Hardy inequalities on metric measure spaces, II: The case p>q
aa r X i v : . [ m a t h . F A ] F e b HARDY INEQUALITIES ON METRIC MEASURE SPACES, II:THE CASE p > q
MICHAEL RUZHANSKY AND DAULTI VERMA
Abstract.
In this note we continue giving the characterisation of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polardecompositions. Since there may be no differentiable structure on such spaces, theinequalities are given in the integral form in the spirit of Hardy’s original inequality.This is a continuation of our paper [RV18] where we treated the case p ≤ q . Herethe remaining range p > q is considered, namely, 0 < q < p , 1 < p < ∞ . Wegive examples obtaining new weighted Hardy inequalities on R n , on homogeneousgroups, on hyperbolic spaces, and on Cartan-Hadamard manifolds. We note thatdoubling conditions are not required for our analysis. Contents
1. Introduction 12. Main results 33. Applications and examples 123.1. Homogeneous groups 123.2. Hyperbolic spaces 143.3. Cartan-Hadamard manifolds 15References 161.
Introduction
In [Har20], Hardy showed his famous inequality Z ∞ b (cid:18) R xb f ( t ) dtx (cid:19) p dx ≤ (cid:18) pp − (cid:19) p Z ∞ b f ( x ) p dx, (1.1)where p > b >
0, and f ≥ Date : February 12, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Hardy inequalities, metric measure spaces, homogeneous group, hyper-bolic space, Riemannian manifolds with negative curvature.The first author was supported in parts by the FWO Odysseus 1 grant G.0H94.18N: Analysisand Partial Differential Equations, by the Methusalem programme of the Ghent University SpecialResearch Fund (BOF) (Grant number 01M01021), and by the EPSRC grant EP/R003025/2. [EE04], Mazya [Maz85, Maz11], Ghoussoub and Moradifam [GM13], Balinsky, Evansand Lewis [BEL15], and references therein. We can also refer to the recent openaccess book [RS19] devoted to Hardy, Rellich and other inequality in the setting ofnilpotent Lie groups.In this paper we show that the inequality (1.1) actually holds in a much moregeneral setting, also with rather general pairs of weights. However, the weights haveto satisfy certain compatibility conditions for such inequalities to hold true, and theseconditions are necessary and sufficient .Several characterisations of weights for the two-weight Hardy inequalities to holdon general metric measure spaces for the case 1 < p ≤ q < ∞ have been obtained in[RV18]. In this paper, complementary to [RV18], we consider the weight characteri-sations for the case 0 < q < p, < p < ∞ . The setting of these papers is rather general: we consider polarisable metric measurespaces , that is, metric spaces ( X , d ) with a Borel measure dx allowing for the following polar decomposition at a ∈ X : we assume that there is a locally integrable function λ ∈ L loc such that for all f ∈ L ( X ) we have Z X f ( x ) dx = Z ∞ Z Σ r f ( r, ω ) λ ( r, ω ) dω r dr, (1.2)for the set Σ r = { x ∈ G : d ( x, a ) = r } ⊂ X with a measure on it denoted by dω = dω r ,and ( r, ω ) → a as r → λ to depend on thewhole variable x = ( r, ω ). The reason to assume (1.2) is that since X does not haveto have a differentiable structure, the function λ ( r, ω ) can not be in general obtainedas the Jacobian of the polar change of coordinates. However, if such a differentiablestructure exists on X , the condition (1.2) can be obtained as the standard polardecomposition formula. In particular, let us give several examples of X for which thecondition (1.2) is satisfied with different expressions for λ ( r, ω ):(I) Euclidean space R n : λ ( r, ω ) = r n − . (II) Homogeneous groups: λ ( r, ω ) = r Q − , where Q is the homogeneous dimensionof the group. Such groups have been consistently developed by Folland andStein [FS82], see also an up-to-date exposition in [FR16].(III) Hyperbolic spaces H n : λ ( r, ω ) = (sinh r ) n − .(IV) Cartan-Hadamard manifolds: Let K M be the sectional curvature on ( M, g ) . A Riemannian manifold (
M, g ) is called a Cartan-Hadamard manifold if it iscomplete, simply connected and has non-positive sectional curvature, i.e., thesectional curvature K M ≤ a ∈ M and denote by ρ ( x ) = d ( x, a ) the geodesic distance from x to a on M . The exponential map exp a : T a M → M is a diffeomorphism,see e.g. Helgason [Hel01]. Let J ( ρ, ω ) be the density function on M , see e.g.[GHL04]. Then we have the following polar decomposition: Z M f ( x ) dx = Z ∞ Z S n − f (exp a ( ρω )) J ( ρ, ω ) ρ n − dρdω, so that we have (1.2) with λ ( ρ, ω ) = J ( ρ, ω ) ρ n − . ARDY INEQUALITIES ON METRIC MEASURE SPACES, II: THE CASE p > q (V) Complete manifolds: Let M be a complete manifold. Let p ∈ M and let C ( p )denote the cut locus of p . Let D p := M \ C ( p ) and S ( p ; r ) := { x ∈ M p : | x | = r } , where | · | is the Riemannian length, where M p stands for the tangent spaceat p . Then for any p ∈ M and any integrable function f on M we have (e.g.see [Cha06, Formula III.3.5, P.123]) the polar decomposition Z M f dV = Z + ∞ dr Z r − S ( p ; r ) ∩ D p f (exp rξ ) √ g ( r ; ξ ) dµ p ( ξ ) (1.3)for some function √ g on D p , where r − S ( p, r ) ∩ D p is the subset of S p obtainedby dividing each of the elements of S ( p, r ) ∩ D p by r , and S p := S ( p ; 1). Here dµ p ( ξ ) is the Riemannian measure on S p induced by the Euclidean Lebesguemeasure on M p . We refer to [Cha06], [Li12, Chapter 4] and [CLN06, Chapter1, Paragraph 12] for more details on this decomposition.Throughout this paper, by A ≈ B we will always mean that the expressions A and B are equivalent.The authors would like to thank Aidyn Kassymov and Bolys Sabitbek for checkingsome calculations in this paper. 2. Main results
We denote by B ( a, r ) the ball in X with centre a and radius r , i.e B ( a, r ) := { x ∈ X : d ( x, a ) < r } , where d is the metric on X . Once and for all we will fix some point a ∈ X , and wewill write | x | a := d ( a, x ) . One of our main results is to characterize the weights u and v for the correspondingHardy inequality to hold on X . For X = R , such result has been proved by Sinnamonand Stepanov [SS96]. Now, we formulate one of our main results: Theorem 2.1.
Suppose < q < p , < p < ∞ and /r = 1 /q − /p . Let X be ametric measure space with a polar decomposition at a. Let u, v > be measurablefunctions positive a.e in X such that u ∈ L ( X \{ a } ) and v − p ′ ∈ L loc ( X ) .Then the inequality (cid:18) Z X (cid:18) Z B ( a, | x | a ) | f ( y ) | dy (cid:19) q u ( x ) dx (cid:19) q ≤ C (cid:26) Z X | f ( x ) | p v ( x ) dx (cid:27) p (2.1) holds for all measurable functions f : X → C if and only if A := (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/p (cid:18) Z B ( a, | x | a ) v − p ′ ( y ) dy (cid:19) r/p ′ u ( x ) dx (cid:19) /r < ∞ . Moreover, the smallest constant C for which (2.1) holds satisfies ( p ′ ) /p ′ q /p (1 − q/p ) A ≤ C ≤ ( r/q ) /r p /p p ′ /p ′ A . MICHAEL RUZHANSKY AND DAULTI VERMA
Before proving the above theorem, we will need to prove several auxiliary facts.Throughout this paper, we will use the following notations: U ( x ) = Z X \ B ( a, | x | a ) u ( y ) dy, (2.2) V ( x ) = Z B ( a, | x | a ) v − p ′ ( y ) dy, (2.3)˜ U ( t ) = Z ∞ t Z Σ ρ λ ( ρ, σ ) u ( ρ, σ ) dσ ρ dρ, (2.4)˜ V ( t ) = Z t Z Σ ρ λ ( ρ, σ ) v − p ′ ( ρ, σ ) dσ ρ dρ, (2.5) U ( ρ ) = Z Σ ρ λ ( ρ, σ ) u ( ρ, σ ) dσ ρ , (2.6) V ( ρ ) = Z Σ ρ λ ( ρ, σ ) v − p ′ ( ρ, σ ) dσ ρ . (2.7) Lemma 2.2.
Let us denote A := (cid:26) Z X U r/q ( x ) V r/q ′ ( x ) v − p ′ ( x ) dx (cid:27) /r . Then A r = ( q/p ′ ) A r . (2.8) Proof.
Using integration by parts, we have A r = Z X U r/p ( x ) V r/p ′ ( x ) u ( x ) dx = Z ∞ ˜ U r/p ( t ) ˜ V r/p ′ ( t ) U ( t ) dt = Z ∞ (cid:18) Z ∞ t U ( ρ ) dρ (cid:19) r/p ˜ V r/p ′ ( t ) U ( t ) dt = − ( q/r ) U r/q ( ∞ ) V r/p ′ ( ∞ ) + ( q/r ) U r/q (0) V r/p ′ (0) + ( q/r )( r/p ′ ) × Z X U r/q ( x ) V r/q ′ ( x ) v − p ′ ( x ) dx = ( q/p ′ ) Z X U r/q ( x ) V r/q ′ ( x ) v − p ′ ( x ) dx = ( q/p ′ ) A r , completing the proof. (cid:3) Lemma 2.3.
Suppose that α, β and γ are non-negative functions and γ is a radialnon-decreasing function of | · | a . If R X \ B ( a, | x | a ) α ( y ) dy ≤ R X \ B ( a, | x | a ) β ( y ) dy for all x ,then R X γα ≤ R X γβ . ARDY INEQUALITIES ON METRIC MEASURE SPACES, II: THE CASE p > q Proof.
Let us denote α ( r ) = Z Σ r λ ( r, σ ) α ( r, σ ) dσ r ,β ( r ) = Z Σ r λ ( r, σ ) β ( r, σ ) dσ r , ˜ γ ( r ) = γ ( x ) , for | x | a = r. Given that, R X \ B ( a, | x | a ) α ( y ) dy ≤ R X \ B ( a, | x | a ) β ( y ) dy , changing to polarcoordinates, we get Z ∞| x | a α ( r ) dr ≤ Z ∞| x | a β ( r ) dr. Using [SS96, Lemma 2.1] which says if α, β , γ are non-negative functions and γ is non-decreasing, and if R ∞ x α ( y ) dy ≤ R ∞ x β ( y ) dy for all x , then R ∞ γα ≤ R ∞ γβ .Therefore, Z X γ ( x ) α ( x ) dx = Z ∞ ˜ γ ( r ) α ( r ) dr ≤ Z ∞ ˜ γ ( r ) β ( r ) dr = Z X γ ( x ) β ( x ) dx, completing the proof. (cid:3) Proposition 2.4.
Suppose that u, b and F are non-negative functions with F non-decreasing such that R X \ B ( a, | x | a ) b ( y ) dy < ∞ for all x = a and R X b ( x ) dx = ∞ . If < q < p < ∞ , F is radial in | x | a , and /r = 1 /q − /p , then (cid:18) Z X F q ( x ) u ( x ) dx (cid:19) /q ≤ ( r/p ) /r (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/q (cid:18) Z X \ B ( a, | x | a ) b ( y ) dy (cid:19) − r/q b ( x ) dx (cid:19) /r × (cid:18) Z X F p ( x ) b ( x ) dx (cid:19) /p . Proof.
Let us denote B ( x ) = Z X \ B ( a, | x | a ) b ( y ) dy, ˜ B ( t ) = Z ∞ t Z Σ ρ λ ( ρ, ω ) b ( ρ, ω ) dω ρ dρ,B ( ρ ) = Z Σ ρ λ ( ρ, ω ) b ( ρ, ω ) dω ρ , ˜ F ( t ) = F ( x ) , for | x | a = t. Applying H¨older’s inequality with indices q/r and q/p , we get
MICHAEL RUZHANSKY AND DAULTI VERMA (cid:18) Z X F q ( x ) u ( x ) dx (cid:19) /q = (cid:18) Z X (cid:18) Z B ( a, | x | a ) U r/p ( y ) B − r/q ( y ) b ( y ) dy (cid:19) q/r F q ( x ) × (cid:18) Z B ( a, | x | a ) U r/p ( y ) B − r/q ( y ) b ( y ) dy (cid:19) − q/r u ( x ) dx (cid:19) /q = (cid:18) Z ∞ (cid:18) Z t ˜ U r/p ( ρ ) ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) q/r ˜ F q ( t ) (cid:18) Z t ˜ U r/p ( ρ ) ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) − q/r U ( t ) dt (cid:19) /q = (cid:18) Z ∞ (cid:18) Z t ˜ U r/p ( ρ ) ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) q/r ˜ F q ( t ) (cid:18) Z t ˜ U r/p ( ρ ) ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) − q/r × U q/r + q/p ( t ) dt (cid:19) /q ≤ (cid:18) Z ∞ (cid:18) Z t ˜ U r/p ( ρ ) ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) U ( t ) dt (cid:19) /r (cid:18) Z ∞ ˜ F p ( t ) (cid:18) Z t ˜ U r/p ( ρ ) ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) − p/r × U ( t ) dt (cid:19) /p . On interchanging the order of integration and using r/p + 1 = r/q , the first factorbecomes (cid:18) Z ∞ ˜ U r/p ( ρ ) ˜ B − r/q ( ρ ) B ( ρ ) (cid:18) Z ∞ ρ U ( t ) dt (cid:19) dρ (cid:19) /r = (cid:18) Z ∞ ˜ U r/q ( ρ ) ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) /r = (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/q (cid:18) Z X \ B ( a, | x | a ) b ( y ) dy (cid:19) − r/q b ( x ) dx (cid:19) /r . To complete the proof we apply Lemma 2.3 to the second factor. We take α ( x ) = (cid:18) R B ( a, | x | a ) U r/p ( y ) B − r/q ( y ) b ( y ) dy (cid:19) − p/r u ( x ), β ( x ) = ( r/p ) p/r b ( x ), and γ ( x ) = F p ( x )in Lemma 2.3. As γ is non-decreasing by assumption, it remains to check that Z X \ B ( a, | x | a ) α ( y ) dy ≤ Z X \ B ( a, | x | a ) β ( y ) dy, ARDY INEQUALITIES ON METRIC MEASURE SPACES, II: THE CASE p > q for all x . Since (cid:18) R t ˜ U r/p ( ρ ) ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) − p/r and ˜ U are non-increasing, Z X \ B ( a, | x | a ) α ( y ) dy = Z ∞| x | a (cid:18) Z t ˜ U r/p ( ρ ) ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) − p/r U ( t ) dt ≤ (cid:18) Z | x | a ˜ U r/p ( ρ ) ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) − p/r Z ∞| x | a U ( t ) dt ≤ (cid:18) Z | x | a ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) − p/r ˜ U − ( | x | a ) Z ∞| x | a U ( t ) dt = (cid:18) Z | x | a ˜ B − r/q ( ρ ) B ( ρ ) dρ (cid:19) − p/r = (cid:18) Z | x | a ˜ B − r/q ( ρ ) d ( − ˜ B ( ρ )) (cid:19) − p/r = (cid:18) ( p/r ) B − r/p ( x ) (cid:19) − p/r = Z X \ B ( a, | x | a ) β ( y ) dy. Finally, by using Lemma 2.3 we get (cid:18) Z X F q ( x ) u ( x ) dx (cid:19) /q ≤ (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/q (cid:18) Z X \ B ( a, | x | a ) b ( y ) dy (cid:19) − r/q b ( x ) dx (cid:19) /r (cid:18) Z X ( r/p ) p/r F p ( x ) b ( x ) dx (cid:19) /p = ( r/p ) /r (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/q (cid:18) Z X \ B ( a, | x | a ) b ( y ) dy (cid:19) − r/q b ( x ) dx (cid:19) /r (cid:18) Z X F p ( x ) b ( x ) dx (cid:19) /p , which completes the proof. (cid:3) Proposition 2.5.
Suppose < p < ∞ and w is a non-negative function satisfying < Z B ( a, | x | a ) w ( y ) dy < ∞ , ∀ x = a, Z X w ( x ) dx = ∞ . (2.9) Then (cid:18) Z X (cid:18) Z B ( a, | x | a ) f ( y ) dy (cid:19) p (cid:18) Z B ( a, | x | a ) w ( y ) dy (cid:19) − p w ( x ) dx (cid:19) /p ≤ p ′ (cid:18) Z X f p ( x ) w − p ( x ) dx (cid:19) /p , (2.10) for all measurable functions f ≥ . Proof.
Let us denote: f ( ρ ) = R Σ ρ λ ( ρ, σ ) f ( ρ, σ ) dσ ρ , w ( ρ ) = R Σ ρ λ ( ρ, σ ) w ( ρ, σ ) dσ ρ . Consider the left hand side of (2.10) and change it into polar coordinates, to get
MICHAEL RUZHANSKY AND DAULTI VERMA (cid:18) Z X (cid:18) Z B ( a, | x | a ) f ( y ) dy (cid:19) p (cid:18) Z B ( a, | x | a ) w ( y ) dy (cid:19) − p w ( x ) dx (cid:19) /p = (cid:18) Z ∞ (cid:18) Z t f ( ρ ) dρ (cid:19) p (cid:18) Z t w ( ρ ) dρ (cid:19) − p w ( t ) dt (cid:19) /p . Now, let us use [SS96, Proposition 2.3] which says that if 1 < p < ∞ and w is anon-negative function satisfying0 < Z x w ( y ) dy < ∞ , ∀ x > , Z ∞ w ( x ) dx = ∞ , then (cid:18) Z ∞ (cid:18) Z t f ( ρ ) dρ (cid:19) p (cid:18) Z t w ( ρ ) dρ (cid:19) − p w ( t ) dt (cid:19) /p ≤ p ′ (cid:18) Z ∞ f p ( t ) w − p ( t ) dt (cid:19) /p . By using H¨older’s inequality to the indices 1 /p and 1 /p ′ , the LHS of (2.10) can beestimated by ≤ p ′ (cid:18) Z ∞ f p ( t ) w − p ( t ) dt (cid:19) /p = p ′ (cid:18) Z ∞ (cid:18) Z Σ t λ ( t, σ ) f ( t, σ ) dσ t (cid:19) p (cid:18) Z Σ t λ ( t, σ ) w ( t, σ ) dσ t (cid:19) − p dt (cid:19) /p = p ′ (cid:18) Z ∞ (cid:18) Z Σ t λ ( t, σ ) f ( t, σ ) w (1 − p ) /p +( p − /p dσ t (cid:19) p (cid:18) Z Σ t λ ( t, σ ) w ( t, σ ) dσ t (cid:19) − p dt (cid:19) /p ≤ p ′ (cid:18) Z ∞ (cid:18) Z Σ t λ ( t, σ ) f p ( t, σ ) w − p dσ t (cid:19)(cid:18) Z Σ t λ ( t, σ ) w p ′ ( p − /p ( t, σ ) dσ t (cid:19) p − × (cid:18) Z Σ t λ ( t, σ ) w ( t, σ ) dσ t (cid:19) − p dt (cid:19) /p = p ′ (cid:18) Z ∞ Z Σ t λ ( t, σ ) f p ( t, σ ) w − p ( t, σ ) dσ t dt (cid:19) /p = p ′ (cid:18) Z X f p ( x ) w − p ( x ) dx (cid:19) /p , completing the proof. (cid:3) Now, we prove our Theorem 2.1 :
Proof.
Set w = v − p ′ . Suppose that inequality (2.1) holds for all f ≥ u and w be L functions such that 0 < u ≤ u and 0 < w ≤ w . We denote˜ u ( ρ ) = Z Σ ρ λ ( ρ, ω ) u ( ρ, ω ) dω ρ , ARDY INEQUALITIES ON METRIC MEASURE SPACES, II: THE CASE p > q ˜ w ( ρ ) = Z Σ ρ λ ( ρ, ω ) w ( ρ, ω ) dω ρ . Let us apply inequality (2.1) to the function f ( x ) = (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/pq (cid:18) Z B ( a, | x | a ) w ( y ) dy (cid:19) r/pq ′ w ( x ) . After changing to polar coordinates and using r ( pq ′ ) + 1 = r ( 1 pq ′ + 1 r ) = r (cid:18) p (1 − q ) + 1 q − p (cid:19) = rp ′ q , we have Z B ( a, | x | a ) f ( y ) dy = Z | x | a Z Σ t λ ( t, σ ) (cid:18) Z ∞ t Z Σ ρ λ ( ρ, ω ) u ( ρ, ω ) dρdω ρ (cid:19) r/pq × (cid:18) Z t Z Σ ρ λ ( ρ, ω ) w ( ρ, ω ) dρdω ρ (cid:19) r/pq ′ w ( t, σ ) dtdσ t = Z | x | a (cid:18) Z ∞ t ˜ u ( ρ ) dρ (cid:19) r/pq (cid:18) Z t ˜ w ( ρ ) dρ (cid:19) r/pq ′ ˜ w ( t ) dt ≥ (cid:18) Z ∞| x | a ˜ u ( ρ ) dρ (cid:19) r/pq Z | x | a (cid:18) Z t ˜ w ( ρ ) dρ (cid:19) r/pq ′ ˜ w ( t ) dt = ( p ′ q/r ) (cid:18) Z ∞| x | a ˜ u ( ρ ) dρ (cid:19) r/pq (cid:18) Z | x | a ˜ w ( ρ ) dρ (cid:19) r/p ′ q = ( p ′ q/r ) (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/pq (cid:18) Z B ( a, | x | a ) w ( y ) dy (cid:19) r/p ′ q . Observe that w = v − p ′ implies that v = w / (1 − p ′ ) = w − p since1 − / (1 − p ′ ) = − p ′ / (1 − p ′ ) = − / (1 /p ′ −
1) = p. We then have (cid:18) Z X ( p ′ q/r ) q (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/p (cid:18) Z B ( a, | x | a ) w ( y ) dy (cid:19) r/p ′ u ( x ) dx (cid:19) /q ≤ (cid:18) Z X (cid:18) Z B ( a, | x | a ) f ( y ) dy (cid:19) q u ( x ) dx (cid:19) /q ≤ C (cid:18) Z X f p ( x ) w − p ( x ) dx (cid:19) /p = C (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/q (cid:18) Z B ( a, | x | a ) w ( y ) dy (cid:19) r/q ′ w p ( x ) w − p ( x ) dx (cid:19) /p ≤ C (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/q (cid:18) Z B ( a, | x | a ) w ( y ) dy (cid:19) r/q ′ w p ( x ) w − p ( x ) dx (cid:19) /p = C (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/q (cid:18) Z B ( a, | x | a ) w ( y ) dy (cid:19) r/q ′ w ( x ) dx (cid:19) /p = C (cid:18) Z X ( p ′ /r )( r/q ) (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/p (cid:18) Z B ( a, | x | a ) w ( y ) dy (cid:19) r/p ′ u ( x ) dx (cid:19) /p = C ( p ′ /q ) /p (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/p (cid:18) Z B ( a, | x | a ) w ( y ) dy (cid:19) r/p ′ u ( x ) dx (cid:19) /p , where the second last equality is integration by parts. Since u and w are in L andare positive, the integral on the right hand side is finite. Therefore, we have( p ′ q/r )( q/p ′ ) /p (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/p (cid:18) Z B ( a, | x | a ) w ( y ) dy (cid:19) r/p ′ u ( x ) dx (cid:19) /r ≤ C. Approximating u and w by increasing sequence of L functions, using( p ′ q/r )( q/p ′ ) /p = ( p ′ )( p ′ ) − (1 /p ) q /p ( q/r ) = ( p ′ ) /p ′ q /p ( q (1 /q − /p )) = ( p ′ ) /p ′ q /p (1 − q/p )and applying the Monotone Convergence Theorem, we conclude that( p ′ ) /p ′ q /p (1 − q/p ) A ≤ C. Suppose now that A < ∞ and, for the moment, that (2.9) holds for w . Set V ( x ) = Z B ( a, | x | a ) w ( y ) dy and apply Proposition 2.4 with b = V − p w and F ( x ) = R B ( a, | x | a ) f ( y ) dy . Let us denote˜ V ( t ) = Z t Z Σ ρ λ ( ρ, σ ) w ( ρ, σ ) dρdσ ρ ,V ( ρ ) = Z Σ ρ λ ( ρ, σ ) w ( ρ, σ ) dσ ρ ,U ( ρ ) = Z Σ ρ λ ( ρ, σ ) u ( ρ, σ ) dσ ρ . ARDY INEQUALITIES ON METRIC MEASURE SPACES, II: THE CASE p > q Also, Z X \ B ( a, | x | a ) b ( y ) dy < ∞ , since Z X \ B ( a, | x | a ) b ( y ) dy = Z X \ B ( a, | x | a ) V − p ( y ) w ( y ) dy = Z ∞| x | a ˜ V − p ( ρ ) V ( ρ ) dρ = ( p ′ /p ) ˜ V − p ( | x | a ) . The conclusion of Proposition 2.4 becomes (cid:18) Z X (cid:18) Z B ( a, | x | a ) f ( y ) dy (cid:19) q u ( x ) dx (cid:19) /q ≤ ( r/p ) /r (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/q (cid:18) Z X \ B ( a, | x | a ) V − p ( y ) w ( y ) dy (cid:19) − r/q V − p ( x ) w ( x ) dx (cid:19) /r × (cid:18) Z X (cid:18) Z B ( a, | x | a ) f ( y ) dy (cid:19) p V − p ( x ) w ( x ) dx (cid:19) /p = ( r/p ) /r (cid:18) Z ∞ (cid:18) Z ∞ ρ U ( t ) dt (cid:19) r/q (cid:18) Z ∞ ρ ˜ V − p ( t ) V ( t ) dt (cid:19) − r/q ˜ V − p ( ρ ) V ( ρ ) dρ (cid:19) /r × (cid:18) Z X (cid:18) Z B ( a, | x | a ) f ( y ) dy (cid:19) p V − p ( x ) w ( x ) dx (cid:19) /p . Using R ∞ s ˜ V − p ( t ) V ( t ) dt = ( p ′ /p ) ˜ V − p ( s ) in the first factor and applying Proposition2.5 to the second factor, we reach the inequality (cid:18) Z X (cid:18) Z B ( a, | x | a ) f ( y ) dy (cid:19) q u ( x ) dx (cid:19) /q ≤ ( r/p ) /r ( p/p ′ ) /q p ′ (cid:18) Z ∞ (cid:18) Z ∞ ρ U ( t ) dt (cid:19) r/q ˜ V ( ρ ) ( p − r/q ˜ V − p ( ρ ) V ( ρ ) dρ (cid:19) /r (cid:18) Z X f p ( x ) v ( x ) dx (cid:19) /p = ( r/p ) /r ( p/p ′ ) /q p ′ (cid:18) Z ∞ (cid:18) Z ∞ ρ U ( t ) dt (cid:19) r/q (cid:18) Z ρ V ( t ) dt (cid:19) r/q ′ V ( ρ ) dρ (cid:19) /r (cid:18) Z X f p ( x ) v ( x ) dx (cid:19) /p = ( r/p ) /r ( p/p ′ ) /q p ′ A (cid:18) Z X f p ( x ) v ( x ) dx (cid:19) /p = ( r/p ) /r ( p/p ′ ) /q p ′ ( p ′ /q ) /r A (cid:18) Z X f p ( x ) v ( x ) dx (cid:19) /p = ( r/q ) /r ( p ′ ) /p ′ p /p A (cid:18) Z X f p ( x ) v ( x ) dx (cid:19) /p . In the fourth last equality, we used r ( p − q − p = r (cid:18) p − q − pr (cid:19) = r (cid:18) p − q − p ( p − q ) pq (cid:19) = rq ′ and in the second last equality, we used Lemma 2.2.To establish sufficiency for general w, we fix positive functions u and w . If w = 0almost everywhere on some ball B ( a, | x | a ) then translating u , w on the left will reducethe problem to the one in which this does not occur. (If w = 0 almost everywhere on X , sufficiency holds trivially). We therefore assume that 0 < Z B ( a, | x | a ) w ( y ) dy, for all x = a . For each n > , set u n = uχ ( B ( a,n )) and w n = min( w, n ) + χ ( X \ B ( a,n )) . Then w n clearly satisfies (2.9), so from previous arguments we have (cid:18) Z X (cid:18) Z B ( a, | x | a ) f ( y ) dy (cid:19) q u n ( x ) dx (cid:19) /q ≤ c (cid:18) Z ∞ (cid:18) Z ∞ ρ ˜ u n ( t ) dt (cid:19) r/p (cid:18) Z ρ ˜ w n ( t ) dt (cid:19) r/p ′ ˜ u n ( ρ ) dρ (cid:19) /r (cid:18) Z X f p ( y ) w − pn ( y ) dy (cid:19) /p , for all f ≥
0. Here c = ( r/q ) /r ( p ′ ) /p ′ p /p . If we take f = g min( w, n ) /p ′ χ ( B ( a,n )) anduse the definitions of u n and w n , the inequality becomes (cid:18) Z B ( a,n ) (cid:18) Z B ( a, | x | a ) g ( y ) min( w, n ) /p ′ dy (cid:19) q u ( x ) dx (cid:19) /q ≤ c (cid:18) Z n (cid:18) Z nρ ˜ u ( t ) dt (cid:19) r/p × (cid:18) Z ρ min( ˜ w, n ) (cid:19) r/p ′ ˜ u ( ρ ) dρ (cid:19) /r (cid:18) Z X g p ( y ) dy (cid:19) /p , for all non-negative g . We let n → ∞ , apply the Monotone Convergence Theoremand substitute f w − /p ′ for g to get the desired inequality and complete the proof. (cid:3) Applications and examples
Homogeneous groups.
Let G be a homogeneous group of homogeneous di-mension Q , equipped with a quasi-norm | · | . For the general description of the setupof homogeneous groups we refer to [FS82] or [FR16]. Particular example of homo-geneous groups are the Euclidean space R n (in which case Q = n ), the Heisenberggroup, as well as general stratified groups (homogeneous Carnot groups) and gradedgroups.In relation to the notation of this paper, let us take a = 0 to be the identity of thegroup G . We can also simplify the notation denoting | x | a by | x | , which is consistentwith the notation for the quasi-norm | · | . Let us consider an example of the power weights u ( x ) = (cid:26) | x | α if | x | < , | x | α if | x | ≥ , v ( x ) = | x | β . Then by Theorem 2.1 the inequality (cid:18) Z X (cid:18) Z B ( a, | x | a ) | f ( y ) | dy (cid:19) q u ( x ) dx (cid:19) q ≤ C (cid:18) Z X | f ( y ) | p v ( x ) dx (cid:19) p ARDY INEQUALITIES ON METRIC MEASURE SPACES, II: THE CASE p > q holds for 0 < q < p, < p < ∞ , if and only if A r = (cid:18) Z X (cid:18) Z X \ B ( a, | x | a ) u ( y ) dy (cid:19) r/p (cid:18) Z B ( a, | x | a ) v − p ′ ( y ) dy (cid:19) r/p ′ u ( x ) dx (cid:19) ≈ Z (cid:18) Z t ρ α + Q − dρ + Z ∞ ρ α + Q − dρ (cid:19) r/p (cid:18) Z t ρ β (1 − p ′ )+ Q − dρ (cid:19) r/p ′ t α + Q − dt + Z ∞ (cid:18) Z ∞ t ρ α + Q − dρ (cid:19) r/p (cid:18) Z t ρ β (1 − p ′ )+ Q − dρ (cid:19) r/p ′ t α + Q − dt < ∞ . Let us consider Z (cid:18) Z t ρ α + Q − dρ + Z ∞ ρ α + Q − dρ (cid:19) r/p (cid:18) Z t ρ β (1 − p ′ )+ Q − dρ (cid:19) r/p ′ t α + Q − dt = Z (cid:18) / ( α + Q ) − t α + Q / ( α + Q ) − / ( α + Q ) (cid:19) r/p (cid:18) t ( β (1 − p ′ )+ Q ) / ( β (1 − p ′ ) + Q ) (cid:19) r/p ′ × t α + Q − dt, which is finite for α + Q < , β (1 − p ′ ) + Q > , ( α + Q ) r/p + ( β (1 − p ′ ) + Q ) r/p ′ + α + Q > , which means α + Q < , β (1 − p ′ ) + Q > , ( α + Q ) r/q + ( β (1 − p ′ ) + Q ) r/p ′ > , since we have r/p + 1 = r (1 /p + 1 /r ) = r (1 /p + 1 /q − /p ) = r/q .Now, consider the other part Z ∞ (cid:18) Z ∞ t ρ α + Q − dρ (cid:19) r/p (cid:18) Z t ρ β (1 − p ′ )+ Q − dρ (cid:19) r/p ′ t α + Q − dt = Z ∞ { ( − t ) ( α + Q ) / ( α + Q ) } r/p { t ( β (1 − p ′ )+ Q ) / ( β (1 − p ′ ) + Q ) } r/p ′ t α + Q − dt, which is finite for α + Q < , β (1 − p ′ ) + Q > , ( α + Q ) r/p + ( β (1 − p ′ ) + Q ) r/p ′ + α + Q < , or for α + Q < , β (1 − p ′ ) + Q > , ( α + Q ) r/q + ( β (1 − p ′ ) + Q ) r/p ′ < . Summarising that we get
Corollary 3.1.
Let G be a homogeneous group of homogeneous dimension Q , equippedwith a quasi-norm | · | . Let < q < p, < p < ∞ , /r = 1 /q − /p, and let α , α , β ∈ R . Assume that α + Q = 0 . Let u ( x ) = (cid:26) | x | α if | x | < , | x | α if | x | ≥ , v ( x ) = | x | β . (3.1) Then the inequality (cid:18) Z G (cid:18) Z B ( a, | x | a ) | f ( y ) | dy (cid:19) q u ( x ) dx (cid:19) q ≤ C (cid:26) Z G | f ( x ) | p v ( x ) dx (cid:27) p (3.2) holds for all measurable functions f : G → C if and only if the parameters satisfy thefollowing conditions: α + Q < , β (1 − p ′ )+ Q > , ( α + Q ) r/q +( β (1 − p ′ )+ Q ) r/p ′ > , ( α + Q ) r/q + ( β (1 − p ′ ) + Q ) r/p ′ < . It is interesting to note that in view of the last two conditions, it is not possible tohave Hardy inequality (3.2) with weights u and v in (3.1) with α = α . This is whywe consider different powers α , α in this example. This is different from the case p ≤ q which was considered as an application in [RV18].The case α + Q = 0 can be treated in a similar way.3.2. Hyperbolic spaces.
Let H n be the hyperbolic space of dimension n and let a ∈ H n . Let us take the weights u ( x ) = (cid:26) (sinh | x | a ) α if | x | < , (sinh | x | a ) α if | x | ≥ , , v ( x ) = (sinh | x | a ) β . We note that A is equivalent to A r ≈ Z (cid:18) Z t (sinh ρ ) α + n − dρ + Z ∞ (sinh ρ ) α + n − dρ (cid:19) r/p (cid:18) Z t (sinh ρ ) β (1 − p ′ )+ n − dρ (cid:19) r/p ′ × (sinh t ) α + n − dt + Z ∞ (cid:18) Z ∞ t (sinh ρ ) α + n − dρ (cid:19) r/p (cid:18) Z t (sinh ρ ) β (1 − p ′ )+ n − dρ (cid:19) r/p ′ × (sinh t ) α + n − dt. In the first part, for α + n − < β (1 − p ′ ) + n > , Z (cid:18) Z t (sinh ρ ) α + n − dρ + Z ∞ (sinh ρ ) α + n − dρ (cid:19) r/p (cid:18) Z t (sinh ρ ) β (1 − p ′ )+ n − dρ (cid:19) r/p ′ × (sinh t ) α + n − dt ≈ Z (cid:18) Z t ( ρ ) α + n − dρ + Z ∞ (exp ρ ) α + n − dρ (cid:19) r/p (cid:18) Z t ( ρ ) β (1 − p ′ )+ n − dρ (cid:19) r/p ′ × ( t ) α + n − dt = Z (cid:18) / ( α + n ) − t α + n / ( α + n ) − (exp 1) α + n − / ( α + n − (cid:19) r/p × (cid:18) t β (1 − p ′ )+ n / ( β (1 − p ′ ) + n ) (cid:19) r/p ′ t α + n − dt, which is finite for,(a) α + n ≥
0, ( β (1 − p ′ ) + n ) r/p ′ + α + n > , (b) α + n <
0, ( α + n ) r/p + ( β (1 − p ′ ) + n ) r/p ′ + α + n > . ARDY INEQUALITIES ON METRIC MEASURE SPACES, II: THE CASE p > q However, we can note that in (a), if α + n ≥
0, then the second condition isautomatically satisfied under our assumption β (1 − p ′ ) + n > . In the second part, for α + n − < , Z ∞ (cid:18) Z ∞ t (sinh ρ ) α + n − dρ (cid:19) r/p (cid:18) Z t (sinh ρ ) β (1 − p ′ )+ n − dρ (cid:19) r/p ′ (sinh t ) α + n − dt ≈ Z ∞ (cid:18) Z ∞ t (exp ρ ) α + n − dρ (cid:19) r/p (cid:18) Z t (exp ρ ) β (1 − p ′ )+ n − dρ (cid:19) r/p ′ (exp t ) α + n − dt = Z ∞ (cid:18) − (exp t ) α + n − / ( α + n − (cid:19) r/p (cid:18) (exp t ) β (1 − p ′ )+ n − / ( β (1 − p ′ ) + n − (cid:19) r/p ′ × ((exp t ) α + n − dt, which is finite for( α + n − r/p + ( β (1 − p ′ ) + n − r/p ′ + α + n − < , which is the same as( α + n − r/q + ( β (1 − p ′ ) + n − r/p ′ < . Corollary 3.2.
Let H n be the hyperbolic space, a ∈ H n , and let | x | a denote thehyperbolic distance from x to a . Let < q < p, < p < ∞ , /r = 1 /q − /p, and let α , α , β ∈ R . Assume that α + n = 0 . Let u ( x ) = (cid:26) (sinh | x | a ) α if | x | < , (sinh | x | a ) α if | x | ≥ , v ( x ) = (sinh | x | a ) β . Then the inequality (cid:18) Z H n (cid:18) Z B ( a, | x | a ) | f ( y ) | dy (cid:19) q u ( x ) dx (cid:19) q ≤ C (cid:26) Z H n | f ( x ) | p v ( x ) dx (cid:27) p holds for all measurable functions f : H n → C if and only if the parameters satisfythe following conditions: α + n − < , β (1 − p ′ ) + n > , ( α + n ) r/q + ( β (1 − p ′ ) + n ) r/p ′ > , ( α + n − r/q + ( β (1 − p ′ ) + n − r/p ′ < . Cartan-Hadamard manifolds.
Let (
M, g ) be a Cartan-Hadamard manifoldand assume that the sectional curvature K M is constant. In this case it is known that J ( t, ω ) is a function of t only. More precisely, if K M = − b for b ≥
0, then J ( t, ω ) = 1if b = 0, and J ( t, ω ) = ( sinh √ bt √ bt ) n − for b >
0, see e.g. [Ngu17]. When b = 0, then letus take the weights u ( x ) = (cid:26) (sinh | x | a ) α if | x | < , (sinh | x | a ) α if | x | ≥ , v ( x ) = (sinh | x | a ) β , then the inequality (2.1) holds for 0 < q < p , 1 < p < ∞ , /r = 1 /q − /p, if andonly if A ≈ (cid:18) Z (cid:18) Z t ρ α + n − dρ + Z ∞ ρ α + n − dρ (cid:19) r/p (cid:18) Z t ρ β (1 − p ′ )+ n − dρ (cid:19) r/p ′ t α + n − dt ++ Z ∞ (cid:18) Z ∞ t ρ α + n − dρ (cid:19) r/p (cid:18) Z t ρ β (1 − p ′ )+ n − dρ (cid:19) r/p ′ t α + n − dt (cid:19) /r < ∞ , which is finite if and only if conditions of Corollary 3.1 hold with Q = n (which isnatural since the curvature is zero).When b >
0, let u ( x ) = (cid:26) (sinh √ b | x | a ) α if | x | < , (sinh √ b | x | a ) α if | x | ≥ , v ( x ) = (sinh √ b | x | a ) β . Then the inequality (2.1) holds for 0 < q < p , 1 < p < ∞ , /r = 1 /q − /p , if andonly if A is finite. We have A ≈ (cid:18) Z (cid:18) Z t (sinh √ bρ ) α ( sinh √ bρ √ bρ ) n − ρ n − dρ + Z ∞ (sinh √ bρ ) α ( sinh √ bρ √ bρ ) n − ρ n − dρ (cid:19) r/p × (cid:18) Z t (sinh √ bρ ) β (1 − p ′ ) ( sinh √ bρ √ bρ ) n − ρ n − dρ (cid:19) r/p ′ × (sinh √ bt ) α ( sinh √ bt √ bt ) n − t n − dt + Z ∞ (cid:18) Z ∞ t (sinh √ bρ ) α ( sinh √ bρ √ bρ ) n − ρ n − dρ (cid:19) r/p × (cid:18) Z t (sinh √ bρ ) β (1 − p ′ ) ( sinh √ bρ √ bρ ) n − ρ n − dρ (cid:19) r/p ′ × (sinh √ bt ) α ( sinh √ bt √ bt ) n − t n − dt (cid:19) /r ≈ (cid:18) Z (cid:18) Z t (sinh √ bρ ) α + n − dρ + Z ∞ (sinh √ bρ ) α + n − dρ (cid:19) r/p × (cid:18) Z t (sinh √ bρ ) β (1 − p ′ )+ n − dρ (cid:19) r/p ′ (sinh √ bt ) α + n − dt + Z ∞ (cid:18) Z ∞ t (sinh √ bρ ) α + n − dρ (cid:19) r/p × (cid:18) Z t (sinh √ bρ ) β (1 − p ′ )+ n − dρ (cid:19) r/p ′ (sinh √ bt ) α + n − dt (cid:19) /r , which has the same conditions for finiteness as the case of the hyperbolic space inCorollary 3.2 (which is also natural since it is the negative constant curvature case). References [BEL15] A. A. Balinsky, W. D. Evans, and R. T. Lewis.
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Michael Ruzhansky:Department of Mathematics: Analysis, Logic and Discrete MathematicsGhent UniversityKrijgslaan 281, Building S8B 9000 Ghent, BelgiumandSchool of Mathematical SciencesQueen Mary University of LondonMile End Road, London E1 4NSUnited Kingdom
E-mail address [email protected]
Daulti Verma:Miranda House CollegeUniversity of DelhiDelhi 110007IndiaandSchool of Mathematical SciencesQueen Mary University of LondonMile End Road, London E1 4NSUnited Kingdom