On Lie groups and hyperbolic symmetry—From Kunze–Stein phenomena to Riesz potentials

Nonlinear Analysis

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Nonlinear Analysis www.elsevier.com/locate/na

On Lie groups and hyperbolic symmetry—From Kunze–Stein phenomena to Riesz potentials William Beckner Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin TX 78712-0257, USA

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Communicated by Enzo Mitidieri Keywords: Lie groups Hyperbolic symmetry

abstract Sharp forms of Kunze–Stein phenomena on SL(2, R) are obtained by using symmetrization and Stein–Weiss potentials. A new structural proof with remarkable simplicity can be given on SL(2, R) which effectively transfers the analysis from the group to the symmetric space corresponding to a manifold with negative curvature. Our methods extend to include the Lorentz groups and n-dimensional hyperbolic space through application of the Riesz–Sobolev rearrangement inequality. A new framework is developed for Riesz potentials on semisimple symmetric spaces and the semi-direct product of groups analogous to the Iwasawa decomposition for semisimple Lie groups. Extensions to higher-rank Lie groups and analysis on multidimensional connected hyperboloids including anti de Sitter space are suggested by the analysis outlined here. © 2015 Elsevier Ltd. All rights reserved.

“Now the upper half-plane is the arena of action of the group SL(2, R) of fractional linear transformations.” [Eli Stein] The challenge for analysis on Lie groups in adapting style and techniques from classical Fourier analysis is initiated in Stein’s lecture at Nice [32] where he asked how Young’s inequality for convolution would be enlarged for a noncompact semisimple Lie group with finite center. Convolution on the group G with left-invariant Haar measure is defined  (f ∗ g)(x) = f (y)g(y −1 x) dy. (1) G

For functions on a unimodular group, an equivalent realization is given by  (f ∗ g)(x) = f (xy −1 )g(y) dy. G

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.na.2015.06.009 0362-546X/© 2015 Elsevier Ltd. All rights reserved.

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A Lie group is called unimodular if the left-invariant measure is also right-invariant—examples include compact groups, semisimple Lie groups and connected nilpotent groups. A characteristic example for a non-unimodular group would be the realization of the group structure on the hyperbolic space Hn . Young’s inequality for a locally compact unimodular Lie group is given by ∥f ∗ g∥Lr (G) ≤ ∥f ∥Lp (G) ∥g∥Lq (G)

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for 1/r = 1/p + 1/q − 1 with p, q, r ≥ 1. For a non-unimodular group the inequality changes (see Weil [34]) ′

∥f ∗ g∥Lr (G) ≤ ∥f ∥Lp (G) ∥∆−1/p g∥Lq (G)

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where ∆ denotes the modular function defined by m(Ex) = ∆(x)m(E) with m denoting Haar measure and 1/p + 1/p′ = 1. This inequality may not incorporate the geometric symmetry of the group as is evident from the example of hyperbolic space Hn where for the Liouville–Beltrami upper half-space representation with coordinates (x, y) ∈ Rn−1 × R+ , Haar measure dν = y −n dx dy and ∆(x, y) = y −(n−1) which is not a radial function. But observe that for radial functions on a non-unimodular group    −1 ϕ dν = ϕ∆ dν = ϕ dνr (5) G

G

G

where dνr is right-invariant Haar measure. Stein observed from his work with Kunze on analytic continuation of group representations that for 1≤p<2 ∥f ∗ g∥L2 (G) ≤ Ap ∥f ∥Lp (G) ∥g∥L2 (G)

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for G being SL(2, R) or any complex classical group. Stein’s conjecture that this estimate should hold for all semisimple Lie groups with finite center was shown by Cowling using representation theory. Our purpose here is to consider the case of SL(2, R) and the Lorentz groups SO(n, 1), and examine how geometric analysis determines the estimate. For his analysis of representation theory of noncompact semisimple Lie groups, Stein emphasized the role played by the nilpotent group N in an Iwasawa decomposition of G. The example of SL(2, R) is instructive. There is an intriguing contrast in viewing the group G and its nilpotent part N as semisimple and thus unimodular (see Harish-Chandra [20]; Stein [32]) while the symmetric space M ≃ G/K will be a Cartan–Hadamard manifold with negative curvature modeled on hyperbolic space with a group structure that is non-unimodular. Our initial arguments are modeled on SL(2, R) where M has a metric structure (e.g., a distance function) and the nilpotent part is abelian. The metric structure of the manifold will be instrumental in determining the analytic approach (see [18]). The strategy is to use computation to draw out structural ideas from patterns in exact model calculations. 1. Rearrangement and symmetry The first objective is to understand the role of symmetry in determining the structure of analysis on the group SL(2, R). Symmetrization is the process of improving the value of a functional by replacing constituent functions with their equimeasurable radial decreasing rearrangements:   |f | f ∗, m |f | > α = m{f ∗ > α}, f ∗ (d) ≥ f ∗ (d + δ). The classical example is the Riesz–Sobolev inequality      f (x)g(x − y)h(y) dx dy  ≤  Rn ×Rn

f ∗ (x)g ∗ (x − y)h∗ (y) dx dy

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Rn ×Rn

which has been instrumental in the study of Sobolev embedding and the determination of optimal constants for Young’s inequality for convolution [2]. This useful result is equivalent to the Brunn–Minkowski inequality

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for the measure of sum sets (see Zygmund’s argument in Hardy, Littlewood and P´olya [19]) m∗ (A + B) ≥ m(A∗ + B ∗ )

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where m denotes Lebesgue measure on Rn , m∗ is inner Lebesgue measure, A and B are measurable sets with finite positive measure and A∗ denotes the rearrangement of the set A as a ball centered at the origin with the same measure as A. The Riesz–Sobolev inequality was extended by Brascamp, Lieb and Luttinger to multilinear functionals [13]:       N m N m      fℓ aℓk xk dx ≤ fℓ∗ aℓk xk dx. (9)  Rmn ℓ=1

Rmn ℓ=1

k=1

k=1

For the group SL(2, R) and its corresponding symmetric space M which can be realized as the twodimensional space H2 , the starting point is to extend the Riesz–Sobolev inequality in this new setting (see Section 6 in [3]; appendix in [6]). Theorem 1.      (f# ∗ g# )(x)h# (x) dx.  (f ∗ g)(x)h(x) dx ≤ G

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G

The differential dx denotes Haar measure on the group, and # denotes an equimeasurable rearrangement of the absolute value of a function on G such that for a Cartan decomposition x = (w, k) the rearranged function is radial in the w variable. Here w ∈ M = SL(2, R)/SO(2) ≃ H2 and k ∈ K = SO(2) which is the maximal compact subgroup of G. This theorem allows the Kunze–Stein inequality to be reduced to functions radial in the w variable and constant in the k variable and thus bi-invariant functions on the group. A function f is bi-invariant if f (k1 xk2 ) = f (x) for all x ∈ G and k1 , k2 ∈ K where K is the maximal compact subgroup of G. The convolution algebra for bi-invariant functions is commutative (see [22, p. 408]). Stein developed a short argument using Harish-Chandra’s positive spherical function and interpolation to show that Kunze–Stein inequality (6) holds when f ∈ Lp (G) is bi-invariant (see [30,32]). Spherical functions have the property that  ϕ(xky) dk = ϕ(x)ϕ(y), x, y ∈ G. (11) K

This method of using Theorem 1 can be adapted for SL(2, C), the Lorentz groups, and any group with an abelian nilpotent part for the Iwasawa decomposition G = KAN . Proof of Theorem 1. Though this result is directed at the group SL(2, R), the proof will be structured so that it is applicable in a more general context. Suppose for the Cartan decomposition of G that M = G/K ≃ Hn . Then the convolution functional for non-negative functions can be expressed as   f (x)g(x−1 y)h(y) dx dy = f (w1 k1 )g(k1−1 w1−1 w2 k2 )h(w2 k2 ) dw1 dw2 dk1 dk2 G×G G×G  = f (w1 , k1 )g(k1−1 w1−1 w2 k1 , k1−1 k2 )h(w2 , k2 ) dw1 dw2 dk1 dk2 . (M,K)×(M,K)

w1−1 w2

The group product will include linear combinations of the nilpotent variables where for an abelian nilpotent subgroup the manifold will be an Euclidean space. In terms of the Iwasawa decomposition, the action of the multiplicative subgroup A will be as coefficients for linear combinations of the nilpotent variables. The strategy will be to apply the Riesz–Sobolev rearrangement inequality for a Euclidean subspace of M ; then make a global change of variables on M and apply Riesz–Sobolev rearrangement again; by the Ljusternik technique one can obtain a sequence of rearrangements on M which converge to a radial decreasing function f# (w, k) on M , equimeasurable with the initial given function |f (w, k)|. Since g# is radial in the

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w variable, g# (k1−1 w1−1 w2 k1 , k1−1 k2 ) = g# (w1−1 w2 , k1−1 k2 ) which for fixed values of the K variable will be a decreasing function of the distance between w1 and w2 in M .  To illustrate this argument with a concrete example, consider the case where M ≃ Hn (for G = SL(2, R), M ≃ H2 ). Here one can use the Liouville–Beltrami upper half-space model for Hn : w = (x, y) ∈ Rn+ ≃ Rn−1 × R+ ≃ SO(n, 1)/SO(n) ≃ Hn with the Poincar´e distance function d(w, w′ ) =

|w − w′ | √ 2 yy ′

and left-invariant Haar measure dν = y −n dx dy. The Riemannian metric on Hn is defined by ds2 = y −2 (dx2 + dy 2 ) and the invariant gradient is given by D = y∇. The group structure of hyperbolic space corresponds to a non-unimodular Lie group that is an extension of the affine “ax + b group”. Hyperbolic space Hn is identified with the subgroup of SL(n, R) given by all matrices of the form   √ 1 x/y n y 0 1/y where x ∈ Rn−1 is represented by a column vector, y > 0 and 1 is the unit (n − 1) × (n − 1) matrix. Such matrices can act via fractional linear transformation on Rn+ ≃ Hn w = x + iyξ ∈ Rn+ −→ for an n×m matrix



A C

B D



Aw + B Cw + D

where A = (n−1)×(n−1) matrix, B = (n−1)×1 matrix, C = 1 ×(n−1) matrix

and D = 1 × 1 matrix with fixed non-zero ξ ∈ Rn−1 . The group action corresponds to the multiplication rule (x, y)(u, v) = (x + yu, yv) for x, u ∈ Rn−1 and y, v > 0. This is a non-unimodular group with the modular function ∆(x, y) = y −(n−1) . The group identity is  0 = (0, 1). The group multiplication w1−1 w2 corresponds to       x u v (x − u) v x 1 (u, v) = − + , = − , (x, y)−1 (u, v) = − , y y y y y y y so the application of Riesz–Sobolev rearrangement is clear for the x, u variables. A Ljusternik sequence of rearrangements in the Euclidean variables for the non-unimodular representation of Hn gives a proof of symmetrization on hyperbolic space Hn . Theorem 2 (Riesz–Sobolev Rearrangement Inequality on Hn ).      (f ∗ g)(w)h(w) dν  ≤ (f ∗ ∗ g ∗ )(w)h∗ (w) dν  Hn

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Hn

where f ∗ denotes the equimeasurable radial decreasing rearrangement of the function |f | on Hn . This theorem allows one to obtain the Brunn–Minkowski inequality on Hn . Let E and F be two measurable subsets of Hn with finite positive measure and E ∗ , F ∗ denoting their rearrangements as geodesic balls in Hn centered at the origin  0 = (0, 1). Let EF denote the set determined by each element of E acting on an element of F through the group operation. Then ν∗ (EF ) ≥ ν(E ∗ F ∗ ).

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A proof will be given in the Appendix including a corresponding derivation of the isoperimetric inequality on Hn : σH∗ (E) ≥ σH (∂E ∗ ).

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Here the lower ∗ indicates “inner measure”. 2. Kunze–Stein phenomena on SL(2, R) Symmetrization provides a direct tool to obtain the Kunze–Stein inequality for a category of semisimple Lie groups. Theorem 3. Let G be any semisimple Lie group with finite center for which inequality (10) holds with f ∈ Lp (G), 1 ≤ p < 2 and g, h ∈ L2 (G). Then for f, g, h ≥ 0, x = (w2 , k2 ) and y = (w1 , k1 ) with dk denoting normalized measure on the compact group K   (f ∗ g)(x)h(x) dx ≤ (f# ∗ g# )(x)h# (x) dx G G = f# (w1 , k1 )g# (w1−1 w2 , k1−1 k2 )h# (w2 , k2 ) dx dy G×G  ≤ F (w1 )G(w1−1 w2 )H(w2 ) dw1 dw2 M ×M   = (F ∗ G)(x)H(x) dx = (G ∗ F )(x)H(x) dx G

G

 = M ×M

G(w1 )F (w1−1 w2 )H(w2 ) dw1 dw2

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   where F (w) = ( K |f# (w, k)|p dk)1/p , G(w) = ( K |g# (w, k)|2 dk)1/2 , H(w) = ( K |h# (w, k)|2 dk)1/2 and these functions are bi-invariant. Proof. Inequality (15) follows from Young’s inequality applied to the integration over the unimodular maximal compact group K. F , G and H though seemingly defined on M = G/K extend to be bi-invariant functions on G. That is, for x = wk ∈ G and k1 , k2 ∈ K F (k1 xk2 ) = F (k1 wkk2 ) = F (k1 wk1−1 k1 kk2 )  1/p −1 ′ p ′ = |f# (k1 wk1 , k1 kk2 k )| dk K



′

p

|f# (w, k )| dk

=

′

1/p = F (w).

K

Hence the map g ∈ L2 (G)

f ∗ g ∈ L2 (G)

g ∈ L2 (G)

F ∗ g ∈ L2 (G)

for f ∈ Lp (G) is controlled by the map

for F ∈ Lp (G) is controlled by the map g ∈ L2 (G)

F ∗ g ∈ L2 (G)

g ∗ F ∈ L2 (G)

for f ∈ Lp (G) and bi-invariant, and the property that the convolution algebra of bi-invariant functions on a semisimple Lie group is applied to exchange the order of convolution: F ∗ G = G ∗ F . 

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Stein’s Lemma. Let f ∈ Lp (G), 1 ≤ p < 2, be a bi-invariant function on a semisimple Lie group G with finite center. Then ∥f ∗ g∥L2 (G) ≤ A∥f ∥Lp (G) ∥g∥L2 (G) .

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Proof. Let ϕ0 be the positive spherical function on G given by  ϕ0 (x) = e−ρH(xk) dk > 0

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K

which satisfies the Harish-Chandra decay estimate 1 1 + = 1. p q

∥ϕ0 ∥Lq (G) < ∞ for q > 2,

H(x) is the Lie algebra component for the A part in the Iwasawa decomposition KAN. Using the identity for spherical functions  ϕ0 (xky) dk = ϕ0 (x)ϕ0 (y) K

and interpolation provides the Kunze–Stein inequality. For k ∈ K    (f ∗ ϕ0 )(x) = f (y)ϕ0 (y −1 x) dy = f (k −1 y)ϕ0 (y −1 kx) dy = f (y)ϕ0 (y −1 kx) dy. G

G

G

Integrate over the maximal compact subgroup K   −1 (f ∗ ϕ0 )(x) = f (y)ϕ0 (y kx) dy dk = f (y)ϕ0 (y −1 ) dy ϕ0 (x) = c ϕ0 (x) G×K

G

with |c| ≤ ∥f ∥Lp (G) ∥ϕ0 ∥Lq (G) . Let T g = f ∗ g for f ≥ 0 and define the multiplication operator Mz g = ϕz0 g for z = σ + iτ , −1 ≤ σ ≤ 1. Consider the operator Sz ∗ M−z T Mz and observe by duality that ∥S1 g∥L∞ (G) ≤ ∥ϕ0 ∥Lq (G) ∥f ∥Lp (G) ∥g∥L∞ (G) ∥S−1 g∥L′ (G) ≤ ∥ϕ0 ∥Lq (G) ∥f ∥Lp (G) ∥g∥L′ (G) . Since |Sz g| ≤ Sσ |g| and S0 g = T g, apply the Riesz interpolation argument to obtain for T g = f ∗ g ∥f ∗ g∥L2 (G) ≤ ∥ϕ0 ∥Lq (G) ∥f ∥Lp (G) ∥g∥L2 (G) . 

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This result can be strengthened by considering explicit variables on SL(2, R) and the application of Stein–Weiss potentials and Pitt’s inequality. Since the analysis here is more directed at the symmetric space M ≃ G/K associated to the non-compact semisimple Lie group G with maximal compact subgroup K, a more straightforward approach would be to show that the convolution algebra of radial functions on M is commutative, and the order of convolution can be exchanged for radial functions. Lemma 1. The convolution of two radial functions on M is a radial function. Proof. Let K be the subgroup of isometries on M that leaves the origin  0 invariant. A function f is radial if f (kwk −1 ) = f (w) for k ∈ K; that is, f is a function of the distance d(w,  0), and f (w) = f (w−1 ). Our notation treats w as a point in the symmetric space and as a group element interchangeably. Suppose f and

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g are radial functions. Consider (f ∗ g)(kwk −1 ) =



f (y)g(y −1 kwk −1 ) dν

M

 = M

 =

f (kyk −1 )g(ky −1 k −1 kwk −1 ) dν  f (y)g(ky −1 wk −1 ) dν = f (y)g(y −1 w) dν

M

M

= (f ∗ g)(w) and the convolution of two radial functions on M is a radial function.



Lemma 2. The convolution algebra for radial functions on M is commutative. Proof. Consider the functional form for f, g, h radial functions   h(x)(f ∗ g)(x) dν = h(x)f (y)g(y −1 x) dν dν M M ×M   = h(x)f (xy)g(y −1 ) dν dν = h(x)f (xy)g(y) dν dν M ×M M ×M  = h(x−1 )f (x−1 y)g(y)∆−1 (x) dν dν M ×M  = h(x−1 )f (y −1 x)g(y)∆−1 (x) dν dν M ×M  = h(x−1 )(g ∗ f )(x)∆−1 (x) dν M   = h(x)(g ∗ f )(x−1 ) dν = h(x)(g ∗ f )(x) dν M

M

and this relation for all h implies (f ∗ g)(x) = (g ∗ f )(x),

x ∈ M.



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The Kunze–Stein result on SL(2, R) can be strengthened by considering explicit variables and the application of Stein–Weiss potentials and Pitt’s inequality. Theorem 4. Let G = SL(2, R) and F ∈ Lp,∞ (H2 ), 1 < p < 2, where F is defined above. Then for f, g, h ≥ 0 and g, h ∈ L2 (G)  (f ∗ g)(x)h(x) dx ≤ cp ∥F ∥Lp,∞ (H2 ) ∥g∥L2 (G) ∥h∥L2 (G) (20) G

Γ (1/p′ ) cp = π Γ (1/p) 



Γ [(1 − 2/p′ )/4] Γ [(1 + 2/p′ )/4]

2 .

This constant is sharp and is not attained. Proof. Use the commutativity for convolution of bi-invariant functions and the resulting formula from Theorem 3   (f ∗ g)(x)h(x) dx ≤ H(w1 )F (w2−1 w1 )G(w2 ) dν1 dν2 . G

H2 ×H2

On H2 the Haar measure dν reduces to 4πdu where u = d2 (w,  0) where d(w,  0) is the Poincar´e distance d(w1 , w2 ) =

|w − w′ | , √ 2 y1 y2

w = (x, y).

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For F ∈ Lp,∞ (H2 ) it suffices to take F (w) = [1/d(w,  0)]2/p . Now the proof of inequality (20) depends on the 2 following fractional integral inequality on H for G, H ≥ 0:  G(w1 )d(w1 , w2 )−2/p H(w2 ) dν1 dν2 ≤ cp ∥G∥L2 (H2 ) ∥H∥L2 (H2 ) . (21) H2 ×H2

¯ y); then the above inequality becomes Let G(w) = y g¯(x, y), H(w) = y h(x,  −1/p  ′ ¯ 2 , y2 ) dx dy ≤ cp ∥¯ ¯ L2 (R2 ) . g¯(x1 , y1 )(y1 y2 )−1/p |x1 − x2 |2 + |y1 − y2 |2 h(x g ∥L2 (R2+ ) ∥h∥ + R2+ ×R2+

Apply Young’s inequality for convolution in the x variables as a map from L2 (R) to L2 (R):    −1/p |x|2 + |y1 − y2 |2 dx = |y1 − y2 |−(2/p−1) (1 + |x|2 )−1/p dx R

R

=

√

Γ (1/p − 1/2) π |y1 − y2 |−(2/p−1) . Γ (1/p)

Now the basic estimate here is reduced to the Stein–Weiss fractional integral on R+ with standard Lebesgue measure. Without loss of generality, set  1/2  1/2 ¯ ¯ y)|2 dx g¯(y) = |¯ g (x, y)|2 dx , h(y) = |h(x, R R  ′ 1 Γ (1/p) −1/p′ −(2/p−1) −1/p ¯ ¯ L2 (R ) . g¯(y1 ) y1 |y1 − y2 | y2 h(y2 ) dy1 dy2 ≤ √ cp ∥¯ g ∥L2 (R+ ) ∥h∥ + π Γ (1/p − 1/2) R+ ×R+ Set α/2 = 1/p′ and observe that 1 − α = 1 − 2/p′ = 2/p − 1; then using the sharp Stein–Weiss potential estimate [4]   2 √ Γ (1/p′ ) 1 Γ [(1 − 2/p′ )/4] Γ (1/p) √ cp = π Γ (1/p − 1/2) Γ [(1 + 2/p′ )/4] π Γ (1/p − 1/2) so that  cp = π

Γ (1/p′ ) Γ (1/p)



Γ [(1 − 2/p′ )/4] Γ [(1 + 2/p′ )/4]

2 .

Since the Stein–Weiss inequality is sharp without extremal functions, the same property must hold for the Kunze–Stein inequality (20).  Since f ∈ Lp (G), G = SL(2, R), implies that F ∈ Lp,∞ (H2 ), this result proves the Kunze–Stein inequality (6). As with the Hardy–Littlewood–Sobolev inequality on Rn , this argument sharpens the control given by Young’s inequality through enabling the calculation of exact constants for fractional integrals. Application of Stein–Weiss techniques allows an extension of the Kunze–Stein inequality for G = SL(2, R) to an index q where either q or its dual q ′ lies in the interval (p, 2): ∥f ∗ g∥Lq (G) ≤ Ap,q ∥f ∥Lp (G) ∥g∥Lq (G) .

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Theorem 5. Let G = SL(2, R) and F ∈ Lp,∞ (H2 ), 1 < p < 2, where F is defined above. For f, g ≥ 0 and g ∈ Lq (G) with the index q or its dual q ′ being in the interval between p and 2 ∥f ∗ g∥Lq (G) ≤ dp,q ∥F ∥Lp,∞ (H2 ) ∥g∥Lq (G)   Γ (1/p′ )Γ (1/2p − 1/2q)Γ (1/2p − 1/2q ′ ) . dp,q = π Γ (1/p)Γ (1/2p′ + 1/2q)Γ (1/2p′ + 1/2q ′ )

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Proof. Proceed as in the proof of Theorem 4 and reduce the estimate to the inequality  −(2/q ′ −1/q) −(2/q−1/p) ¯ g¯(y1 )y1 |y1 − y2 |−(2/p−1) y2 h(y2 ) dy1 dy2 R+ ×R+

Γ (1/p) 1 ¯ q′ dp,q ∥¯ g ∥Lq (R+ ) ∥h∥ ≤√ L (R+ ) . π Γ (1/p − 1/2) This estimate is given by Theorem 2 for n = 1 in [8]. Observe that it is not required that the powers of y1 and y2 not both be negative but only for y1−α y2−β that 0 < α + β = (2/q ′ − 1/p) + (2/q − 1/p) = 2/p′ < 1, and for α < 1/q ′ , β < 1/q which requires both q, q ′ be larger than p. Then   √ Γ (1/p′ )Γ (1/2p − 1/2q)Γ (1/2p − 1/2q ′ ) 1 Γ (1/p) √ dp,q = π Γ (1/p − 1/2)Γ (1/2p′ + 1/2q ′ )Γ (1/2p′ + 1/2q) π Γ (1/p − 1/2)   Γ (1/p′ )Γ (1/2p − 1/2q)Γ (1/2p − 1/2q ′ ) dp,q = π . Γ (1/p)Γ (1/2p′ + 1/2q ′ )Γ (1/2p′ + 1/2q) Again this inequality is sharp with no extremal functions since it is obtained by application of a Stein–Weiss integral on the diagonal Lq → Lq .  The emphasis here is to develop the connection between Kunze–Stein phenomena, Riesz potentials, the Hardy–Littlewood–Sobolev inequality and Stein–Weiss integrals—and analysis on the symmetric space and non-unimodular groups. Other approaches utilizing maximal functions and interpolation on the full group were treated by Ionescu (see [23]). But by reducing the analysis on SL(2, R) to the product manifold SO(2) × H2 and radial decreasing functions on H2 through application of the Riesz–Sobolev rearrangement inequality, a purely elementary integration argument can be given to show the natural result suggested by Ionescu’s theorem: L2,1 (H2 ) ∗ L2,1 (H2 ) ⊂ L2,∞ (H2 ) ∩ L∞ (H2 )

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which strengthens understanding for convolution on SL(2, R) while utilizing only qualitative features of the geometric structure. A similar integration argument extends this result to n-dimensional hyperbolic space. 3. Kunze–Stein Phenomena on SL(2, C) The Lie group SL(2, C) is an instructive example since its Cartan decomposition is similar to that for SL(2, R); namely the maximal compact subgroup is SU (2) and    a 0 1 z G/K ≃ ≃ H3 , a > 0, z ∈ C. 0 a−1 0 1 This is an equivalent but different representation for H3 than outlined above in Section 1. But when the dimension is larger than two, the growth of balls in Hn is variable between large balls and small balls. Consider radial variables in Hn and let u = [d(w,  0)]2 ; then Haar measure restricted to radial functions is given by dν =

(n/2 − 1) (4π)n/2  u(1 + u) du Γ (n/2)

and F ∈ Lp,∞ (Hn ) means that  ∗

F (u) ≤ A u

−n/2



1

t(1 + ut)

n/2 − 1

−1 1/p dt .

0

Observe that for max{n/2, (n/2 − 1)p} < c < min{(n − 1), np/2}, f ∗ (u) ≤ Au−c/ρ and the techniques utilizing Stein–Weiss estimates from the previous section can be applied to show that for 1 < p < ∞ ∥F ∗ G∥L2 (Hn ) ≤ B∥F ∥Lp,∞ (Hn ) ∥G∥L2 (Hn ) .

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Such an inequality by the nature of the approximation for the integral kernel is not sharp and can be improved if the concept for Stein–Weiss integral can be enlarged. For w ∈ Hn , and u = [d(w,  0)]2 , define for 1 < p < 2 −1/p  1  n/2 − 1 . Kn (w) = u−n/2p s(1 + us) ds 0

The growth of geodesic balls in hyperbolic space can be represented by hypergeometric functions, but there does not seem to be a more elementary form for Kn (w) when the integer n is not even. Theorem 6. Let G = SL(2, C) and F ∈ Lp,∞ (H3 ), 1 < p < 2, where F is defined as in Theorem 3. Then for f, g, h > 0 and g, h ∈ L2 (G)  (f ∗ g)(x)h(x) dx ≤ cp ∥F ∥Lp,∞ (H3 ) ∥g∥L2 (G) ∥h∥L2 (G) (26) G    cp = 8 K3 x2 + sinh2 (t) dx dt. R2 ×R

The constant is sharp and not attained. Proof. Follow the outline of the argument as given in Theorem 4   (f ∗ g)(x)h(x) dx ≤ H(w1 )F (w1 w2−1 )G(w2 ) dν1 dν2 . H3 ×H3

G

F (w1 w2−1 )

K(w1 w2−1 )

Here take = = K[d2 (w1 , w2 )] with a slight modification in notation. Apply Young’s inequality in the x variable so that now we have     K3 d2 (w1 w2−1 ) dx1 = 4y1 y2 K3 (x2 + v1,2 ) dx = 4y1 y2 ϕ(v1,2 ), v1,2 = (y1 − y2 )2 /(4y1 , y2 ) R2

R2

√ g(y1 ) = 2 v1



1/2 |G(x + v1 )| dx , 2

2

√ h(y2 ) = 2 v2

1/2 |H(x + v2 )| dx



R2

2

2

R2

2

2

where v1 = (y1 − 1) /4y1 and v2 = (y2 − 1) /4y2 . The objective is to show  1/2   −3 2 −3 4 g(y1 )y1 y2 ϕ(v1,2 )h(y2 )(y1 y2 ) dy1 dy2 ≤ cp |g| y dy R+ ×R+

R+

2 −3

|h| y

1/2 dy

.

R+

Then letting  g¯(y1 ) =

1/2 |G(x + v1 )| dx , 2

R2

2

¯ 2) = h(y



1/2 |H(x + v2 )| dx 2

2

R2

the required inequality becomes  ¯ L2 (R ) ¯ 2 ) 1 dy1 d2 ≤ 4cp ∥¯ g ∥L2 (R+ ) ∥h∥ 16 g¯(y1 ) ϕ(v1,2 )h(y + y1 y2 R+ ×R+ where R+ now denotes the multiplicative group with Haar measure y −1 dy. Then by a simple application of Young’s inequality     −1 cp /4 = ϕ(v1,2 )y1 dy1 = K3 x2 + (y − 1)2 /4y dx y −1 dy 2 R+ R ×R+   2  =2 K3 x + sinh2 (t) dx dt.  R2 ×R

This argument extends naturally to determine a Stein–Weiss integral for n-dimensional hyperbolic space.

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Theorem 7. For non-negative G and H in L2 (Hn ) and 1 < p < 2    H(w1 )Kn d2 (w1 , w2 ) G(w2 ) dν1 dν2 ≤ cp ∥G∥L2 (Hn ) ∥H∥L2 (Hn ) Hn ×Hn    cp = 2n Kn x2 + sinh2 (t) dx dt.

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Rn−1 ×R

The constant is sharp and not attained. Proof. Follow the steps of the previous argument. The Riesz–Sobolev inequality on Hn (Theorem 2) allows G and H to be radial decreasing on Rn . Apply Young’s inequality in the x variable: let     √ Kn [x2 + v1,2 ] dx Kn d2 (w1 w2−1 ) dx1 = (2 y1 y2 )n−1 Rn−1

Rn−1

√

−n−1

v1,2 = (y1 − y2 )2 /(4y1 y2 )

= (2 y1 y2 )

ϕ(v1,2 ), 1/2 √ g(y1 ) = (2 y1 )(n−1)/2 |G(x2 + v1 )|2 dx , Rn−1  1/2 √ (n−1)/2 2 2 |H(x + v2 )| dx h(y2 ) = (2 y1 ) 

Rn−1

where v1 = (y1 − 1)2 /4y1 and v2 = (y2 − 1)2 /4y2 . The objective is to show  1 1 2n−1 g(y1 )(y1 y2 )(n−1)/2 ϕ(v1,2 )h(y2 )(y1 y2 )−(n−1) dy1 dy2 y1 y2 R+ ×R+  1/2  1/2 ≤ cp |g|2 y −n dy |h|2 y −n dy . R+

R+

Then letting  g¯(y1 ) =

1/2 |G(x + v1 )| dx , 2

2

Rn−1

¯ 2) = h(y



1/2 |H(x + v2 )| dx 2

2

Rn−1

the required inequality becomes  n−1 ¯ 2 ) 1 dy1 1 dy2 ≤ cp ∥¯ ¯ L2 (R ) 2 g¯(y1 ) ϕ(v1,2 )h(y g ∥L2 (R+ ) ∥h∥ + y1 y2 R+ ×R+ where R+ denotes the multiplicative group with Haar measure y −1 dy. Then using Young’s inequality again     cp /2n−1 = ϕ(v1,2 )y1−1 dy1 = Kn x2 + (y − 1)2 /4y dx y −1 dy R+ Rn−1 ×R+   2  =2 Kn x + sinh2 (t) dx dt.  Rn−1 ×R

4. Kunze–Stein phenomena on Lorentz groups Structural complexity for matrix groups can be expressed by multiple geometric representations of a manifold or symmetric space. In principle, such a framework allows quite varied techniques from analysis to be brought to bear on the objective of obtaining “sharp estimates”. For SL(2, C) two models described above realize the manifold as a homogeneous space under the group action. Another possibility would be to utilize the vector space character of the matrix though identification of the group structure will require more effort. The symmetric space associated with SL(2, C) can be identified with the collection of two dimensional Hermitian matrices with determinant one:   p0 + p3 p1 − ip2 p0 1 + p¯ · σ ¯= , p20 − p¯2 = 1 p1 + ip2 p0 − p3

W. Beckner / Nonlinear Analysis

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where σ ¯ denotes the Pauli spin matrices   0 1 , σ1 = 1 0

 0 σ2 = i

(

)

–

 1 σ3 = 0

 −i , 0

 0 . −1

This manifold corresponds to the two sheeted hyperboloid in R4 , {p20 − |¯ p|2 = 1}, and is a double copy of the realization given for H3 above since it includes both ±1. In terms of classical geometry, this representation is natural from the conformal equivalence of the two-sheeted hyperboloid in Rn+1 with Rn and S n . The natural group action on this manifold is the Lorentz group, with the associated symmetric space SO(3, 1)/SO(3), viewed as the group of transformations that leave the form pq = p0 q0 − p¯q¯ invariant—for Λ ∈ SO(3, 1), then     1 0 1 0 T Λ Λ= . 0 −1 0 −1  Analysis will be restricted to the positive hyperboloid H+ ⊂ R4+ with p0 = 1 + p¯2 which corresponds to the identity component so that the manifold is simply connected. As with the Poincar´e metric for the Liouville–Beltrami upper half-space model of Hn , a “distance function” is used which is not a metric d2 (p, q) = pq − 1 ≥ 0. Using notation that takes into account the Lorentz metric structure   1 0 , p2 = p20 − p¯2 0 −1 then 0 ≤ 2d2 (p, q) = −(p − q)2 = |¯ p − q¯|2 − |p0 − q0 |2 ≤ |¯ p − q¯|2 . Observe that this distance is monotone increasing in terms of the separation between two points. The invariant measure on H+ is given by dν(p) = 2δ(1 − p2 ) dp = 

1 1 + p¯2

d¯ p.

With this geometric representation, the variables are on an equal footing and there are no “straight lines” for n ≥ 2 which precludes the possibility of a Riesz–Sobolev rearrangement inequality. This is a simple reflection of the manifold’s curvature and a similar obstruction occurs for the sphere. But it is possible to obtain a “symmetrization lemma” using “two-point symmetrization” as with the sphere if one function is radial  2 ; then 1 + p ¯ decreasing in terms of the distance. More generally, let p = (p0 , p¯) ∈ H+ ⊂ Rn+1 for p = 0 + Theorem 8 (Symmetrization Lemma on H+ ). For non-negative F, G ∈ S(H+ ), K a non-negative monotone decreasing function on R+ , and with F ∗ , G∗ denoting equimeasurable radial decreasing rearrangements of F, G:   F (p)G(q)K(pq − 1) dν dν ≤ F ∗ (p)G∗ (q)K(pq − 1) dν dν. (28) H+ ×H+

H+ ×H+

Proof. The first approach is to use the “two-point symmetrization lemma” (see Appendix in [10]). For the variables p¯ ∈ Rn , the ball has reflection symmetry so one can apply the two-point symmetrization argument for a half-plane decomposition and gain improvement; then reorganize the variables by using a Lorentz transformation on H+ and repeat the two-point argument. A sequence of such iterations will converge to the required radial decreasing rearrangements: Fm (p) −→ F ∗ (p),

Gm (p) −→ G∗ (p).

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An alternative method for n ≥ 2 is to write for p¯ = |¯ p|ξ, q¯ = |¯ q |η with ξ, η ∈ S n−1   pq − 1 = p0 q0 − 1 − |¯ p| |¯ q | + |¯ p| |¯ q |(1 − ξ · η) . Then for fixed values of |¯ p|, |¯ q |, the kernel K is a decreasing function of the distance between two points on n−1 the sphere S . Now apply the symmetrization lemma for the sphere and gain improvement; then use a Lorentz transformation on H+ to change variables and again apply the rearrangement lemma for the sphere to gain more improvement. As before, a sequence of iterated functions is obtained that will converge to radial decreasing functions on H+ .  5. Fractional integrals on symmetric spaces The examples treated above for Kunze–Stein phenomena on SL(2, R), SL(2, C) and the Lorenz groups SO(n, 1)/SO(n) suggest an elementary formulation for fractional integrals on symmetric spaces that captures the flavor of Riesz potentials and Stein–Weiss integrals on Euclidean space. Let M be a simply-connected symmetric space with a homogeneous group structure determined by a semi-simple Lie group with finite center and its maximal compact subgroup K—M ≃ G/K. Suppose that the metric structure on M is expressed by a suitable distance function d(w1 , w2 ) which measures the distance between two points w1 and w2 though not necessarily by direct use of a metric. For 1 < p < 2, let  1/p KM (w) = Vol(Bw ) where Bw is a ball centered at the origin  0 with radius d(w,  0). Then the following sharp inequality is obtained using Young’s inequality (4): Theorem 9. For f, g ∈ L2 (M ) and 1 < p < 2   g(w)(f ∗ Km )(w) dν = M

  f (w1 )Km d2 (w1 , w2 ) g(w2 ) dν dν

M

≤ cp ∥f ∥L2 (M ) ∥g∥L2 (M )  cp =

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∆−1/2 KM (w) dν.

M

This constant is sharp and not attained. Proof. This estimate follows directly from Young’s inequality (4). No assumption is made here to characterize those symmetric spaces for which the constant cp is finite.  Remarks on Theorem 9. Intuition for the scope and potential application of this inequality comes entirely from the example where M corresponds to n dimensional simply-connected hyperbolic space Hn where       cp = ∆−1/2 Kn d2 (w,  0) dν = 2n Kn x2 + sinh2 (t) dx dt. Hn

Rn−1 ×R

A conceptual framework for how such an estimate could be developed is contained in the following “mock” chain (the value c is a generic constant):   ∆−1/2 KM [d2M ] dν ≃ c ∆−1/2 ∆−1/2 KM [d2A + d2N ] dν M AN   2 2 −1 ≃c KM [dA + dN ]∆ dν ≃ c KM [d2A + d2N ] dνR AN AN  ≃c KM [d2A + d2N ] da dm < ∞. A×N

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The argument for finiteness of the integral would depend on three factors: (a) KM is locally integrable for small distance and p > 1; (b) global integrability on the nilpotent part depends on p < 2; and (c) distance grows exponentially for A since the non-compact manifold M has negative curvature and this fact assures global integrability on A. More generally, if a Riesz–Sobolev rearrangement inequality can be obtained for the symmetric space M , then a stronger result can be shown: ∥f ∗ g∥L2 (M ) ≤ ∥f ∥Lp,∞ (M ) ∥g∥L2 (M ) ,

1 < p < 2.

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In this case, control by the kernel is equivalent to the norm ∥f ∥Lp,∞ (M ) . A second issue that lies at the forefront of analysis for convolution estimates is how to model “Riesz potentials” on a non-compact manifold with negative curvature—perhaps either in terms of inverse fractional powers of the volume form or the distance, or by integral operators related to fractional measures of smoothness, or to fundamental solutions of intrinsic second-order differential operators. On flat Euclidean space these options are essentially identical. Observe that for radial decreasing rearrangement  −1/p f ∗ ≤ c vol(ball) ⇐⇒ f ∈ Lp,∞ (M ).

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As evident from arguments developed here, the choice of “Riesz potential” will depend upon context. For background reference, papers [4–10,12,11,14,27,31] treat aspects of equimeasurable rearrangement, symmetrization, Riesz potentials and Stein–Weiss integrals while [1,15–17,21,24–26,29,33,35,36] provide examples for concrete calculations on Lie groups and Cartan–Hadamard manifolds. Acknowledgments The question treated here was described to me by Eli Stein almost forty years ago. I would like to thank Guozhen Lu and Chris Sogge for their warm support, and Emmanuel Hebey for arranging my participation in the Workshop Analyse stochastique et th´eorie du potentiel at Saint-Priest de Gimel in September 2002 where the conceptual framework for the SL(2, R) argument was presented. I would like to thank Michel Ledoux and Stephen Semmes for encouraging the development of some ideas expressed in the Appendix, and Tony Knapp for emphasizing the importance of the indefinite orthogonal group for understanding the structure of analysis on semisimple Lie groups. I appreciate the assistance of Enzo Mitidieri. Further details concerning the extension of this program to a larger class of non-unimodular Lie groups corresponding to manifolds with negative curvature will appear in a forthcoming paper. Appendix A.1. Proof of Young’s inequality for non-unimodular groups The original argument by W.H. Young transfers naturally to this case (see [37]). For f, g ≥ 0 consider the form   (f ∗ g)(x) = f (y)g(y −1 x) dy = f (xy)g(y −1 ) dy G G      ′ −1 −1 = f (xy )g(y)∆(y) dy = f (xy −1 )∆(y)−1/p g(y)∆(y)−1/p dy G G   p/r  q/r   p/q′   ′ ′ q/p −1 −1/p −1/p′ = f (xy )∆(y) g(y)∆(y) f (xy −1 )∆(y)−1/p g(y)∆(y)−1/p dy. G

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Using the fact that 1/r = 1/p + 1/q − 1 is equivalent to 1/r + 1/p′ + 1/q ′ and apply H¨older’s inequality to obtain the upper bound for the integral   1/r  1/q′    q/p′ ′ q ′ f (xy −1 )p ∆(y)−1 g(y)∆(y)−1/p dy f (xy −1 )∆(y)−1 dy ∥g∆−1/p ∥Lq (G) G  G    p −1 p −1 p p f (xy ) ∆(y ) dy = f (xy) dy = f (y) dy = ∥f ∥Lp (G) . G

G

G

Then 1/r   p/q′  q/p′    ′ ′ q (f ∗ g)(x) ≤ ∥f ∥Lp(G) ∥g∆−1/p ∥Lq (G) f (xy −1 )p ∆(y)−1 g(y)∆(y)−1/p dy G

and for 1/r = 1/p + 1/q − 1 with p, q, r ≥ 1 ′

∥f ∗ g∥Lr (G) ≤ ∥f ∥Lp (G) ∥g∆−1/p ∥Lq (G) . A.2. Brunn–Minkowski inequality for Lie groups Let G be a locally compact non-compact Lie group with a metric structure, and m denotes left-invariant Haar measure on G. Suppose there exists a notion of equimeasurable radial rearrangement of measurable sets that is compatible with the group structure—namely, the Riesz–Sobolev rearrangement inequality holds for positive functions:   (f ∗ g)(x)h(x) dx ≤ (f ∗ ∗ g ∗ )(x)h∗ (x) dx (32) G

G

 ≥

(f ∗ ∗ g)(x)h∗ (x) dx

(33)

G

where the first integral is finite. Here f ∗ denotes the equimeasurable radial decreasing rearrangement of f , and h∗ denotes 1/k ∗ where k = 1/h; and f, g, h ≥ 0. Inequality (33) implies that for f, g ≥ 0 ∥f ∗ g∥r ≥ ∥f ∗ ∗ g ∗ ∥r ,

0 < r < 1.

(34)

Then  G

|f ∗ g|r dx ≥



|f ∗ ∗ g ∗ |r dx,

0
G

and letting r → 0 results in     m support(f ∗ g) ≥ m support(f ∗ ∗ g ∗ ) .

(35)

Let A and B be bounded sets of finite Haar measure. Consider  (χA ∗ χB )(x) = χA (y)χB (y −1 x) dy G

and observe that support(χA ∗ χB ) ⊂ AB

(36)

where AB denotes the set {xy ∈ G : x ∈ A and y ∈ B}. Then the Brunn–Minkowski inequality on G is obtained from (35): m∗ (AB) ≥ m [support(χA ∗ χB )] ≥ m [support(χA∗ ∗ χB ∗ )] = m(A∗ B ∗ )

(37)

where m∗ denotes inner Haar measure. Observe that as particular cases, (1) on Rn the Brunn–Minkowski inequality is equivalent to the Riesz–Sobolev inequality and (2) the Riesz–Sobolev inequality on hyperbolic space Hn implies the Brunn–Minkowski inequality on Hn .

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A.3. Isoperimetric inequality To continue with the assumptions made in Section 2, an additional hypothesis concerns the growth rate for small balls which will be used to define surface measure. Let {Bε } be a decreasing family of balls of small volume centered at the origin with m(Bε ) = vol(Bε ) ≃ cΣ m

as ε → 0

(38)

where the group G is viewed as a homogeneous symmetric space with a metric structure. The value of m will be considered to be the local dimension of G so that locally this corresponds to an m-dimensional Riemannian manifold which is shrinking to zero in size. For a measurable set A with finite measure and rectifiable boundary, the surface area of A will be defined, possibly up to a normalization factor depending on the group G, by m(Bε A) = m(A) + εσ(∂A) + o(Σ ).

(39)

Then the Brunn–Minkowski inequality will give the isoperimetric inequality: σ(∂A) ≥ σ(∂A∗ )

(40)

where A∗ is a ball centered at the origin with the same volume as A. If the boundary of A is not measurable with respect to a suitable definition of Hausdorff measure, then a notion of inner surface measure could be used σ∗ (∂A) ≥ σ(∂A∗ ). On n-dimensional hyperbolic space Hn , the “two-point symmetrization lemma” from [10] can be used to show that gradient norms decrease under symmetrization       ϕ |Df ∗ | dν ≤ ϕ |Df | dν (41) Hn

Hn

where D = y∇ and ϕ convex with ϕ(0) = 0. Then for ϕ linear, the isoperimetric inequality (40) follows immediately. An argument to show how the decrease of the gradient norm (41) can be obtained directly from the coarea formula and the isoperimetric inequality is given in the Appendix on “symmetrization and gradient estimates” in [7]. As with the case of Euclidean space Rn , the isoperimetric inequality on Hn can be expressed as a Sobolev estimate. Erhard Schmidt’s isoperimetric inequality on the hyperbolic plane (see Osserman review [28]) is the classic example that provided the first indication that inequalities that measure smoothness would have additional structure, particularly in the setting of negative curvature:  4πAH (D) + AH (D)2 ≤ LH (∂D). (42) Here D is a domain in H2 with rectifiable boundary ∂D and AH , LH are respectively hyperbolic area and length. This estimate corresponds to the hyperbolic Sobolev inequality    2  2 1/2 2 2 1 2 4π ∥f ∥L (H ) + ∥f ∥L (H ) ≤ |Df | dν. (43) H2

It suffices to obtain this estimate from the isoperimetric inequality (42) by using elementary L2 functional  geometry for simple functions f of the form cK χAK with cK > 0 and the sets {Ak } mutually disjoint:      2  2 1/2 2  2 1/2 4π ∥f ∥L2 (H) + ∥f ∥L1 (H2 ) = 4π cK χAK dν + cK ν(AK ) ≤



cK

√

2 1/2 1/2 2  4π ν(AK ) + cK ν(AK )

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 1/2 cK 4πν(AK ) + ν(AK )2   ≤ cK LH (∂AK ) = |Df | dν. ≤



H2

The proof is completed by taking appropriate limits. It would be more efficient to simply observe that the left-hand side is a norm; set |||f ||| = [∥f ∥22 + ∥f ∥21 ]1/2 , then  1/2 |||f + g||| = ∥f + g∥22 + ∥f + g∥21  2  2 1/2 ≤ ∥f ∥2 + ∥g∥2 + ∥f ∥1 + ∥g∥1 1/2  1/2  ≤ ∥f ∥22 + ∥f ∥21 + ∥g∥22 + ∥g∥21 = |||f ||| + |||g|||. This result extends to the general case for n-dimensional simply connected hyperbolic space Hn : let D be a domain with finite volume volH (D) and rectifiable boundary ∂D; σH denotes surface measure and volE (B) is the Euclidean volume of the unit ball in Rn ; then  1/n σH (∂D) volE (B) (n − 1)n + nn . (44) ≤ volH (D) volH (D) This inequality not only extends Erhard Schmidt’s two-dimensional isoperimetric inequality but captures in a quantitative fashion the isoperimetric paradigm that has been used to signify “hyperbolic phenomena” – namely the characteristic exponential growth of balls as a function of the radius — which follows directly from the fundamental isoperimetric relation (n − 1)volH (D) ≤ σH (∂D).

(45)

The isoperimetric inequality (43) can be obtained from a Sobolev inequality that combines the natural Riemannian Sobolev inequality corresponding to Euclidean space and the L1 spectral estimate for Hn . Theorem 10. For f ∈ S(Hn ) and q = n/(n − 1), n ≥ 2   n 1/n n   ≤ (n − 1)n ∥f ∥L1 (Hn ) + nn vol A E(B) ∥f ∥Lq (Hn )

|Df | dν.

(46)

Hn

This inequality is sharp, but no extremal function exists for n > 2. Proof. The inequality (46) is reduced to radial variables by symmetrization (that is, using equimeasurable geodesic radial decreasing rearrangement). It suffices to show this estimate for simple functions. Let f be   ∗ a simple non-negative function; then f can be written as a finite sum fK so that f ∗ = fk where fk = αk χEk , αk > 0 and Ek∗ is a geodesic ball centered at the origin with ν(Ek ) = ν(Ek∗ ). Now observe that the left-hand side is a norm: for c1 , c2 > 0     n  n 1/n n  n 1/n c1 ∥f + g∥1 + c2 ∥f + g∥q ≤ c1 ∥f ∥1 + ∥g∥1 + c2 ∥f ∥q + ∥g∥q  1/n  1/n ≤ c1 ∥f ∥n1 + c1 ∥f ∥nq + c1 ∥g∥n1 + c2 ∥g∥nq . Then 

 n  n 1/n (n − 1)n ∥f ∥L1 (Hn ) + nn volE (B) ∥f ∥Lq (Hn )   n  n 1/n ≤ (n − 1)n ∥fk ∥L1 (Hn ) + nn volE (B) ∥fk ∥Lq (Hn ) k

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=



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 ∗ n n 1/n  (n − 1)n ∥fK ∥L1 (Hn ) + nn volE (B) ∥fk∗ ∥Lq (Hn )

k

≤

 K

Hn

|Dfk∗ | dν =



|Df ∗ | dν ≤

Hn

 |Df | dν Hn

where the first inequality follows from Minkowski’s inequality and the norm property, the second inequality depends on the estimate holding for the characteristic function of a geodesic ball and the third inequality uses the fact that radial symmetrization decreases gradient norms. Restricted to radial variables, the measure dν is expressed in terms of the variable u = [d(w,  0)]2 as dν =

 n −1 (4π)n/2  u(1 + u) 2 du Γ (n/2)

and |Df | =



 df    u + u2  . du

For the basic estimate in Theorem 10 to hold for the characteristic function of a ball, it is required that the following elementary inequality hold for r > 0 and n ≥ 2:  n   r n−1  r   n −1   n −1  n(n−1)/2 n u(1 + u) 2 du (n − 1) u(1 + u) 2 du + ≤ r(1 + r) . (47) 2 0 0 For n = 2 this estimate becomes an identity so it suffices to show the result for n ≥ 3. The inequality is determined by a convexity argument in the following lemma, and that demonstration will complete the proof of the theorem.  Lemma. For α ≥ 3/2 and r > 0  2α   r 2α−1  r  α−1  α−1  α(2α−1) (2α − 1) u(1 + u) du + α u(1 + u) du ≤ r(1 + r) . 0

(48)

0

Proof. By explicit computation, one finds that the inequality holds for α = 2 and is an identity for α = 1. Make the change of variables w = u(1 + u) and ρ = r(1 + r); then du = (1 + 4w)−1/2 dw and the required estimate becomes  2α   ρ 2α−1  ρ α−1 −1/2 α−1 −1/2 (2α − 1) w (1 + 4w) dw + α w (1 + 4w) dw ≤ ρα(2α−1) ; 0

0

by rescaling this becomes 2α   ρ 2α−1   1  ρ α−1 w (1 + w)−1/2 dw + α wα−1 (1 + w)−1/2 dw ≤ ρα(2α−1) . α− 2 0 0 Asymptotic behavior shows that the inequality holds near ρ = 0 and ρ = ∞. To gain insight, consider the case α = 3/2. Let  ρ  3   ρ  2 w 3 w 3 Λ0 (ρ) = ρ − dw − dw ; 1+w 2 0 1+w 0 then  ρ  2    ρ  w ρ 3 w ρ Λ′0 (ρ) = 3ρ2 − 3 dw −3 dw 1 + w 1 + ρ 2 1 + w 1 + ρ 0 0  ρ =3 Λ1 (ρ) 1+ρ

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where Λ1 (ρ) = ρ

3/2





ρ



1+ρ− 0

w dw 1+w

2

3 − 2



ρ



0

w dw; 1+w

then    ρ w ρ 3 ρ 3  1 dw − ρ(1 + ρ) + ρ3/2 (1 + ρ)−1/2 − 2 2 2 1+w 1+ρ 2 1+ρ 0     ρ 1 ρ w = 3(1 + ρ) + ρ − 3 − 4 dw 2 1+ρ 1 + w 0     ρ ρ w =2 ρ− dw ≥ 0. 1+ρ 1+w 0

Λ′1 (ρ) =

Observe for ρ ≥ 0 Λ′1 (ρ) ≥ 0 =⇒ Λ1 (ρ) ≥ 0 =⇒ Λ′0 (ρ) ≥ 0 =⇒ Λ0 (ρ) ≥ 0 and the required estimate holds for α = 3/2. Then general case follows from similar steps. Let 2α  2α−1  1 − αFα (ρ) Λ0 (ρ) = ρα(2α−1) − α − Fα (ρ) 2 with  ρ Fα (ρ) = wα−1 (1 + w)−1/2 dw; 0

then 2α−1  1 Fα (ρ) ρα−1 (1 + ρ)−1/2 Λ′0 (ρ) = α(2α − 1)ρα(2α−1)−1 − α(2α − 1) α − 2  2α−2 − α(2α − 1) αFα (ρ) ρα−1 (1 + ρ)−1/2 = α(2α − 1)ρα−1 (1 + ρ)−1/2 Λ1 (ρ) with Λ1 (ρ) = ρ2α(α−1)



(1 + ρ) −



α−

2α−2 2α−1  1 − αFα (ρ) . Fα (ρ) 2

Then  1 Λ′1 (ρ) = 2α(α − 1)ρ2α(α−1)−1 1 + ρ + ρ2α(α−1) (1 + ρ)−1/2 2  2α−2  2α−3 1  1 − (2α − 1) α − α− Fα (ρ) ρα (1 + ρ)−1/2 − 2α(α − 1) αFα (ρ) ρα−1 (1 + ρ)−1/2 2 2 = ρα−1 (1 + ρ)−1/2 Λ2 (ρ) where    2(α−1)   1  (α−1)(2α−1) 1  ρ α−3/2 w Λ2 (ρ) = (2α − 1) α − ρ − α− w dw 2 2 0 1+w      

2α−3

ρ

wα−1 (1 + w)−1/2 dw

+ 2α(α − 1) ρα(2α−3) − α

.

0

Each term in braces is non-negative for α ≥ 3/2 so Λ2 (ρ) ≥ 0 and for ρ ≥ 0 Λ2 (ρ) ≥ 0 =⇒ Λ′1 (ρ) ≥ 0 =⇒ Λ1 (ρ) ≥ 0 =⇒ Λ′0 (ρ) ≥ 0 =⇒ Λ0 (ρ) ≥ 0 which demonstrates that the estimate holds for all α ≥ 3/2 and the proof of the Lemma is complete.



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A.4. Hardy–Littlewood–Sobolev inequality on hyperbolic space Conformal equivalence between Euclidean space and the two-sheeted hyperboloid requires that there will be a natural Hardy–Littlewood–Sobolev inequality on hyperbolic space expressed in terms of a potential defined by inverse powers of the distance (see [3]). By using the upper half-space model for Hn , a simple change of variables provides this result. For 0 < λ < n and n ≥ 2, let   −λ |x|2 + |y − 1|2 . ψλ (w) = d(w,  0) , d(w,  0) = √ 2 y Theorem 11. Let 1 < p < 2, n ≥ 2 and λ = 2n/p′ ; then the following two inequalities are equivalent: ∥F ∗ ψλ ∥Lp′ (Hn ) ≤ 2λ Ap ∥F ∥Lp (Hn )  −λ   |x| ∗ f  p′ n ≤ Ap ∥f ∥Lp (Rn ) L (R )      2 −1 Γ n p1 − 12 ′ Γ (n) p Ap = π n/p Γ (n/p) Γ (n/2)

(49) (50)

with the property that extremals exist for the second inequality but not for the first. Proof. Make the change of variables F = y n/p f and it follows that (50) implies (49). For the opposite implication, let f be any smooth function with compact support on Rn and move the support to lie in a half-space using translation. Then reverse the change of variables used above. The competing symmetry of the two manifolds cannot be used directly to determine the extremals for (50). But axial symmetry in Rn provides a different equivalence with hyperbolic space that can be used to determine extremals for (50) (see [5]).  References [1] A.O. Barut, Electrodynamics and Classical Theory of Fields and Particles, Macmillan, 1964. [2] W. Beckner, Inequalities in Fourier analysis, Ann. of Math. 102 (1975) 159–182. [3] W. Beckner, Geometric inequalities in Fourier analysis, in: Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton University Press, 1995, pp. 36–68. [4] W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995) 1897–1905. [5] W. Beckner, Sharp inequalities and geometric manifolds, J. Fourier Anal. Appl. 3 (1997) 825–836. [6] W. Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. 129 (2001) 1233–1246. [7] W. Beckner, Estimates on Moser embedding, Potential Anal. 20 (2004) 345–359. [8] W. Beckner, Pitt’s inequality with sharp convolution estimates, Proc. Amer. Math. Soc. 136 (2008) 1871–1885. [9] W. Beckner, Multilinear embedding estimates for the fractional Laplacian, Math. Res. Lett. 19 (2012) 175–189. [10] W. Beckner, Embedding estimates and fractional smoothness, Int. Math. Res. Not. (2014) 390–417. [11] W. Beckner, Functionals for multilinear fractional embedding, Acta Math. Sin. (Engl. Ser.) 31 (2015) 1–28. [12] W. Beckner, Multilinear embedding and Hardy’s inequality, in: Some Topics in Harmonic Analysis, in: Advanced Lectures in Mathematics, vol. 34, 2015, pp. 1–26. [13] H.J. Brascamp, E.H. Lieb, J.M. Luttinger, A general rearrangement inequality for multiple integrals, J. Funct. Anal. 17 (1974) 227–237. [14] H. Federer, Geometric Measure Theory, Springer, 1996. [15] G.B. Folland, E.M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, 1982. [16] S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, third ed., Springer, 2004. [17] R. Gilmore, Lie Groups, Lie Algebras, and Some of their Applications, John Wiley, 1974. [18] M. Gromov, Metric Structures on Riemannian Spaces, Birkh¨ auser, 1999. [19] G.H. Hardy, J.E. Littlewood, G. P´ olya, Inequalities, Cambridge University Press, 1967. [20] Harish-Chandra, Spherical functions on a semisimple Lie group I, Amer. J. Math. 80 (1958) 241–310. [21] E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Amer. Math. Soc. (2000). [22] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962. [23] A.D. Ionescu, An endpoint estimate for the Kunze–Stein phenomenon and related maximal operators, Ann. of Math. 152 (2000) 259–275.

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