “·” is the scalar product in Hn . This group can be viewed as the nilpotent part of Iwasawa decomposition of Lorentz group Sp(n + 1, 1), which is the isometry 5 Note here that we need take a little care to choose a proper inner product on the Lie algebra to make G with this law to be a group of H-type.

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group of associated rank one hyperbolic symmetric space over quaternions, identified with homogeneous   space Sp(n + 1, 1)/ Sp(n + 1)Sp(1) . We denote the homogeneous dimension and norm by Q = 4n + 6,

1

|u| = |(q, w)| = (|q|4 + |w|2 ) 4 ,

associated to the group dilation δu = (δq, δ 2 w) for any δ > 0 and use du = dqdw, the Lebesgue measure, for the invariant Haar measure. Then the group distance of two elements u = (q, w), v = (q ′ , w′ ) is defined naturally by  1 dG (u, v) = |v −1 u| = |q − q ′ |4 + |w − w′ + 2 Im q · q ′ |2 4 . The left-invariant vector fields corresponding to one-parameter real coordinates subgroups are given by Xj0 =

∂ ∂ ∂ ∂ + 2qj1 + 2qj2 + 2qj3 , ∂qj0 ∂w1 ∂w2 ∂w3

Xj1 =

∂ ∂ ∂ ∂ − 2qj0 − 2qj3 + 2qj2 , ∂qj1 ∂w1 ∂w2 ∂w3

Xj2 =

∂ ∂ ∂ ∂ + 2qj3 − 2qj0 − 2qj1 , 2 ∂qj ∂w1 ∂w2 ∂w3

Xj3 =

∂ ∂ ∂ ∂ − 2qj2 + 2qj1 − 2qj0 , 3 ∂qj ∂w1 ∂w2 ∂w3

Tk =

∂ , ∂wk

1 ≤ j ≤ n, 1 ≤ k ≤ 3,

all of which form together a basis of the Lie algebra and a second-order left-invariant differential operator, the sublaplacian,6 is defined by  1 L=− (Xjk )2 , 4 1≤j≤n, 0≤k≤3

which is independent of the choice of basis and hypoelliptic from a famous theorem of H¨omander. The fundamental solution of L was proved, by analogy with Folland’s argument,7 to be     Q Q−6 2 2 −5 Γ Q−2 Γ 4 4 L−1 (u, v) = d2−Q (u, v). (2.1) Q G −1 2 π An analogue was proved in [34] for general H-type, heuristically believed to be the biggest group class with this property. The Cayley transform and Peter–Weyl decomposition. We denote the quaternionic sphere by S = {ζ = (ζ ′ , ζn+1 ) ∈ Hn × H : |ζ ′ |2 + |ζn+1 |2 = 1} ∼ = S4n+3 endowed with the Lebesgue surface measure dζ. The quaternionic Heisenberg group G is then equivalent to the punctured sphere S \ {o} with south pole o = (0, . . . , 0, −1), through the (boundary) Cayley transform, which and its inversion are defined by8 C:

G −→ S \ {o}

u = (q, w) −→ ζ = (ζ ′ , ζn+1 ) =



6 A counterpart of the Euclidean Laplacian −∆. 7 In [15], Folland proved a similar formula on the Heisenberg group. 8 We use the left quotient q = (q ′ )−1 q. q′

2q 1 − |q|2 + w , 2 1 + |q| − w 1 + |q|2 − w

 ,

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C −1 : S \ {o} −→ G ζ = (ζ ′ , ζn+1 ) −→ u = (q, w) =



1 − ζn+1 ζ′ , −Im 1 + ζn+1 1 + ζn+1

 ,

(2.2)

with Jacobian determinant  − Q |JC (u)| =2Q−3 (1 + |q|2 )2 + |w|2 2 =2−3 |1 + ζn+1 |Q .

(2.3) n

This transform is a generalization of the stereographic projection on R and is actually conformal. We define a distance on S by 1

1

dS (ζ, η) = 2− 2 |1 − ζ · η| 2 .

(2.4)

It is interesting to note that there exists the following relation between distances on the sphere S and group G,  − 1  − 1 dS (ζ, η) = (1 + |q|2 )2 + |w|2 4 (1 + |q ′ |2 )2 + |w′ |2 4 dG (u, v) 1

3

1

=2 Q −1 |JC (u)| 2Q |JC (v)| 2Q dG (u, v).

(2.5)

The Cayley transform and sphere distance both have generalized counterparts on H-type groups, see [4,2],9 and [31] is also a nice book for reference. If we take complex coordinates 1 2 ζ = (ζ11 , ζ12 ; · · · ; ζn1 , ζn2 ; ζn+1 , ζn+1 ) ∈ C2n+2

with ζj1 , ζj2 ∈ C for 1 ≤ j ≤ n + 1,

then we can write the sphere distance (2.4) in the following complex form 1

1

dS (ζ, η) = 2− 2 (|1 − ζ ·C (η 1 , η 2 )|2 + |ζ ·C (−η 2 , η 1 )|2 ) 4 , where we use the complex product “·C ”. Note that the distance dS is not invariant under the action of complex unitary group U (2n + 2), so the associated integral operator with kernel dS (ζ, η) is not diagonal with respect to the irreducible complex bigraded spherical harmonics decomposition of L2 (S). However, this is not strange as the quaternion unitary group Sp(n + 1) is much smaller than U (2n + 2). Actually, the maximal compact subgroup (stabilize the origin point) of the isometry group Sp(n + 1, 1) of the quaternionic (ball-model) symmetric space is Sp(n+1)Sp(1), which leaves invariant and acts transitively on the boundary sphere and its subgroup leaving the north pole fixed is isomorphic to Sp(n)Sp(1), then we can realize the    2 2 ∼ spherical principle series representation of Sp(n + 1)Sp(1) on L (S) = L Sp(n + 1)Sp(1)/ Sp(n)Sp(1) . We need to decompose further the real spherical harmonic subspace into more Sp(n + 1)Sp(1)-irreducible subspaces as the action of Sp(n + 1)Sp(1) is contained in O(4n + 4). Going to details, we first have (real) O(4n + 4)-irreducible decomposition  L2 (S) = Hk , k≥0

with Hk being the space of k-homogeneous harmonic polynomials in real variables, and for quaternionic case, as Hk is also invariant under the action of Sp(n + 1)Sp(1), then we can decompose Hk further into Sp(n + 1)Sp(1)-irreducible parts, and finally the Sp(n + 1)Sp(1)-irreducible decomposition is given by  L2 (S) = Vj,k , (2.6) j≥k≥0

9 [4] defined a general “distance” on the boundary of solvable extension in a unifying H-type setting, d (ζ, η) = 2− 21 |P ζ − η| 21 , S η with Pη being the orthogonal projection operator onto Tη(2) ⊕ Rη, where Tη(2) is the second layer of tangent space at point η. This function dS was proved to be a distance for and only for I-type, when the Cayley transform is conformal with respect to the C-C metric (see also [2] for Sobolev spaces and related operators).

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where Vj,k ⊂ Hj+k is called (quaternionic) “(j, k)-bispherical harmonic” subspace, which is generated by the action of Sp(n + 1)Sp(1) on the zonal spherical harmonic10 given by (Theorem 3.1 (4) in [33])  j+k  [   2 ] j+k−l   Zj,k (ζ) ∼ ProjH  (−1)l (2Reζn+1 )j+k−2l |ζn+1 |2l  l l=k



sin(j − k + 1)φ (2n−1,j−k+1) cosj−k θPk (cos 2θ), sin φ

(2.7)

with |ζn+1 | = cos θ, Reζn+1 = cos θ cos φ (θ ∈ [0, π2 ], φ ∈ [0, π]), where ProjH is the projection operator (2n−1,j−k+1) onto harmonic subspace H consisting of all harmonic polynomials and Pk (z) is a Jacobi polynomial.11 See e.g. [33] for the classical spherical harmonic realization of spherical principle series. In other words, we get the Peter–Weyl decomposition of L2 (S) under the representation action of Sp(n + 1)Sp(1). Using the explicit formula for the reproducing kernel of bispherical harmonic subspaces, we will give a quaternionic analogue of the Funk–Hecke formula, which is useful in proving a critical quadratic ISV inequality for sharp HLS inequality by computing the associated eigenvalues. The sublaplacian and conformal sublaplacian on S are defined by   2 1 L′ = − |C∗ Xjk |−1 C∗ Xjk , D = L′ + n(n + 1), 4 1≤j≤n,0≤k≤3

where C∗ is the induced tangent map of the Cayley transform. The operator D is an analogue of the Gellertype sublaplacian on the complex sphere [22]. The fundamental solution and spectrum w.r.t (2.6) of D are given by         Q−6 Q+2 Q+2 Γ k + Γ Γ j + − 1 Γ Q−2 4 4 4 4 ,   d2−Q (ζ, η), λj,k (D) =  (2.8) D−1 (ζ, η) = Q Q S −1 Q−2 Q−2 2 2 2 π Γ j+ 4 Γ k+ 4 −1 and from the definitions, we have the following relation between the sublaplacian L on G and conformal sublaplacian D on S,   Q−2 Q+2 L (23 |JC |) 2Q (F ◦ C) = (23 |JC |) 2Q (DF ) ◦ C, ∀F ∈ C ∞ (S). (2.9) See e.g [2] for a proof in unifying I-type setting. Note that the constant 23 comes from the difference between the standard volume elements we use here and those associated with the standard contact forms for G and S. The sub-(and conformal sub-) laplacians have one kind of generalization: intertwining operators. Intertwining operators on G and S: general intertwining operators are defined for principle (complementary) series representations of semisimple Lie groups (of real rank one), here we concern the Lorentz group Sp(n + 1, 1). See e.g. [35,33,12] and that of SU (n + 1, 1) was also studied in [5] for Heisenberg groups in analysis language. Denote Aut(G) the set of all conformal transformations12 on G, any element of which is composition of translations, rotations (on q), dilations and inversion   q w σinv : (q, w) → − 2 ,− . (2.10) |q| − w |q|4 + |w|2 η 10 We call elements in V j,k (bi)spherical harmonics and define a zonal harmonic Zj,k (ζ) in Vj,k to be the unique element, modulo a constant, that is invariant under the action of the subgroup fixing η, which is isomorphic to Sp(n)Sp(1). Uniqueness is guaranteed by [37]. When η = (0, . . . , 1), we denote it by Zj,k (ζ). Besides, the projection kernel onto Vj,k denoted by Zj,k (ζ, η) or Zj,k (ζ · η¯) is nothing but a zonal harmonic modulo a constant. 11 Jacobi polynomials {P (α,β) } α β n∈N is a family of orthogonal polynomials associated to weight (1 − z) (1 + z) on interval [−1, 1]. n 12 A conformal transformation is a diffeomorphism preserving the contact structure.

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It is a celebrated theorem that Aut(G) ∼ = Sp(n + 1, 1) and like the Cayley transform, this inversion is a 13 boundary extension of an isometry on related rank one symmetric space. For the quaternionic sphere, we denote similarly the set of all conformal transformations on S by Aut(S) = {τ = C ◦ σ ◦ C −1 : σ ∈ Aut(G)}. Before going on, it is necessary to define the (inhomogeneous Folland–Stein) Sobolev space of exponent (d, 2)(d > 0), denoted by W d,2 here, to be the completion of C ∞ (S) (or D(G)) w.r.t the norm ∥f ∥W d,2 = d d ∥(I + L′ ) 2 f ∥2 (or ∥(I + L) 2 f ∥2 ), where D(G) means the smooth function space of compact support on G. d It is obvious that ∥ · ∥W d,2 (S) is equivalent to ∥D 2 · ∥2 . Then for d ∈ (0, Q), we define the intertwining operator Ad on S to be any operator satisfying   Q−d Q+d (2.11) |Jτ | 2Q (Ad F ) ◦ τ = Ad |Jτ | 2Q (F ◦ τ ) , for all F ∈ C ∞ (S), τ ∈ Aut(S) and |Jτ | is the Jacobian determinant of τ . The definition states that operator Ad intertwines with two principle (complementary) series representations πd , π−d of Sp(n + 1, 1), Q+d with πd (τ ) : F → |Jτ | 2Q F ◦ τ . It was first proved in [33] (with more representation language) that the intertwining operator is diagonal w.r.t to the Sp(n + 1)Sp(1)-irreducible bispherical harmonic decomposition (2.6), and its spectrum is, modulo a constant dependent of d, uniquely given by,     Q+d Γ j + Q+d − 1 Γ k + 4 4   . λj,k (Ad |Vj,k ) =  (2.12) Q−d Q−d Γ j+ 4 Γ k+ 4 −1 We also simulate in Appendix A the calculus of [5] to get the equivalence between this spectral decomposition and the intertwining relation (2.11). Then the operator can be extended onto Folland–Stein–Sobolev space d W 2 ,2 (S). If we choose the d-dependent constant to be 1, then by (1) in Lemma 5.5 (corresponds to α = Q−d 4 ), the fundamental solution of Ad is given by ′ d−Q A−1 (ζ, η) d (ζ, η) = cd dS

(2.13)

with c′−1 = d

Q−d 2 +1

  Q π 2 −1 Γ d2   , Q−d Γ Γ Q−d − 1 4 4 2 

(2.14)

and this result is also listed in [2] and implicitly contained in [33]. The corresponding intertwining operator Ld on G can be defined similarly to be any operator satisfying   Q+d Q−d |Jσ | 2Q (Ld f ) ◦ σ = Ld |Jσ | 2Q (f ◦ σ) , (2.15) for all σ ∈ Aut(G), f ∈ D(G), or equivalently by a similar relation with Ad to (2.9) using the Cayley transform   Q−d Q+d Ld (23 |JC |) 2Q (F ◦ C) = (23 |JC |) 2Q (Ad F ) ◦ C, (2.16) for all F ∈ C ∞ (S). Take {Φkλ }k∈N,λ∈Im H∼ functions on G, which are =R3 to be the Sp(n)-spherical   ∂ ∂ ∂ 14 normalized joint radial eigenfunctions of sublaplacian L and T = ∂w1 , ∂w2 , ∂w3 , with joint spectrum   2(k + n)|λ|, iλ , and their formulas are given by 2

Φkλ (q, w) = eiλ ·R w−|λ| |q| L2n−1 (|λ| |q|2 ), k

(2.17)

13 For general H-type, we have similar inversion on the solvable extension—N A-group, which is proved to be an isometry (on the rank one symmetric space) if and only if the group is of I-type. For inversion on H-type and I-type group, see e.g. [36,13]. 14 We use “radial” here to mean that the function only depends on |q| and w.

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3 where λ ·R w = j=1 λj wj and L2n−1 is the kth classical Laguerre polynomial15 of order 2n − 1. Then we k can define the spherical Fourier transform “fˆ” on G using spherical functions (2.17), which acts on radial functions in L1 (G) as  ˆ: f −→ fˆ(λ, k) = f (u)Φkλ (u). (2.18) G

Generalized spherical functions and Fourier transform were studied on groups of H-type and their solvable extensions by Damek and Ricci [14]. From the intertwining relation (2.15) and using functional calculus of the spherical Fourier transform (2.18), we have that the operator Ld is, modulo a constant, uniquely given by     Q+d −1 Γ k + − 1 L + 2+d 4 d Γ |2T | d 4 ˆ  2 2   .  (2.19) f (λ, k), or L = |2T | L f (λ, k) = |2λ| d d Γ |2T |−1 L + 2−d Γ k + Q−d − 1 4 4 d

Then the operator can also be extended to the Folland–Stein–Sobolev space W 2 ,2 (G) as before. The fundamental solution of Ld is given by d−Q L−1 (u, v) d (u, v) = cd dG

(2.20)

cd = 2Q−d−3 c′d ,

(2.21)

with

where c′d is defined in (2.14). We can move the argument on the sphere directly onto the group to prove (2.19) and (2.20), which is also contained in [12], where similar operators on the nilpotent subgroups of all rank one semisimple Lie groups were studied. The intertwining operators are one kind of generalization of fractional (conformal) sublaplacians, whose second order counterparts are just the (conformal) sublaplacian defined above respectively on the sphere d and group, and especially on the Euclidean space and sphere, Ld = (−∆) 2 and Ad is the spherical picture d obtained from (−∆) 2 by the stereographic projection. We can also try to consider the intertwining operator at d = Q, then we may get corresponding analogue of Beckner–Onofri inequality, which is not included in this paper. Above results were all discussed for the Heisenberg group (and complex sphere) in [5] and can also be derived from the theory of Knapp–Stein intertwining operators in [35], both in the compact and noncompact pictures. 3. Main results Return to the HLS inequality, we now can state our main sharp HLS and related HLS-type theorems. The HLS with general exponents states (in the Hermitian form) that      f (u)g(v) λ 1 1   dudv + + = 2,   . ∥f ∥p ∥g∥q , −1 λ  G×G |u v|  p q Q which also have a fractional integral operator form: ∥f ∗ |u|−λ ∥q′ . ∥f ∥p . Define the associated functional      f (u)g(v)   I(f, g) :=  dudv  (∥f ∥p ∥g∥q ),   G×G |u−1 v|λ 15 Laguerre polynomials {Lα } −x α x on (0, ∞). n n∈N is a family of orthogonal polynomials w.r.t weight e

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then our extremal problem is to ask whether there is a non-vanishing function pair (f∗ , g∗ ), s.t., I(f∗ , g∗ ) =

sup

f ∈Lp , g∈Lq , f,g̸≡0

I(f, g),

and if exists, is it unique, in what sense and furthermore can we figure out all of these function pairs using elementary functions? We call this a priori function pair an extremizer pair and for the quadratic functional I(f ) := I(f, f ), we call f∗ an (quadratic) extremizer if I(f∗ ) = sup0̸≡f I(f ). Any sequence {fj } s.t. I(fj ) converges to the supremum is called an extremizing sequence while the finite supremum of the functional is called the sharp constant. For this HLS functional, we have the existence of extremizer pairs and can 2Q figure out all the extremizers for symmetric exponents p = q = 2Q−λ . We will use the same notations for counterparts on the sphere. 3.1. Sharp HLS inequalities Theorem 3.1 (Sharp HLS on the Quaternionic Heisenberg Group). Let 4 ≤ λ < Q = 4n + 6, p = ∀ f, g ∈ Lp (G),      f (u)g(v)   dudv  ≤ Cλ ∥f ∥p ∥g∥p    G×G |u−1 v|λ

2Q 2Q−λ ,

then

(3.1)

with sharp constant 2nλ Q

  2 π 2n+2 Γ Q−λ |S|1− p 2     Cλ = −1 Γ 2Q−λ Γ 2Q−λ 4 4 λ  Q−2  Qλ  Q−4  1− Q  Q−λ  ! Γ 2 2 2 π   ,  = Q−8 2Q−λ 2Q−λ 2 2 Γ − 1 Γ 4 4 21−

where |S| =

2π 2n+2 (2n+1)!

(3.2)

is the surface area of S. Moreover, all extremizers are given by  − 2Q−λ 1 f ∼ g ∼ (|JC ◦ σ| |Jσ |) p ∼  |q|2 + w − 2q0 · q¯ + r0  2 ,

(3.3)

with σ ∈ Aut(G), q0 ∈ Hn , r0 ∈ H, satisfying Rer0 > |q0 |2 , and we can choose the following correspondence: σ = Dδ0 ◦ Lu0 or Dδ ◦ C −1 ◦ Aξ ◦ C, where Dδ0 , Dδ are dilations, Lu0 is a left translation and Aξ is a ξ 2 − 21 , u0 = (q0 , − Im r0 ) rotation in Sp(n + 1) s.t. A−1 ξ (0, . . . , 0, 1) = |ξ| , with parameters δ0 = (Re r0 − |q0 | )  1∓|ξ| 2q0 r0 −1 and ξ = ( r0 +1 , r0 +1 ), δ = 1±|ξ| . Using the Cayley transform, we can give the sphere edition of last theorem: with the relation (2.5) between two distances on G and S, we have the following theorem, an equivalent sphere edition of sharp HLS inequality theorem, through the following correspondence between functions f on G and f˜ on S, 1 f˜(ζ) = f (C −1 ζ)|JC −1 | p .

Theorem 3.2 (Sharp HLS on the Quaternionic Sphere). Let 4 ≤ λ < Q = 4n + 6, p = ∀ f, g ∈ Lp (S),      f (ζ)g(η)   dζdη   ≤ Cλ′ ∥f ∥p ∥g∥p  S×S dλS (ζ, η) 

(3.4) 2Q 2Q−λ ,

then

(3.5)

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with sharp constant Cλ′ = 2

(4n+3)λ Q



  2 |S|1− p Γ Q−λ 2     = Γ 2Q−λ Γ 2Q−λ −1 4 4      Qλ  λ Q−4 Q−λ Q−2 2 2 !Γ 2 2 2 2π     , =  Q−4 2Q−λ 2Q−λ ! Γ Γ − 1 2 4 4 1+ λ 2

2

where |S| =

2π 2n+2 (2n+1)!

π

2n+2

(3.6)

is the surface area of S. Moreover, all extremizers are given by 1 ¯ − 2Q−λ 2 f ∼ g ∼ |Jτ | p ∼ |1 − ξ · ζ| ,

(3.7)

with τ ∈ Aut(S), ξ ∈ Hn+1 , |ξ| < 1, and we can choose the following correspondence: τ = C ◦ Dδ0 ◦ Lu0 ◦ C −1 or C ◦ Dδ ◦ C −1 ◦ Aξ , where Dδ0 , Dδ are group  dilations, Lu0 is a group left translation, with parameters

n+1 √ n+12| , u0 = ( 1+ξξ , − Im 1−ξ δ0 = |1+ξ 1+ξn+1 ), δ = n+1 ′

1−|ξ|

=

1±|ξ| 1∓|ξ| ,

and Aξ is a rotation in Sp(n + 1) s.t. A−1 ξ (0, . . . , 0, 1)

ξ |ξ| .

At first, we give several small remarks about the two theorems. • Extremizer for (3.1) and (3.5) exists for all 0 < λ < Q and even nonsymmetric exponents. Several a bit standard methods can be used to prove the existence while compactness is the basic idea. We will still borrow the ingenious argument from [19] and give the proof for λ ≥ 4. • The large conformal symmetry group of the HLS inequality (3.1) consists of left-translations, dilations, 2Q rotations (on q variable) and inversion: f (u) → f (σinv u)|u|− p (a Kelvin-type transform), where σinv is the inversion transform (2.10). In other words, the inequality is invariant under the conformal action 1 f → f ◦ σ|Jσ | p , ∀σ ∈ Aut(G). Similarly, from the correspondence (3.4), the HLS inequality on the sphere 1 (3.5) is invariant under the conformal action f → f ◦ τ |Jτ | p , ∀τ ∈ Aut(S). We can also see these from the following relations: 1

dG (σ(u), σ(v)) = (|Jσ (u)| |Jσ (v)|) 2Q dG (u, v), dS (τ (ζ), τ (η)) = (|Jτ (ζ)| |Jτ (η)|)

1 2Q

dS (ζ, η),

∀σ ∈ Aut(G) ∀τ ∈ Aut(S).

(3.8)

• Moreover, in Theorem 3.1, modulo only constant multiples, left-translations and dilations, extremizer exists uniquely, i.e., sharp equality (3.1) holds if and only if  − 2Q−λ 4 f = g = (1 + |q|2 )2 + |w|2 . Similarly, in Theorem 3.2, modulo only constant multiples, quaternionic rotations, and group dilations under the conjugate action of the Cayley transform, extremizer exists uniquely, i.e., sharp equality (3.5) holds if and only if f = g = 1. • For finding the explicit formula for extremizers, the core we want in this note, we select a special extremizer  which is proved to be constant for λ > 4 and be in direct sum of subspaces V0,0 j≥k≥2 Vj,k (see (2.6)) for λ = 4. Fortunately, for λ = 4, we can recur to the Euler–Lagrange equation of first variation to restrict further the extremizer to be in V0,0 which consists of constant functions. We can also see from the original

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functional of the inequality. For λ < 4, we say constant 1 is a “weak local” extremizer and guess it be global. We leave the proofs of the two main theorems in later sections and want to give some related sharp inequalities first. 3.2. Related sharp Sobolev-type inequalities We can also write above sharp HLS inequalities in dual CIS inequalities16 concerning intertwining operators, noticing that the fundamental solutions of intertwining operator Ad on S and Ld on G are a constant multiple of respectively dS (ζ, η)d−Q and dG (u, v)d−Q . See (2.13) and (2.19) and here we set d ∈ (0, Q). Corollary 3.3 (Sharp CIS Inequality). (1) Let 0 < d ≤ Q − 4, p′ =

2Q Q−d ,

d

then ∀f ∈ W 2 ,2 (G), 

f¯Ld f ≥ C˜d ∥f ∥2p′ ,

(3.9)

G

with sharp constant 

4

Q 2 −1

− Qd

2 π  C˜d = (cd CQ−d )−1 =   Q − 2 ! 2

   Γ Q−d 4 −1    , Q+d Γ Q+d Γ − 1 4 4

Γ



Q−d 4

(3.10)

and all extremizers  − Q−d f ∼  |q|2 + w − 2q0 · q¯ + r0  2 ,

(3.11)

with q0 ∈ Hn , r0 ∈ H, satisfying Rer0 > |q0 |2 . d 2Q (2) Let 0 < d ≤ Q − 4, p′ = Q−d , then ∀f ∈ W 2 ,2 (S),  S

f¯Ad f ≥ C˜d′ ∥f ∥2p′ ,

(3.12)

with sharp constant    − Qd  Q Γ Q−d Γ Q−d 2 −1 4 4 −1 2π     , =  Q Q+d Q+d − 2 ! Γ Γ − 1 2 4 4 

′ C˜d′ = (c′d CQ−d )−1

(3.13)

and all extremizers ¯− f ∼ |1 − ξ · ζ| 16 We call this fractional Sobolev inequality (FS) in [41].

Q−d 2

,

(3.14)

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with ξ ∈ Hn+1 , |ξ| < 1. ′ Constant cd , c′d are defined by (2.21) and (2.14) in preliminary section, and CQ−d , CQ−d are given by (3.2) and (3.6) in Theorems 3.1 and 3.2. It is obvious that we only need to consider the inequality for real-valued, or even nonnegative functions. Besides, we can easily checked that there exists a similar conformal symmetry group, i.e. the inequalities are 1 invariant under the action of the transformations f → f ◦ σ|Jσ | p′ , ∀σ ∈ Aut(G) (or Aut(S)). Also, as in [19], we can give the endpoint limit case of sharp HLS inequality at λ = Q, using standard functional differentiation argument. The endpoint case corresponds to a LogS inequality. We list it in the sphere framework. For another endpoint λ = 0, we lack the sharp HLS inequality of small exponent to obtain corresponding sharp Log-HLS inequality. Corollary 3.4 (Sharp LogS Inequality). ∀f ≥ 0 ∈ L2 LogL(S), normalized by   |f (ζ) − f (η)|2 dζdη ≥ C f 2 log f 2 , dQ (ζ, η) S S×S S

 S

f 2 = |S|, (3.15)

with sharp constant 2 2 +3 π 2 −1    , Q QΓ Q 4 −1 Γ 4 Q

C=

Q

(3.16)

and some extremizers ¯ − Q2 , f ∼ |1 − ξ · ζ|

(3.17)

satisfying normalized condition with ξ ∈ Hn+1 , |ξ| < 1. Now we give a simple proof of Corollaries 3.3 and 3.4. First, we assume the validity of Theorems 3.1 and 3.2. Proof of Corollary 3.3. Because of relationship between Ld and Ad and similar arguments, we only prove, for example, on the group. We may restrict functions to be positive, however, our proof goes for all complex ones. Note the exponent p′ in this dual theorem is the conjugate of that p in the main theorems given in last subsection under the correspondence λ + d = Q. We use ⟨·, ·⟩ for L2 -inner product with associated measure in the context, then by Plancherel formula, Cauchy–Schwarz inequality and properties (2.19) and (2.20) of intertwining operators, we have    21    − 21 2  Q+d Q+d  2 Γ k + − 1 Γ k + − 1 4 4 d d 2   fˆ, |2λ| 2    gˆ ⟨f, g⟩ = fˆ, gˆ = |2λ| 2  Q−d Q−d Γ k+ 4 −1 Γ k+ 4 −1    −1  ≤ fˆ, L gˆ, L df d g    d−Q ∗ g .  = fˆ, L gˆ, cd |u| df So, we have the following equivalent form   2 |⟨f, g⟩| ≤ cd ⟨f, Ld f ⟩ g, |u|d−Q ∗ g , and from the Lp norm characterization and sharp HLS inequality (3.1), we have   g(u)g(v) 2 ¯ ∥f ∥p′ ≤ cd f Ld f sup dudv Q−d (u, v) ∥g∥p =1 G G×G dG

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f¯Ld f,

= cd CQ−d G



and “=” holds if and only if g is an extremizer for the sharp HLS inequality (3.1) and g ∼ f p −1 , i.e. together with Theorem 3.1, this inequality is sharp with sharp constants (3.10) and all extremizers are given by (3.11) in the corollary. Corollary 3.3 is then proved.  Actually, from the obvious opposite direction, we can see the equivalence of the sharp CIS-type inequality and HLS inequality. In applications, it depends on cases to choose for convenience which formula to use. Proof of Corollary 3.4. Step 1. Inequality (3.15). From the normalized assumption,  2 f 2 (ζ) + f 2 (η) dζdη = 2Cλ′ |S| p . λ dS (ζ, η) S×S Subtracting two multiples of the sharp HLS inequality (3.5) and taking limitation λ → Q, we get   2  |f (ζ) − f (η)|2 dζdη ≥ lim 2Cλ′ |S| p − ∥f ∥2p Q λ→Q dS (ζ, η) S×S 2

|S| p − ∥f ∥2p 2 2 +3 π 2n+2    lim =  . Q λ→Q Q−λ Γ Q 4 −1 Γ 4 Q

It is easy to check17 2

2

|S| p − ∥f ∥2p p − 2 |S| p − ∥f ∥2p λ→Q = −−−→ Q−λ Q−λ p−2

   2   2 f log f 2 f log f 2 2 S − − = S . Q 2 Q

Constant (3.16) is got, then the theorem is proved after checking equality can be achieved by any function of (3.17), which is the limit of extremizers (3.7) for the sharp HLS inequality (3.5) as λ → Q. Step 2. Sharpness. Actually, if we denote Q

h ∼ |1 − ξ · η¯|− 2 , both satisfying the normalized condition, then  12  2 |S| hp , hλ =  4 hp S

hλ ∼ |1 − ξ · η¯|−

2Q−λ 2

,

hλ ⇒ h(λ → Q)

and we have  S×S

 2  |hλ (ζ) − hλ (η)|2 dζdη = 2Cλ′ |S| p − ∥hλ ∥2p . λ dS (ζ, η)

As λ → Q, by careful checking of the commutation of integral and limitation, this integral inequality converges to (3.15), replacing f by h, because of the uniformly convergence hλ ⇒ h. Another way to prove √ the sharpness of (3.15) is using the extremizing function18 f ϵ (ζ) = 1 − ϵ + ϵ Re ζn+1 formerly used by [19] for the Heisenberg group problem.  Remark. What about uniqueness? This is open. Take any extremizer h for (3.15), then if we can prove some approximation of h to the extremizing submanifold Eλ of the sharp HLS (3.5), consisting of all extremizers 17 For any 1 < p < 2, function |f |p log |f |, the derivative of |f |p relative to p, is controlled by |f |2 log |f | when |f | & 1 and by |f | log1/2 |f | when |f | . 1. 18 Here we compute the difference functional using Lemma 5.5 as both the two sides in (3.15) vanish as ϵ → 0 to order ϵ2 .

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(3.3), e.g. the Lp convergence dp (h, Eλ ) → 0, then h can only be the form of (3.17). Actually, in [41], we have a stability like  f (ζ)f (η) ′ 2 dζdη &λ d2p (f, Eλ ), Cλ ∥f ∥p − (3.18) S×S dS (ζ, η) and if the constant in the right side have low bound or is infinitesimal with low order < 1 as λ → Q, then we have the uniqueness conclusion for our LogS inequality, after replacing f with h, but unfortunately the best constant for (3.18) is still unknown. The rest of this paper is given to the proof of our main results of sharp HLS inequalities on the quaternionic group and sphere (Theorems 3.1 and 3.2). In order to express clearly the idea thread, we split it into steps. 4. Existence of extremizers For completeness, we here prove a partial result for our purpose using an ingenious Frank–Lieb argument in [19], which can be easily transformed to our case, considering compactness argument, Fatou lemma and a refined HLS. Positive definiteness is very important here to simplify the problem to a L2 estimate. Concentration compactness from [40] is an alternative method, which works perfectly for general nonsymmetric exponents, see [24] for the Heisenberg group case. Existence of extremizers for λ ≥ 4. We now demonstrate Frank–Lieb method on group G, considered to be a relatively simpler way. First, we note that the kernel — negative power of distance |u|−λ is positive definite for λ > 4 and semi-positive definite for λ = 4; Actually, it is the fundamental solution of intertwining −λ operators L−1 , with d = Q − λ, see (2.20) and we will also compute the eigenvalues of the distance d ∼ |u| kernel in Lemma 5.5 in Section 5.4. So, the maximum problem can be considered in the quadratic form (for f ∼ g). Now we assume λ > 4 as λ = 4 case comes from continuity argument.19 Lemma 4.1. Let 4 < λ < Q, then there exists a positive definite, real-valued, even and homogeneous of order 2Q

− Q+λ kernel k ∈ LwQ+λ satisfying 2 |u|−λ = k ∗ k. Proof. Fix d = Q − λ, then from the positive definiteness and relation (2.19), we have for λ > 4 that  −1 as operators |u|−λ ∗ ∼ Ld 2 , which is again positive definite and self-adjoint, and the corresponding convolution kernel      Q+λ Γ · + 2−d Γ · + 4+d − 12 − 12 − Q+λ −1 − 4 8   ,  k ∼ Ld δ0 ∼ Ld L d |u| 2 ∼ m(|2T | L)|u| 2 , where m(·) = 2 Γ · + 2+d Γ · + 4−d 4 8 and δ0 is the standard Dirac function at zero. From the commutation of m(|2T |−1 L) with inversion and Q+λ dilation, it transmits the even and homogeneous nature of |u|− 2 to k, which is also real-valued from the self-adjointness. The weak Lebesgue property comes from the Marcinkiewicz interpolation and the following multiplier lemma due to M¨ uller, Ricci and Stein [42]. We only need to check the Marcinkiewicz condition for the spectral multiplier m(·) of the essentially self-adjoint operator |2T |−1 L:           ′  . 1, (x∂x )l log m(x) = (x∂x )l φ t − x + 1 (log Γ) (t)dt   l 2 R 19 We find the formula of extremizers in our main theorem is continuous w.r.t λ, but pay attention that this only tells from the result for λ > 4 that for λ = 4 the corresponding functions are (but not necessarily) extremizers. Then are they right all the extremizers? The answer is yes, which completes the proof of our range and we will see this later.

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where    1/2 φ(t) = −1/2   0

|t| ≤ d/8 d/8 < |t| ≤ d/4 otherwise

and

1 (log Γ) (t) = log t − −2 2t ′

Then this lemma is proved after above easy computation.

 R+

sds (s2

+

t2 )(e2πs

− 1)

.



Lemma 4.2. The multiplier m(·) is strong-type (p, p) for any 1 < p < ∞, i.e., ∥m(|2T |−1 L)f ∥p . ∥f ∥p

∀f ∈ S(G),

if m satisfies the following Marcinkiewicz condition |(x∂x )l log m(x)| .l 1

∀x ∈ [n, ∞), l ∈ N.

Then the sharp problem changes to that of 2Q is the conjugate exponent. (4.1) λ Here L2 is a special Hilbert space with good compactness. When using the compactness argument for above inequality, we should be alert as the weak-limit of extremizing sequence is easy to vanish. So, the following “quaternionic edition” of an enforced estimate called “refined HLS” inequality kills this bug perfectly. See [3] for the analogue on the Heisenberg group, which was first used by G´erard [23] to prove the existence of extremizers of the Sobolev inequality on Rn . ∥f ∗ k∥q . ∥f ∥2 ,

2Q λ ,k

q = p′ =

−1

∼ Ld 2 δ, then   Q−λ Q λ λ Q −βL ∥f ∗ k∥q . ∥f ∥2 sup β 4 ∥e βL(f ∗ k)∥∞ ,

Lemma 4.3. Let 4 ≤ λ < Q, q =

β>0

with e−βL being the heat semigroup of the sublaplacian. Proof. We borrow the idea from the proof of the Euclidean and Heisenberg group analogue. First we prove the following inequality d

d

∥g∥q . ∥L 2 g∥2 (Bgλ ) Q where Bgλ := supβ>0 β 4 ∥e−βL βLg∥∞ , then the lemma is proved from the L2 bound ∥L 2 (f ∗ k)∥2 . ∥f ∥2 , which is again got by the spectral multiplier theorem— Lemma 4.2. For the inequality of g, we write g ∈ S(G) into an integral of the heat flow of the sublaplacian operator,   ∞  ∞ A −tL g= e Lgdt = + e−tL Lgdt = g1A + g2A , d

λ

0

0

A

leaving A a constant to be fixed later. From the following pointwise estimate    A    d d d d d   A −1 −tL 1− |g1 | =  (−t) 2 e (−tL) 2 L 2 f dt . A 2 |M (L 2 g)|  0   ∞ λ λ |g2A | .Bgλ t−(1+ 4 ) dt . A− 4 Bgλ , A

where M is the maximal function operator and the first estimate follows from the rapidly decreasing of the   Qd d λ d d heat kernel of the sublaplacian, we have after taking A = Bgλ M (L 2 g) that |g| . (Bgλ ) Q (M (L 2 g)) Q . Then the lemma is proved from the (p, p)-boundness of M . For the property of the maximal function operator on groups, see e.g. [44]. 

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We remark here that the g inequality is actually a special case of an embedding relation between Lebesgue, rQ homogeneous Sobolev and Besov norms; more precisely, for any 1 ≤ r ≤ ∞, 0 < s < Q r , q = Q−rs , we have 1− sr

sr

Q ∥f ∥q . ∥f ∥W˙ s,r ∥f ∥ Qs− Q . (G) r B˙ ∞,∞

Then our lemma follows from the following characterization of the Besov norm −s ∥f ∥B˙ ∞,∞ ∼ sup t 2 ∥(tL)k e−tL f ∥∞ ∼ Bf2s s

t>0

∀ k ∈ N+ , s > 0,

and see [20] for the definition and properties of the Besov norms and spaces on general stratified Lie groups, where both continuous and discrete wavelet characterization was specially given. We now give a compactness lemma due to Brezis and Lieb, and see e.g. [39] for its application to classical HLS. Lemma 4.4. For a norm-bounded sequence {fj } ⊂ Lp on a measure space, i.e., ∥fj ∥p . 1, if pointwise convergence happens, i.e., fj → f , a.e., then we have a refined Fatou-type formula for Lp norm    j→∞  |fj |p − |f |p − |fj − f |p  − −−→ 0.

∥f ∗k∥

Proposition 4.5. Let {fj } be any normalized extremizing sequence, i.e., ∥fj ∥2 = 1 and I(fj ) := ∥fj j ∥2 q converges to the supremum of this functional, then after moving onto a subsequence, there exist two sequences Q/2 2 {uj } ⊂ G and {δj } ⊂ R+ , s.t. after conformal transformation fj → δj fj (δj u−1 j ·), we get a new strong L convergent sequence, with the limit being an extremizer. Proof. Step 1. Under the conformal actions of dilations and left translations, we can get a new sequence fj , which weakly converges to a non-vanishing function f . This follows from refined HLS— Lemma 4.3. λ

Actually, Bgλj & 1 with gj = fj ∗ k, then there exist {uj } , {δj }, s.t., |δj4 e−δj L δj Lgj (uj )| & 1. If we denote by φ the Schwartz kernel associated to e−L L, then we have λ

|δj4 fj ∗ k ∗ φ

1

δj2

(uj )| & 1,

where φδ is the dilation given by δ −Q φ(δ −1 ·), and denote Dδ f = f (δ·), then    λ λ −Q 4 4 δj fj ∗ k ∗ φ 12 (uj ) = δj fj ∗ D − 12 D 12 k ∗ φ (uj ) = δj 4 fj ∗ D − 12 (k ∗ φ)(uj ) δj δ δj δj  Q  j   1 = δj4 fj uj δj2 v (k ∗ φ)(v −1 )dv =: fuδjj (k ∗ φ)˜. G

G

δ

Here f˜ denotes inversion f˜(v) := f (v −1 ) and we will still use fj for the new sequence fujj in last equality, which is still normalized. From Alaoglu’s compactness principle, we know there exists a function f ∈ L2 , 2Q

weak L2

s.t., fj −−−−−→ f and it follows from the weak Young inequality and the fact k ∈ LwQ+λ (Lemma 4.1) that j

k ∗ φ ∈ L2 , which tells then          fj (k ∗ φ)˜ →  f (k ∗ φ)˜ & 1     G

and finally f ̸≡ 0.

G

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Step 2. For above new subsequence, we have fj ∗ k → f ∗ k, a.e. For this, we recall from the proof Q+λ of Lemma 4.1 that k ∼ m(|2T |−1 L)|u|− 2 and T := m(|2T |−1 L) is an operator of strong-type (p, p), especially L2 → L2 , so {T fj } is again weak compact and the weak limit is nothing but T f . Then the problem is equivalent to T fj ∗ |u|−

Q+λ 2

→ T f ∗ |u|−

Now, we prove a stronger20 Lrloc convergence for any r < k1 = χ|u|≥ϵ |u|−

Q+λ 2

,

Q+λ 2

2Q λ .

,

a.e.

Denote

k2 = χ|u|≤ϵ |u|−

Q+λ 2

,

then ∥T (fj − f ) ∗ k∥Lrloc ≤ ∥T (fj − f ) ∗ k1 ∥Lrloc + ∥T (fj − f ) ∗ k2 ∥Lrloc . For the first term, the fact that k1 ∈ L2 and the weak L2 convergence give T fj ∗ k1 → T f ∗ k1 and then Lrloc convergence comes from the (bounded) dominated convergence theorem; for the second term, it 2Q 2r follows from Young inequality that ∥T (fj − f ) ∗ k2 ∥r ≤ ∥T (fj − f )∥2 ∥k2 ∥r∗ , where r∗ = r+2 < Q+λ and ∥k2 ∥r∗ . ϵ

2Q−λr r+2

ϵ→0

−−−→ 0. L2

Step 3. Furthermore, fj −−→ f . Now we borrow Lemma 4.4 and get  2/q ∥fj ∗ k∥2q ∥f ∗ k∥qq + ∥(fj − f ) ∗ k∥qq + o(1) C := supremum ← = ∥fj ∥22 ∥f ∥22 + ∥fj − f ∥22 + o(1) ≤

∥f ∗ k∥2q + ∥(fj − f ) ∗ k∥2q + o(1) ≤ C + o(1), ∥f ∥22 + ∥fj − f ∥22 + o(1)

from which we easily see f is an extremizer as all inequalities above reach equality. However, we have a stronger conclusion that ∥f ∥2 = 1 and ∥fj − f ∥2 → 0, which follows from the equality condition of the first inequality above. Here for clarity, we use contradiction argument. Assume ∥f ∥22 = 1 − ϵ with ϵ > 0, then put the fact that f is an extremizer and fj − f is also an extremizing sequence into ∥fj ∗ k∥qq = ∥(fj − f ) ∗ k∥qq + ∥f ∗ k∥qq + o(1), we get a strict inequality, i.e., the formula  q/2 C q/2 = (1 − ϵ)C + (ϵC)q/2 + o(1), is impossible for any positive ϵ. Now the proposition is proved and therefore from the conjugate relation of the sharp problem (4.1) and sharp HLS (3.1), there also exists for the sharp HLS an extremizer (f ∗ k)q−1 , where f is the extremizer we get in this proposition. The question for (3.5) is also answered.  Now, for Theorems 3.1 and 3.2, it suffices to figure out all extremizers. We proved an uniqueness, modulo the conformal symmetry group and also compute the explicit formula of all extremizers. 5. Uniqueness and formulas of extremizers Fix λ ≥ 4. For simplicity of computation, we deal with the sharp inequality on the framework of quaternionic sphere. From (1) in Lemma 5.5 or the fundamental solution of intertwining operators (2.13), we know that the integral kernel dS (ζ, η)−λ is positive definite for λ > 4 and semi-positive for λ = 4. Therefore 20 Lr convergence implies pointwise a.e. convergence for a subsequence. So does local Lebesgue convergence using a classical diagonally choosing method.

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we can restrict the sharp problem to f = g case, especially for λ > 4, for which, by standard argument, we can further restrict the extremizer to be a complex multiple of a positive real-valued function. We conclude these in the following lemma. Lemma 5.1. For λ > 4, any extremizer pair (f, g) of the sharp HLS is given by f ∼ g ∼ h, where h is a positive quadratic extremizer. For λ = 4, any quadratic extremizer f ∼ h, where h is a positive one.   Proof. We set ⟨·, ·⟩λ := · ∗ d−λ S , · , which is positive definite for λ > 4, then for extremizer pair (f, g), we have I(f, g) ≤ (I(f )I(g))1/2 with equality if and only if f ∼ g. Besides, for λ ≥ 4, any quadratic extremizer must be a complex multiple of a positive one. For any f = a + ib, with a, b being respectively the real and the image part, ⟨f, f ⟩λ = ⟨a, a⟩λ + ⟨b, b⟩λ ≤ ⟨|f |, |f |⟩λ by the symmetry of the integral kernel √ and Cauchy–Schwarz inequality, which gives I(f ) ≤ I(|f | = a2 + b2 ) with equality if and only if for a.e.     (ζ, η) ∈ S × S, f (η) = 0 or a(ζ), b(ζ) = c a(η), b(η) with c > 0, which implies finally after fixing a  

point b(η) ̸= 0 (otherwise b ≡ 0, then f = a with fixed sign) that f = a(η) b(η) + i sign b(η) |b|. So, modulo a complex constant, the quadratic extremizer must be real-valued and non-negative. Now, we compute the first variation around any non-negative extremizer h,    ∂  −p p−1 I(h + tϕ) = 2∥h∥−2 h ∗ d−λ , ϕ , ∀ϕ ∈ Lp , p S − ⟨h, h⟩λ ∥h∥p h ∂t t=0 which gives the Euler–Lagrange equation from the critical property of extremizers  h(η) hp−1 (ζ) ∼ λ dη, ¯| 2 S |1 − ζ · η which tells h is positive a.e. Then the lemma is proved.

(5.1)



Now, we will first care the quadratic extremizer for λ ≥ 4. 5.1. Zero center-mass condition The sharp HLS inequality on group is invariant under translation, dilation and constant multiple, while the sphere-edition is preserved by the action of quaternionic unitary group Sp(n+1). So, through the Cayley transform, we can intuitionally try breaking the huge conformal symmetry by assuming that the extremizer satisfies a certain zero center-mass condition. Indeed, the conformal invariant group plays a critical role here, while is absent for most of other inequalities. Lemma 5.2. For any extremizer of the sharp HLS inequality (3.5), there exists a conformal transformation ˜ = |Jγ −1 | p1 h ◦ γ −1 , we get another extremizer h ˜ satisfying zero γ s.t. by the conformal action γ : h → h center-mass condition, i.e.,  ˜ p (ζ)dζ = 0. ζ|h| (5.2) S

Proof. Step 1. For any integrable function f with nonzero mean, there is a conformal transformation γ on  S, s.t., S γf = 0. For this, we define a conformal map on S \ {A−1 o}: δ γA = A∗ ◦ C ◦ Dδ ◦ C −1 ◦ A

(5.3)

with any A ∈ Sp(n + 1) and dilation Dδ for δ > 0 on G. Actually, take ξ = A−1 (0, . . . , 0, 1), then the δ δ map γA is (δ, ξ)-determinate, but independent of the choice of A, so we can use notation γξδ = γA : from

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diagram  2q 1 − |q|2 + w , 1 + |q|2 − w 1 + |q|2 − w   2δq 1 − δ 2 (|q|2 − w) A∗ δ Dδ ◦C −1 C 2 −−−−−→ (δq, δ w) → , −−→ γξ (ζ) 1 + δ 2 (|q|2 − w) 1 + δ 2 (|q|2 − w) A



ζ− →

(5.4)

and unitary property of A 1 − |q|2 + w , ζ · ξ¯ = Aζ · (0, . . . , 0, 1) = 1 + |q|2 − w we give the explicit form γξδ (ζ) =

2 ¯ ¯   2δ ¯ + 1 + ζ · ξ − δ (1 − ζ · ξ) ξ ζ − (ζ · ξ)ξ 2 2 ¯ ¯ ¯ ¯ 1 + ζ · ξ + δ (1 − ζ · ξ) 1 + ζ · ξ + δ (1 − ζ · ξ)

(5.5)

with boundary limits δ→0

δ→1

γξδ (ζ) −−−→ ξ,

γξδ (ζ) −−−→ ζ.

(5.6)

Note that as δ → 1, the convergence is uniform in (ξ, ζ) on total S × S, and as δ → 0, the convergence is uniform on any slightly small subset ¯ ≥ ϵ > 0}. E = {(ζ, ξ) ∈ S × S : |1 + ζ · ξ| For any f with

 S

f = 1, take  F (rξ) = S

γξ1−r (ζ)f (ζ)dζ,

defined originally on unit ball minus origin point, then we have limits both at origin and on boundary, uniformly in ξ:  r→0 r→1 F (rξ) −−−→ ζf (ζ)dζ, F (rξ) −−−→ ξ. (5.7) ξ

ξ

S

r→r0

The notation −−−→ means the convergence is uniformly in ξ when r → r0 . Actually, the r → 0 case is a ξ

direct consequence of the δ → 1 case in (5.6), while the r → 1 case comes from the δ → 0 case in (5.6) and    |F (rξ) − ξ| ≤ + |γξ1−r (ζ) − ξ|f (ζ)dζ E|S

[(S×S)\E]|S

 =

dS (ζ,−ξ)≥ϵ

 +

dS (ζ,−ξ)≤ϵ



|γξ1−r (ζ) − ξ|f (ζ)dζ

=I1 + I2 , ϵ→0

r→1

with I2 −−−→ 0 and I1 −−−→ 0. So, F can be extended to be a continuous function on closed unit ball in ξ 4n+4

ξ

Hn+1 ≃ R and F (ξ) = ξ on the sphere. By Brouwer’s fixed point theorem, F has at least one zero point,  i.e., there exists a pair (δ0 , ξ0 ) s.t. S γξδ00 (ζ)f (ζ)dζ = 0. Step 2. Take f = |h|p with h the (positive) extremizer, then from above arguments, there exists a conformal   ˜ p (ζ)dζ = 0. transformation γ := γξδ00 , s.t., S γ(ζ)|h|p (ζ)dζ = 0. Change the variables, then we get S ζ|h| ˜ is still Besides, from the conformal symmetry, see the second item in the remark following Theorem 3.2, h an (positive) extremizer for (3.5).  The zero center-mass condition is critical here in restricting our extremizers into a small, particular function class.

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5.2. Second variation and inverse inequality We now return to prove the main theorem. Now, from Lemma 5.2, we can assume the positive quadratic extremizer h satisfies (5.2). Then we will try to prove h can only be constant function. We first compute the second variation around h of the functional I(f ) associated to the sharp HLS inequality (3.5)     ∂ 2  I(h + tϕ) = 2∥h∥−2−p ⟨ϕ, ϕ⟩λ ∥h∥pp − (p − 1) ⟨h, h⟩λ hp−2 , |ϕ|2 , p  2 ∂t t=0   for any ϕ ∈ Lp , hp−1 , ϕ = 0, where we use the same notation ⟨·, ·⟩λ for the inner product as that in Lemma 5.1 and the extremizer gives non-positive second variation, so     ϕ(ζ)ϕ(η) h(ζ)h(η) p h − (p − 1) hp−2 |ϕ|2 ≤ 0 (5.8) λ dζdη λ dζdη 2 2 ¯| ¯| S S×S |1 − ζ · η S S×S |1 − ζ · η   for all ϕ satisfying S hp−1 ϕ = 0. Substitute ϕ(ζ) = hζj1 , hζj2 (1 ≤ j ≤ n + 1), all satisfying S hp−1 ϕ = 0, into the second variation inequality (5.8) and summing the results, then we get h(ζ)



n+1  j=1

S×S

(ζj1 ηj1 + ζj2 ηj2 )h(η)

|1 − ζ · η¯|



h(ζ)h(η)

dζdη ≤ (p − 1)

λ 2

λ

S×S

|1 − ζ · η¯| 2

dζdη.

From the symmetry of left side integrand on (ζ, η) and ζ¯ ·C η + η¯ ·C ζ = ζ¯ · η + η¯ · ζ, we have   h(ζ)(ζ¯ · η + η¯ · ζ)h(η) h(ζ)h(η) dζdη ≤ 2(p − 1) λ λ dζdη. |1 − ζ · η¯| 2 ¯| 2 S×S S×S |1 − ζ · η

(5.9)

In Sections 5.4 and 5.5, we use quaternionic analogue of Funk–Hecke formula to prove that extremizer satisfying (5.9) can only be constant (Proposition 5.6). Concerning this involves very complicated computation, we leave it in independent subsections. Actually, we claim here: For any 4 ≤ λ < Q = 4n + 6 and for h, which is the extremizer discussed above, we have the following inverse second variation inequality (ISV) h(ζ)(ζ¯ · η + η¯ · ζ)h(η)

 S×S

|1 − ζ · η¯|

λ 2



h(ζ)h(η)

dζdη ≥ 2(p − 1)

λ

S×S

|1 − ζ · η¯| 2

dζdη.

(5.10)

Moreover, “=” holds only when h ≡ constant. 5.3. Sharp constants and formulas of extremizers Assume the claim holds, then we have proved an uniqueness of the extremizer. We need to take care for λ = 4 case. The claim tells that any quadratic extremizer can only be almost constant. Then we said quadratic extremizer is still the only case for pair. Lemma 5.3. For λ = 4, any extremizer pair (f, g) for I(f, g) is quadratic, i.e. f ∼ g. Proof. We consider on the sphere. From Lemma 5.2 and the claim, we know any quadratic extremizer 1/2 1/2 f ∼ |Jτ |1/p , τ ∈ Aut(S). Given an extremizer pair (f, g), the fact ⟨f, g⟩λ ≤ ⟨f, f ⟩λ ⟨g, g⟩λ gives I(f, g) ≤ (I(f )I(g))1/2 , so (f, f ) and (g, g) are again extremizer pairs and can only be the form of quadratic extremizer, i.e., we may assume f = 1 and g = |Jτ |1/p for some τ ∈ Aut(S). But from the equality condition  of Cauchy–Schwarz inequality, when restricted onto the positive definite subspace V := j≥k=0 Vj,k , we

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have f |V ∼ g|V . However, by simple computation, g|V1,0 = 0 if and only if g ∼ 1. Besides, we can also see from the Euler–Lagrange equation for non-quadratic functional I(f, g), p−1 f ∗ d−λ , S ∼g

p−1 g ∗ d−λ . S ∼f

So, any extremizer pair for λ ≥ 4 should be also quadratic.



We can give the sharp constants and all extremizers now with the a priori claim. 1. Extremizers for (3.1) and (3.5). For λ ≥ 4, from Lemmas 5.1–5.3 and the claim, we know that all extremizer pairs on the sphere are given by f ∼ g ∼ |Jτ |1/p , where τ is any conformal transformation in Aut(S), and correspondingly all extremizers on the group are given by f ∼ g ∼ |JC ◦ σJσ |1/p , σ ∈ Aut(G) from the relation (3.4) and definitions of the conformal groups. The explicit formulas now follows from our calculation in Appendix B. Note that some transformations are assimilated in extremizer formulas. 2. Sharp constants for (3.1) and (3.5). Using the fact that constant 1 is an extremizer, we have from the λ

formula of λ0,0 (K14 ) in (5.16) Lemma 5.5 that the sharp constants   λ 2 2 λ 1 1− p 4 λ Cλ′ = 2 2 |S|1− p dη = |S| 0,0 K1 λ S |1 − ηn+1 | 2   2π 2n+2 Γ 2n − λ2 + 3 2 λ 1− p 2     = 2 |S| Γ 2n − λ4 + 2 Γ 2n − λ4 + 3      Qλ  λ Q−4 Q−λ Q−2 2 2 !Γ 2 2 2 2π     , =  2Q−λ Q−4 2Q−λ ! Γ Γ − 1 2 4 4 and then Cλ = 2



 =

(4n+3)λ Q

π

Q−2 2

2

Q−8 2

Cλ′

 Qλ

  2π 2n+2 Γ 2n − λ2 + 3     =2 |S| Γ 2n − λ4 + 2 Γ 2n − λ4 + 3   1− Qλ   Q−λ Q−4 Γ ! 2 2  .   2Q−λ 2Q−λ Γ −1 Γ 4 4 − 2nλ Q

2 1− p

The two main theorems are then proved after we check the claim at the end of Section 5.2, which tells that any extremizer h that satisfying zero center-mass condition can only be constant. In the remaining part of this paper, we focus on proving this and before that we need to introduce the Funk–Hecke formula. 5.4. Quaternionic Funk–Hecke formula and eigenvalues In this subsection, we are going to derive a quaternionic analogue of classical real or complex Funk–Hecke theorems, which concerns the integral operators associated with kernel of the form K(ζ · η¯). We then give eigenvalues of integral operators with two useful kernels. We left the proof of the claim to next subsection. Lemma 5.4 (Quaternionic Funk–Hecke Formula). Let K be a function on the unit ball in H, s.t. the following integral exists, e.g. by assuming K ∈ L1 (B(0, 1)). Then any integral operator on S with kernel of the form

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K(ζ · η¯) is diagonal w.r.t the decomposition (2.6), and the eigenvalue on Vj,k is given by  π2 2π 2n k! (2n−1,j−k+1) sin4n−1 θ cosj−k+3 θPk (cos 2θ) dθ λj,k (K) = (j − k + 1)(k + 2n − 1)! 0  sin(j − k + 1)φ × K(cos θu) du, sin φ |u|=1

(5.11)

with Re u = cos φ (φ ∈ [0, π]) and du is the Lebesgue measure on S3 . Proof. From Schur’s lemma and the irreducibility of bispherical subspace Vj,k , we see the integral operator K with kernel K(ζ · η¯) is diagonal w.r.t the decomposition (2.6) with eigenvalues denoted by λj,k . Now, we µ mj,k compute the eigenvalues. Assume {Yj,k }µ=1 is a normalized orthogonal basis of Vj,k , where mj,k = dim Vj,k , then as footnoted in the definition of zonal harmonics in the preliminary section and in abuse of notation, the reproducing kernel of projection operator onto Vj,k is given by mj,k



µ µ ¯ = Z η (ζ). (η) = Zj,k (ζ · η¯) = Zj,k (η · ζ) Yj,k (ζ)Yj,k j,k

µ=1

By the definition of spectrum λj,k , 

¯ dη = λj,k Zj,k (1), K(ζ · η¯)Zj,k (η · ζ)

S

which implies λj,k =

−1 Zj,k (1)

−1 = Zj,k (1)



¯ dη K(ζ · η¯)Zj,k (η · ζ)

S

 K(ηn+1 )Zj,k (ηn+1 ) dη.

(5.12)

S

In polar coordinates for η = (η1 , η2 , . . . , ηn+1 ), η1 = u1 sin θn sin θn−1 . . . sin θ1 , |ui | = 1, ui = (u1i , u2i , u3i , u4i ), η2 = u2 sin θn sin θn−1 . . . cos θ1 , u4i = sin φ3i sin φ2i sin φ1i , ηi = . . . u3i = sin φ3i sin φ2i cos φ1i , ηn = un sin θn cos θn−1 , u2i = sin φ3i cos φ2i , ηn+1 = un+1 cos θn , u1i = cos φ3i , with θi ∈ [0, π2 ], φ3i , φ2i ∈ [0, π], φ1i ∈ [0, 2π], ∀1 ≤ i ≤ n + 1, we have the invariance measure on the sphere (11.7.3 (2) in [45]) dη =

n 

(sin4i−1 θi cos3 θi dθi )

i=1

n+1 

duj ,

duj = sin2 φ3j sin φ2j dφ3j dφ2j dφ1j .

(5.13)

j=1

Putting the formula (2.7) for zonal harmonics21 and invariance measure (5.13) together into (5.12), and from the formulas |S4n−1 | =

21 Z

j,k (ζ)

2π 2n , (2n)!

only depends on ζn+1 as η is the north pole.

(2n−1,j−k+1)

Pk

(1) =

(k + 2n − 1)! k!(2n − 1)!

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(22.2.1 in [1]), finally we get λj,k =

2π 2n k! (j − k + 1)(k + 2n − 1)!



π 2

0

(2n−1,j−k+1)

sin4n−1 θn cosj−k+3 θn Pk  ×

S3

K(cos θn un+1 )

(cos 2θn ) dθn

sin(j − k + 1)φ3n+1 dun+1 .  sin φ3n+1

In order to prove the claim (Proposition 5.6), it suffices to compute eigenvalues of kernels for functions of two forms K1α (q) = |1−q|−2α , K2α (q) = |q|2 |1−q|−2α , noting that ζ · η¯+η· ζ¯ = 2 Re ζ · η¯ = 1+|ζ · η¯|2 −|1−ζ · η¯|2 . Define22 the following Gamma functions series for any real numbers pair (a, b, c) satisfying c > a + b by A(a, b, c) :=

 Γ(µ + a)Γ(µ + b) Γ(a)Γ(b)Γ(c − a − b) = . µ!Γ(µ + c) Γ(c − a)Γ(c − b)

(5.14)

µ≥0

Lemma 5.5 (Eigenvalues). Given −1 < α < (a, b, c) :=

Q 4

and denote after fixing an integer pair j ≥ k ≥ 0

  Q j + α, k + α − 1, j + k + − 1 . 2

(5.15)

(1) The eigenvalues of integral operators with kernels K1α (ζ · η¯) = |1 − ζ · η¯|−2α are given by 2π 2n+2 A(a, b, c) Γ(α)Γ(α − 1)   − 2α Γ Q 2 Γ(j + α) Γ(k + α − 1)    . = 2π 2n+2 Q Γ(α)Γ(α − 1) Γ j + − α Γ k + Q − α − 1 2 2

λj,k (K1α ) =

(5.16)

(2) The eigenvalues of integral operators with kernels K2α (ζ · η¯) = |ζ · η¯|2 |1 − ζ · η¯|−2α are given by α λj,k (K2α ) = Cj,k λj,k (K1α ),

(5.17)

with α Cj,k

 = 1 − (α − 2)(c − a − b)

 1 1 c−a−b+1 + − (α − 2) . (a − 1)(c − a) (b − 1)(c − b) (a − 1)(b − 1)(c − a)(c − b) (5.18)

In the singular points α = 0, 1, 2, above formulas can be viewed as limits, fixing j and k. Proof. (1) Putting K = K1α into Lemma 5.4, we get  π2 4π 2n+1 k! (2n−1,j−k+1) α λj,k (K1 ) = sin4n−1 θ cosj−k+3 θPk (cos 2θ) dθ (j − k + 1)(k + 2n − 1)! 0  π   × (1 + cos2 θ − 2 cos φ cos θ)−α cos(j − k)φ − cos(j − k + 2)φ dφ. 0

22 The second equality property for A(a, b, c) comes from 15.1.1 and 15.1.20 in [1].

(5.19)

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Using Gegenbauer polynomials, we have the following fact ((5.11) in [19] from [1])  π (1 + cos2 θ − 2 cos φ cos θ)−α cos(j − k)φ dφ 0

π

=

Γ2 (α)



cos|j−k|+2µ θ

µ≥0

Γ(µ + α)Γ(µ + |j − k| + α) , µ!(µ + |j − k|)!

(5.20)

which, from (5.19) and j ≥ k ≥ 0, gives λj,k (K1α ) =

=:

 Γ(µ + α) 4π 2n+2 k! (5.21) 2 (j − k + 1)(k + 2n − 1)!Γ (α) µ! µ≥0   π2 Γ(µ + |j − k| + α) (2n−1,j−k+1) sin4n−1 θ cos2(j−k)+3+2µ θPk × (cos 2θ) dθ (µ + |j − k|)! 0   π2 Γ(µ + |j − k| + 2 + α) (2n−1,j−k+1) 4n−1 2(j−k)+5+2µ sin θ cos θPk (cos 2θ) dθ − (µ + |j − k| + 2)! 0 4π 2n+2 k! (I1 − I2 ). (j − k + 1)(k + 2n − 1)!Γ2 (α)

(5.22)

Using the following Rodrigues’ formula ((22.11.1) in [1]), (2n−1,j−k+1)

Pk

(t) =

k (−1)k −(2n−1) −(j−k+1) d (1 − t) (1 + t) {(1 − t)2n+k−1 (1 + t)j+1 }, 2k k! dtk

and by changing variable cos 2θ = t and integrating by parts, we get  π2 (2n−1,j−k+1) sin4n−1 θ cos2(j−k)+3+2µ θPk (cos 2θ) dθ 0

=

(−1)k 2−(µ+j+2n+2) k!

= χµ≥k = χµ≥k = χµ≥k



1

−1

(1 + t)µ

dk {(1 − t)2n+k−1 (1 + t)j+1 } dt dtk

 2−(µ+j+2n+2) µ! 1 (1 + t)µ+j−k+1 (1 − t)2n+k−1 dt k!(µ − k)! −1 µ!B(µ + j − k + 2, 2n + k) 2k!(µ − k)! µ!Γ(µ + j − k + 2)Γ(2n + k) , 2k!(µ − k)!Γ(µ + j + 2n + 2)

(5.23)

where B is the Beta function. Inserting (5.23) into (5.21), we get I1 =

1  Γ(µ + α)Γ(µ + |j − k| + α) µ!Γ(µ + j − k + 2)Γ(2n + k) , 2 µ!(µ + |j − k|)! k!(µ − k)!Γ(µ + j + 2n + 2) µ≥k

I2 =

1 2

 µ+1≥max{k,1}

Γ(µ + α)Γ(µ + |j − k| + 2 + α) (µ + 1)!Γ(µ + 1 + j − k + 2)Γ(2n + k) . (5.24) µ!(µ + |j − k| + 2)! k!(µ + 1 − k)!Γ(µ + 1 + j + 2n + 2)

Now, we get the difference of the two terms. For k ≥ 1, I1 − I2 =

1  µ!Γ(µ + j − k + 2)Γ(2n + k) 2 k!(µ − k)!Γ(µ + j + 2n + 2) µ≥k   Γ(µ + α)Γ(µ + |j − k| + α) Γ(µ − 1 + α)Γ(µ + |j − k| + 1 + α) × − µ!(µ + |j − k|)! (µ − 1)!(µ + |j − k| + 1)!

M. Christ et al. / Nonlinear Analysis 130 (2016) 361–395

=

387

1  µ!Γ(µ + j − k + 2)Γ(2n + k) (α − 1)(j − k + 1)Γ(µ + α − 1)Γ(µ + j − k + α) 2 k!(µ − k)!Γ(µ + j + 2n + 2) µ!(µ + j − k + 1)! µ≥k

=

(α − 1)(j − k + 1)Γ(2n + k)  Γ(µ + α − 1)Γ(µ + j − k + α) 2k! (µ − k)!Γ(µ + j + 2n + 2) µ≥k

(α − 1)(j − k + 1)Γ(2n + k)  Γ(µ + k + α − 1)Γ(µ + j + α) = . 2k! µ!Γ(µ + j + k + 2n + 2) µ≥0

For k = 0, as computation above I1 − I2 =

(α − 1)(j + 1)Γ(2n)  Γ(µ + α − 1)Γ(µ + j + α) Γ(2n)Γ(α)Γ(j + α)(j + 1) + 2 µ!Γ(µ + j + 2n + 2) 2Γ(j + 2n + 2) µ≥1

(α − 1)(j + 1)Γ(2n)  Γ(µ + α − 1)Γ(µ + j + α) = . 2 µ!Γ(µ + j + 2n + 2) µ≥0

So, together with (5.14), (5.15) and (5.21), we get 2π 2n+2 A(a, b, c) Γ(α)Γ(α − 1)   − 2α Γ Q 2 Γ(j + α) Γ(k + α − 1)    . = 2π 2n+2 Γ(α)Γ(α − 1) Γ j + Q − α Γ k + Q − α − 1 2 2

λj,k (K1α ) =

The first part of the theorem is then proved. (2) Putting K = K2α into Lemma 5.4, we get  π2 4π 2n+1 k! (2n−1,j−k+1) λj,k (K2α ) = sin4n−1 θ cosj−k+5 θPk (cos(2θ)) dθ (j − k + 1)(k + 2n − 1)! 0  π   × (1 + cos2 θ − 2 cos φ cos θ)−α cos(j − k)φ − cos(j − k + 2)φ dφ. 0

(5.25)

Compare this with (5.19), (5.21) and (5.23), we see µ there is now substituted by µ + 1, so repeating the same computation in (1), we obtain an analogue of (5.21), with  1 Γ(µ + α)Γ(µ + |j − k| + α) (µ + 1)!Γ(µ + 1 + j − k + 2)Γ(2n + k) I1 = 2 µ!(µ + |j − k|)! k!(µ + 1 − k)!Γ(µ + 1 + j + 2n + 2) µ+1≥max{k,1}

1 I2 = 2

 µ+2≥max{k,2}

Γ(µ + α)Γ(µ + |j − k| + 2 + α) (µ + 2)!Γ(µ + 2 + j − k + 2)Γ(2n + k) . µ!(µ + |j − k| + 2)! k!(µ + 2 − k)!Γ(µ + 2 + j + 2n + 2)

(5.26)

Then for k ≥ 2, λj,k (K2α ) =

 (µ + k)(µ + j + 1)Γ(µ + k + α − 2)Γ(µ + j + α − 1) 2π 2n+2 Γ(α)Γ(α − 1) µ!Γ(µ + j + k + 2n + 2) µ≥0

 Γ(µ + k + α − 1)Γ(µ + j + α) 2π 2n+2 (µ + k)(µ + j + 1) = . (5.27) Γ(α)Γ(α − 1) µ!Γ(µ + j + k + 2n + 2) (µ + k + α − 2)(µ + j + α − 1) µ≥0

Noting that using the notation (5.15) (µ + k)(µ + j + 1) (µ + a − 1 − (α − 2))(µ + b − 1 − (α − 2)) = (µ + k + α − 2)(µ + j + α − 1) (µ + a − 1)(µ + b − 1)   1 1 1 = 1 − (α − 2) + − (α − 2) , µ+a−1 µ+b−1 (µ + a − 1)(µ + b − 1)

(5.28)

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so, from (5.14), we have α λj,k (K2α ) = Cj,k λj,k (K1α ),

(5.29)

with α Cj,k = 1 − (α − 2)(c − a − b)   1 1 c−a−b+1 × + − (α − 2) . (a − 1)(c − a) (b − 1)(c − b) (a − 1)(b − 1)(c − a)(c − b)

(5.30)

For k < 2(k = 0, 1), we only prove for k = 1 as we can check similarly that the special cases can be formally integrated into the general formula. We denote by (I1 − I2 )i the obvious terms for Kiα (i = 1, 2), see (5.24) and (5.26), then α (I1 − I2 )2 = Cj,k (I1 − I2 )1 + (α − 1)(j + 1)(α − 2)   Γ(α − 1)Γ(j + α) Γ(α)Γ(j + α − 1) Γ(α − 1)Γ(j + α − 1) + − (α − 2) × Γ(j + 2n + 3) Γ(j + 2n + 3) Γ(j + 2n + 3)   Γ(j + 2) (α − 1)jΓ(α)Γ(j + α) Γ(α)Γ(j + α − 1) − − Γ(j + 2n + 3) (j + 1)! (j − 1)! α = Cj,k (I1 − I2 )1 .

The second part of the theorem is proved.



5.5. Proof of the claim and Theorems 3.1 and 3.2 For simplicity, we first prove a proposition about a general quadratic ISV-type inequality, which is almost equivalent to the claim, with α = λ4 . Proposition 5.6 (Quadratic ISV Inequality). Let 1 ≤ α < Q 4 , then for any f on S, s.t. the following integrals exist, we have   2α f (ζ)(ζ¯ · η + η¯ · ζ)f (η) f (ζ)f (η) dζdη ≥ dζdη, (5.31) Q 2α |1 − ζ · η¯| ¯|2α S×S S×S |1 − ζ · η 2 −α and here α ≥ 1 is sharp23 ; moreover, when α > 1, equality holds if and only if f is a constant function, and  when α = 1, equality holds if and only if f ∈ V0,0 j≥k≥2 Vj,k . Proof. We always first consider the problem for a test function, and then extends to bigger function spaces and all singular points of Gamma functions are interpreted as limits here. From ζ · η¯ + η · ζ¯ = 1 + |ζ · η¯|2 − |1 − ζ · η¯|2 , to prove the quadratic inequality in the proposition, it suffices to prove (not necessary) that     2α 2 2 λj,k (K1α ) ∥fj,k ∥2 , (5.32) λj,k (K1α ) + λj,k (K2α ) − λj,k (K1α−1 ) ∥fj,k ∥2 ≥ Q − α j≥k≥0 j≥k≥0 2  where f = j≥k≥0 fj,k (fj,k ∈ Vj,k ) is the bispherical harmonic decomposition (2.6) of f . From the formulas of eigenvalues in Lemma 5.5, we can check (5.32) by meticulous computation and for completion, we do it here. When α > 1, from Lemma 5.5 and recalling that λj,k (K1α ) > 0 (∀ j ≥ k ≥ 0), (5.32) holds if (not only if) we can prove that, ∀ j ≥ k ≥ 0,   1 1 c−a−b+1 2α 2 − (α − 2)(c − a − b) + + ≥ Q , (a − 1)(c − a) (b − 1)(c − b) (a − 1)(b − 1)(c − a)(c − b) 2 −α 23 For any α < 1, there exists a function, s.t., the inequality fails.

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which is  (α − 2) as c − a − b =

Q 2

1 1 c−a−b+1 + + (a − 1)(c − a) (b − 1)(c − b) (a − 1)(b − 1)(c − a)(c − b)

 ≤

Q 2

2 , −α

(5.33)

− 2α > 0.

For α > 2, it suffices to check   2 1 c−a−b+1 1  − + ≥ (a − 1)(c − a) (b − 1)(c − b) (a − 1)(b − 1)(c − a)(c − b) (α − 2) Q 2 −α which becomes, after substituting (a, b, c) (see (5.15)),     (k − 1)(α − 2) + (j + 1) Q j(α − 2) + k Q 2 − α + kj 2 − α + (j + 1)(k − 1)    +     Q Q (α − 2) Q (α − 2) Q 2 − α (α − 2 + k) 2 − α + j 2 − α (α − 2 + j + 1) 2 − α + k − 1 Q

≥ (α − 2 + k)



Q 2

2

− 2α + 1

− α + j (α − 2 + j + 1)



Q 2

. −α+k−1

(5.34)

To prove (5.34), first note that the denominator of left side is smaller than that in the right side, i.e.,        Q Q Q − α ≤ min (α − 2 + k) − α + j , (α − 2 + j + 1) −α+k−1 , (α − 2) 2 2 2 which is equivalent to  (k − 1)(α − 2) + (j + 1)

 Q − α + (j + 1)(k − 1) ≥ 0, 2

Q and we easily see this as the left side is bigger than Q 2 − α − 1 − (α − 2) = 2 − 2α + 1 ≥ 1. Then compare the numerators, and we find that the sum of numerators in the left side is larger than that in the right side, i.e.,   Q Q (j + k − 1)(α − 2) + (k + j + 1) − α + kj + (j + 1)(k − 1) ≥ − 2α + 1, (5.35) 2 2

which is just 

   Q Q −3 j+ − 1 k + 2jk ≥ 0. 2 2

This inequality is obvious and become equality if and only if j = k = 0, and in this special case, inequalities above (5.33)–(5.35) all become equalities. So, (5.32) holds and reaches equality only when fj,k = 0, ∀ (j, k) ̸= (0, 0), and therefore, for α > 2, recalling V0,0 = {constant functions}, we have proved that (5.31) holds and becomes equality if and only if f is a constant function. For 1 < α < 2, for (5.33), it suffices to prove the inverse of (5.34),     Q j(α − 2) + k Q − α + kj (k − 1)(α − 2) + (j + 1) − α + (j + 1)(k − 1) 2 2    +     Q Q (α − 2) Q (α − 2) Q 2 − α (α − 2 + k) 2 − α + j 2 − α (α − 2 + j + 1) 2 − α + k − 1 Q

≤ (α − 2 + k)



Q 2

2

− 2α + 1

− α + j (α − 2 + j + 1)



Q 2

. −α+k−1

For k ≥ 1, it is obvious by checking sign that a strict inequality holds. For k = 0, it is also obvious and “=” holds if and only if j = 0, checking sign of the first term and using (5.35) for j ≥ 1. The α = 2 case is obvious from similar or limitation argument. So, Proposition 5.6 is first proved for α > 1.

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When α ≤ 1, from Lemma 5.5, (5.32) holds if we can prove, ∀ j ≥ k ≥ 0,      j(α − 2) + k Q (k − 1)(α − 2) + (j + 1) Q 2 − α + kj 2 − α + (j + 1)(k − 1)      +    Q Q (α − 2) Q (α − 2) Q 2 − α (α − 2 + k) 2 − α + j 2 − α (α − 2 + j + 1) 2 − α + k − 1  Q − 2α + 1 Γ(k + α − 1) 2    − ≤ 0. (5.36) Q Q Γ(α − 1) − α + j (α − 2 + j + 1) −α+k−1 (α − 2 + k) 2

2

For α = 1, (5.36) holds and equality holds if and only if k = j = 0 or k ≥ 2. When α < 1, we find it still Q

−1

4 2 holds for k = 0 and equality holds if and only if j = 0. For k = 1, it still holds for j ≤ jα = 32 − Q 4 + 1−α , reaches equality if j = jα (if jα ∈ N), and especially holds strictly for j = 1, while the opposite strict inequality holds for j > jα . Note that jα ∈ (1, ∞) and is strict increasing for α ∈ (0, 1). For k ≥ 2, the opposite inequality holds strictly. So, we have proved that (5.32) and whence (5.31) holds for α = 1, but fails for some functions in the case 0 < α < 1.

Then Proposition 5.6 is proved for α ≥ 1, which is a sharp range.



Proof of the claim. The λ > 4 case is already contained in Proposition 5.6. For λ = 4, from Lemma 5.5, we λ know that the eigenvalue λj,k (|1−ζ·¯ η |− 2 ) vanishes for k ≥ 1 and is positive for k = 0, so from Euler–Lagrange equation (5.1) and Proposition 5.6, h can only be constant. Another way to see this is considering directly the original inequality as for any positive extremizer h satisfying zero center-mass condition, we have   h(ζ)h(η) h0 (ζ)h0 (η) dζdη = dζdη, but ∥h∥2p ≥ ∥h0 ∥2p , λ λ d (ζ, η) d (ζ, η) S S S×S S×S  1 where h0 = |S| h (mean of h on S) is the projection of h onto V0,0 .  S Proof of Theorems 3.1 and 3.2. The claim is now proved and together with Lemmas 5.1–5.3 tell that all extremizer pairs on the sphere is given by f ∼ g ∼ |Jτ |1/p , τ ∈ Aut(S), and the sharp constants and explicit formulas are proved in Section 5.3 and Appendix B.  6. A remark for small λ Weak local extremizer for λ < 4. For λ < 4, we primarily believe constant function should still be a global extremizer as we failed to find any counterexample and maybe strengthened analysis is needed. But what the extremizer could be is still unknown by us so far. However, we remark substitutively that constant function is indeed a weak local extremizer. Actually, we proved that (5.8) holds strictly for h ≡ 1 and all ϕ   satisfying S ϕ = 0, S ϕ(ζ)ζdζ = 0. Note the second assumption of ϕ comes from the conformal symmetry  group. If we decompose it w.r.t (2.6), ϕ = j,k Yj,k with Yj,k ∈ Vj,k , then Y0,0 = Y1,0 = 0, and it suffices to prove  2 α (λα j,k − (p − 1)λ0,0 )|S| ∥Yj,k ∥2 < 0, j,k

where

λα j,k

is the eigenvalues of operator |1 − ζ · η|−2α with α = λ4 . From (5.16), we can check that α λα j,k − (p − 1)λ0,0

is positive when j = k = 0, zero when j = 1, k = 0, and strictly negative otherwise. So, we have proved a strictly negative second variation and therefore “weak local” maximum at constant function 1. A stronger concept of local extremizer means, any function h is called a local extremizer of functional I(f ), if there exists a small δ > 0, s.t, in the δ-neighborhood of h, I(h) reaches a (local) maximum. Unfortunately, we have no uniformly norm expansion near 1.

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391

Acknowledgments The work is partially done during An Zhang’s visit as a visiting student researcher in the department of mathematics, UC, Berkeley. He would like to thank especially the department for hospitality. Michael Christ is supported in part by NSF grant DMS-0901569. Heping Liu is supported by National Natural Science Foundation of China under Grant No. 11371036 and the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 2012000110059. An Zhang is funded by China Scholarship Council under Grant No. 201306010009. Appendix A. Intertwining operators We give in this appendix a proof of the spectrum characterization of the intertwining operators defined on the sphere at the end of Section 2. We use the idea of [5] from [33]. Proposition A.1. Any operator Ad on the sphere S satisfies the intertwining property, i.e.,   Q+d Q−d |Jτ | 2Q (Ad F ) ◦ τ = Ad |Jτ | 2Q (F ◦ τ ) ,

(A.1)

for any F ∈ C ∞ (S) and τ ∈ Aut(S), if and only if it is diagonal with respect to the bispherical harmonic decomposition with its spectrum, modulo a constant, almost uniquely given by     Γ j + Q+d Γ k + Q+d 4 4 −1   . λj,k (Ad |Vj,k ) =  Γ j + Q−d Γ k + Q−d 4 4 −1 Proof. Sufficiency. From the eigenvalues of the kernel K = dS (ζ, η)d−Q , it suffices to prove this kernel satisfies an inverse intertwining relation, i.e.,   Q−d Q+d K |Jτ | 2Q F ◦ τ = |Jτ | 2Q (KF ) ◦ τ, for any conformal transformation and test function, which is obvious from changing variables and the relations (3.8): the left side equals  Q+d dd−Q (ζ, η)|Jτ (η)| 2Q (F ◦ τ )(η)dη S S  Q+d d−Q (τ (ζ), η)|Jτ (τ −1 (η))| 2Q F (η)|Jτ −1 (η)|dη = |Jτ −1 (τ (ζ))Jτ −1 (η)| 2Q dd−Q S S  Q−d = |Jτ (ζ)| 2Q dd−Q (τ (ζ), η)F (η)dη, S S

which is the right side. Necessity. Again the diagonal comes from Schur’s lemma and we now compute the spectrums. Choose δ > 0, and from the diagram (5.4), we have   2δζ ′ 1 + ζn+1 − δ 2 (1 − ζn+1 ) −1 ′ τδ = C ◦ Dδ ◦ C : ζ = (ζ , ζn+1 ) → τδ (ζ) = , . 1 + ζn+1 + δ 2 (1 − ζn+1 ) 1 + ζn+1 + δ 2 (1 − ζn+1 ) We choose the special zonal spherical harmonics for the intertwining relation (A.1) F = Zj,k ,

τ = τδ ,

a

Fδa (ζ) := |Jτδ (ζ)| Q (Zj,k ◦ τδ )(ζ),

where Zj,k is given in (2.7) and the Jacobian of τδ is given by   −Q |Jτδ | = (2δ)−1 1 + ζn+1 + δ 2 (1 − ζn+1 )  .

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Then for operator Ad , we always have Q+d 2

λj,k Fδ

Q−d 2

= Ad Fδ

.

(A.2)

Note that by comparing the highest order of Re ζn+1 , |ζn+1 |, we have  ∂  F a = Aa Zj+1,k + B a Zj,k+1 + Remainder, ∂δ δ=1 δ

(A.3)

where we omit the subscripts j, k of A and B just for simplicity. Then from this property and differentiation of (A.2) with regards to δ at point δ = 1, we have λj,k A

Q+d 2

Zj+1,k + λj,k B

Q+d 2

Zj,k+1 = λj+1,k A

Q−d 2

Zj+1,k + λj,k+1 B

Q−d 2

Zj,k+1 ,

which gives Q+d

A 2 λj+1,k = Q−d , λj,k A 2

Q+d

λj,k+1 B 2 = Q−d . λj,k B 2

(A.4)

Now we compute A, B. We first give several formulas j−k [   2 ] sin(j − k + 1)φ j−k j−k j−k−2i i j−k−i = ai cos φ with ai = (−1) 2j−k−2i , sin φ i i=0   i  n−i n   n+α n+β x+1 x−1 Pnα,β (x) = for x ∈ R, i n−i 2 2 i=0   2n + α + β n α,β Pn (2x − 1) = x + ..., n   n 2n + α + β n−1 α,β ′ (Pn ) (2x − 1) = x + .... 2 n

As only ζn+1 is related, in abuse of notation, we denote by just ζ this last quaternion coordinate, then  j−k   [ 2 ]  ∂    F a = a Re ζ  aj−k (Re ζ)j−k−2i |ζ|2i  Pk2n−1,j−k+1 (2|ζ|2 − 1) i ∂δ δ=1 δ i=0

+

j−k [ 2 ]

   aj−k χi< j−k 2(j − k − 2i) 2(Re ζ)2 − |ζ|2 − 1 (Re ζ)j−k−2i−1 |ζ|2i i

i=0

2

 + χi>0 2i(Re ζ)j−k−2i+1 |ζ|2i−2 (|ζ|2 − 1) Pk2n−1,j−k+1 (2|ζ|2 − 1) 

 j−k [ 2 ]   j−k  + ai (Re ζ)j−k−2i |ζ|2i  (Pk2n−1,j−k+1 )′ (2|ζ|2 − 1) i=0

and after assuming that j − k is positive even and comparing the coefficients of the highest order terms (Re ζ)j−k |ζ|2k and (Re ζ)|ζ|j+k in (A.3), we have     2n + j + k j−k 2n + j + 1 + k j+1−k a a0 (a + 2j) = a0 A , k k        2n + j + k 2n + j + 1 + k 2n + j + k + 1 j−k a j+1−k (a + j + k)aj−k − 2a = A a + B a aj−k−1 j−k j−k−2 j−k j−k−2 , k k k+1 2 2 2 2

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393

which gives Aa =

2n + j + 1  a j+ , 2n + j + k + 1 2

Ba =

  k+1 a k+ −1 2n + j + k + 1 2

and from the recursion relation (A.4), we proved the proposition.



Appendix B. The Jacobian determinants of conformal transformations and formulas of extremizers Proposition B.1. The Jacobian of conformal transformations on the sphere and group are given by  −Q ¯ −Q , |Jτξ | ∼ |1 − ξ · ζ| |JC◦(σqr0 ) | ∼  |q|2 + w − 2q0 · q¯ + r0  , 0

(B.1)

for any ξ ∈ Hn+1 satisfying |ξ| < 1, and q0 ∈ Hn , r0 ∈ H satisfying Re r0 > |q0 |2 . The correspondence between the conformal transformation and its parameters is not unique, but we can choose τξ = C ◦Dδ ◦C −1 ◦Aξ , where  ξ −1 r0 2 −2 δ = 1±|ξ| 1∓|ξ| and Aξ ∈ Sp(n + 1) s.t. Aξ (0, . . . , 0, 1) = |ξ| , and σq0 = Dδ0 ◦ Lu0 , where δ0 = (Re r0 − |q0 | ) and u0 = (q0 , − Im r0 ). Moreover, all extremizers for the sharp HLS are 1/p power of the Jacobians above. 1

Proof. We can prove directly for the Jacobians, but we want more attention on extremizers. A similar computation was done for the Heisenberg group and complex sphere in [32,5]. Extremizers on the sphere. On the one hand, for any extremizer h of (3.5), after the conformal action in ˜ is a constant function, which means h = |Jγ | p1 is the only possible24 form of any extremizer, Lemma 5.2, h with γ given by (5.3). To compute, first note |Jγ | ∼ |JC −1 (Aζ)| |JC (Aγ(ζ))|, then from the formula (2.3) for Jacobian of the Cayley transform  − 2Q−λ 4 (1 + |δq|2 )2 + |δ 2 w|2 h ∼ 2 2 2 (1 + |q| ) + |w|   2Q−λ  1 + |δq|2 ± δ 2 w − 2  =  1 + |q|2 ± w   − 2Q−λ  δ 2 − 1 1 − |q|2 ± w  2 ∼ 1 − 2 δ + 1 1 + |q|2 ∓ w  ¯ − 2Q−λ 2 = |1 − ξ · ζ| , 2

with (q, w) = C −1 (Aζ), ξ = A−1 (0, . . . , δδ2 −1 +1 ), satisfying |ξ| < 1. On the other hand, given any |ξ| < 1, we can inverse the above process: there exists one A ∈ Sp(n + 1) s.t. Aξ = |ξ|(0, . . . , 0, 1), so, through the Cayley transform, we have  − 2Q−λ 1 1  1 − |q|2 ± w  2 − 2Q−λ −1  p p 2 |JC | |1 − ξ · (A ζ)| = |JC | 1 − |ξ| 1 + |q|2 ∓ w   − 2Q−λ 4 (1 + |δq|2 )2 + |δ 2 w|2 2 2 2 − 2Q−λ ∼ ((1 + |q| ) + |w| ) 4 2 2 2 (1 + |q| ) + |w| = ((1 + |δq|2 )2 + |δ 2 w|2 )− with (q, w) = C −1 ζ, δ 2 = of function

1±|ξ| 1∓|ξ| ,

2Q−λ 4

,

which tells from the relation formula (3.4) that the counterpart on the group |1 − ξ · (A−1 ζ)|−

24 Actually they are, as constants are proved to be extremizers.

2Q−λ 2

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is just the δ-dilation of a particular extremizer which corresponds to a constant function on the sphere. Because of the rotation-invariant of (3.5), any function with formula (3.7) is indeed an extremizer. Then, combine the two direction arguments, we have proved that functions with formulas (3.7) are right all the extremizers for (3.5) and the first formula in (B.1) is then proved. Moreover, we note that the set of Jacobians of all conformal transformations γξδ in Lemma 5.2 ergod the set of Jacobians |Jγ |, which tells the correspondence part of the proposition on the sphere. Extremizers on the group. For any |ξ| < 1, 1 p

¯ − 2Q−λ 2

|JC | |1 − ξ · ζ|

  − 2Q−λ 2  2q 1 − |q|2 + w   ∼ ((1 + |q| ) + |w| ) , 1 − ξ ·   1 + |q|2 − w 1 + |q|2 − w  − 2Q−λ   2  ∼ 1 + |q|2 + w − (ξ1 , ξ2 , . . . , ξn ) · (2q) − ξn+1 · (1 − |q|2 − w)  − 2Q−λ ∼  |q|2 + w − 2q0 · q¯ + r0  2 , 2 2

2 − 2Q−λ 4

n+1 1 ,...,ξn ) 2 , r0 = 1−ξ with q0 = (ξ1+ξ 1+ξn+1 , satisfying Rer0 > |q0 | . Then, because of the bijection between {ξ : |ξ| < 1} n+1 and {(q0 , p0 ) : Rep0 > |q0 |2 }, we have proved that all extremizers of (3.1) are explicitly given by (3.3). Moreover, we remark that all Jacobians |JC◦σ | are ergoded by dilations and left translations from |JC |, with 1 the correspondence σqr00 = Dδ0 ◦ Lu0 , where δ0 = (Rer0 − |q0 |2 )− 2 , u0 = (q0 , −Imr0 ). Actually, for any δ0 > 0, u0 = (q0 , w0 ) ∈ G,  − Q 2 2 2 |JC (δ0 (u−1 ¯)|2 2 0 u))| ∼ (1 + |δ0 (q − q0 )| ) + |δ0 (w − w0 − 2Imq0 · q  −Q = 1 + |δ0 (q − q0 )|2 + δ02 (w − w0 − 2Imq0 · q¯)  −Q ∼ |q|2 + w − 2q0 · q¯ + δ0−2 + |q0 |2 − w0  .

Then we get another part of this proposition and Theorems 3.1 and 3.2.



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