Lp SELF IMPROVEMENT OF GENERALIZED POINCARE INEQUALITIES IN SPACES OF HOMOGENEOUS TYPE

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Lp SELF-IMPROVEMENT OF GENERALIZED POINCARE INEQUALITIES IN SPACES OF HOMOGENEOUS TYPE NADINE BADR, ANA JIMENEZ-DEL-TORO, AND JOSE MARIA MARTELL Abstract. In this paper we study self-improving properties in the scale of Lebesgue spaces of generalized Poincare inequalities in spaces of homogeneous type. In con- trast with the classical situation, the oscillations involve approximation of the iden- tities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization. The techniques used do not involve any classical Poincare or Sobolev-Poincare inequalities and therefore they can be used in general settings where these estimates do not hold or are unknown. We apply our results to the case of Riemannian manifolds with doubling volume form and assum- ing Gaussian upper bounds for the heat kernel of the semigroup e?t∆ with ∆ being the Laplace-Beltrami operator. We obtain generalized Poincare inequalities with oscillations that involve the semigroup e?t∆ and with right hand sides containing either ? or ∆1/2. 1. Introduction In analysis and PDEs we can find various estimates that encode self-improving prop- erties of the integrability of the functions involved. For instance, the John-Nirenberg inequality establishes that a function in BMO, which a priory is in L1loc(R n), is indeed exponentially integrable which in turn implies that is in Lploc(R n) for any 1 ≤ p <∞.

  • large constant

  • therefore µ

  • poincare inequalities

  • self-improving properties

  • lebesgue spaces

  • any classical

  • lp self


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´ LpSELF-IMPROVEMENT OF GENERALIZED POINCARE INEQUALITIES IN SPACES OF HOMOGENEOUS TYPE
´ ´ ´ NADINE BADR, ANA JIMENEZ-DEL-TORO, AND JOSE MARIA MARTELL
Abstract.In this paper we study self-improving properties in the scale of Lebesgue spacesofgeneralizedPoincare´inequalitiesinspacesofhomogeneoustype.Incon-trast with the classical situation, the oscillations involve approximation of the iden-tities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization. The techniques used do not involve any classicalPoincar´eorSobolev-Poincar´einequalitiesandthereforetheycanbeusedin general settings where these estimates do not hold or are unknown. We apply our results to the case of Riemannian manifolds with doubling volume form and assum-ing Gaussian upper bounds for the heat kernel of the semigroupetΔwith Δ being theLaplace-Beltramioperator.WeobtaingeneralizedPoincare´inequalitieswith oscillations that involve the semigroupetΔand with right hand sides containing eitherror Δ1/2 .
1.Introduction
In analysis and PDEs we can find various estimates that encode self-improving prop-erties of the integrability of the functions involved. For instance, the John-Nirenberg inequality establishes that a function in BMO, which a priory is inLcol1(Rn), is indeed exponentially integrable which in turn implies that is inLlpoc(Rn) for any 1p <. Another situation where functions self-improve their integrability comes from the clas-sical (p, platiiynr´einequ)-PoincaRn,n2, 1p < n, ZQ|ffQ|pdxC `(Q)ZQ. |rf|pdx It is well-known that this estimate yields that for any functionfLlpoc(Rn) with rfLlpoc(Rn) , ZQ|ffQ|pdx1/pC `(Q)ZQ|rf|pdx1/p wherep=npnp. Againfgains integrability properties, since the previous inequality givesfLlpoc(Rn involve the). Both they situations have something in common:
Date: July 28, 2010. 2010Mathematics Subject Classification.46E35 (47D06, 46E30, 42B25, 58J35). Key words and phrases.nrek,sleflespmi-viroprngeropestiSemigroups,heat-e´racinPoedizaleren,g SobolevandHardyinequalities,pseudo-Poincar´einequalities,dyadiccubes,weights,good-λinequal-ities, Riemannian manifolds. The second and third authors are supported by MEC Grant MTM2007-60952. The third author was also supported by CSIC PIE 200850I015. We warmly thank P. Auscher for his interest and helpful discussions. 1
´ ´ ´ 2 NADINE BADR, ANA JIMENEZ-DEL-TORO, AND JOSE MARIA MARTELL oscillation of the functions on some cubeQviaffQ [FPW], general versions of. In these estimates are considered. They start with inequalities of the form (1.1)Z− |ffQ|dxa(Q, f), Q whereais a functional depending on the cubeQ, and sometimes on the functionf. There,theauthorspresentageneralmethodbasedontheCalder´on-Zygmundtheory and the good-λinequalities introduced by Burkholder and Gundy [BG] that allows them to establish that under mild geometric conditions on the functionala, inequality (1.1) encodes an intrinsic self-improvement onLptype forp >1. On the other hand, in [Ma1] a new sharp maximal operator associated with an approximation of the identity{St}t>0is introduced: Q3xZQ|fStQ MS#f(x) = supf|dy, wheretQis a parameter depending on the side-length of the cubeQ operator. This allows one to define the spaceBM OS, for which the John-Nirenberg inequality also holds (see [DY]). In this way, starting with an estimate as (1.1) where the oscillation ffQis replaced byfStQf, anda(Q, f) =Ca self-improving property is obtained. This new way of measuring the oscillation allows one to define new function spaces as the just mentionedBM OSof [DY] and the Morrey-Campanato associated with an approximation of the identity of [DDY], [Tan]. In [Jim] and [JM] self-improving properties related to this new way of measuring oscillation are under study. The starting estimate is as follows (1.2)ZQ|fStQf|dxa(Q, f),
withStbeing a family of operators (e.g., semigroup) with fast decay kernel. By anal-ogyto(1.1),wewillrefertotheseestimatesasgeneralizedPoincar´einequalities.The caseaboth in the Euclidean setting an also in spaces ofincreasing, considered in [Jim] homogeneous type, yields local exponential integrability of the new oscillationfStf. In [JM] functionals satisfying a weaker`r-summability condition (seeDrbelow) are studied in the Euclidean setting. In this caseLr,local integrability of the oscillation is obtained. In this paper we continue the study in [Jim], [JM] considering (1.1) in the setting of the spaces of homogeneous type for functionals satisfying some summability con-ditions. We obtain estimates in weak Lebesgue spaces with the oscillationfStf in the left hand side and an expansion ofaover dilations of balls on the right hand side. These expansions, that already appeared in [Jim], [JM], have fast decay coeffi-cients and are natural due to the lack of localization of the operatorsSt. The proofs are more technically involved since the setting is less friendly. However, we are able toobtainapplicationsinsettingswhereonemaylackofPoincar´einequalities.That is the case of some Riemannian manifolds assuming only doubling volume form and Gaussian upper bounds for the heat kernel associated to the semigroup generated by the Laplace-Beltrami operator. In order to present these applications, which are the main motivation of the general results presented here, we need to introduce some notation, see Section 4.3 for more details.
´ LpSELF-IMPROVEMENT OF GENERALIZED POINCARE TYPE INEQUALITIES
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LetMbe a complete non-compact connected Riemannian manifold withdits geo-desic distance. Assume that volume formµis doubling and letnbe its doubling order (see (2.1) below). ThenMequipped with the geodesic distance and the volume form µis a space of homogeneous type. Let Δ be the positive Laplace-Beltrami operator onMgiven by hΔf, gi=Zrf∙ rg dµ M whereris the Riemannian gradient onMandis an inner product onT M. We assume that the heat kernelpt(x, y) of the semigroupetΔhas Gaussian upper bounds if for some constants >c, C0 and allt >0, x, yM, pt(x, y)µ(B(,xCt))ecd2(ty,x).(U E) We defineq˜+as the supremum of thosep(1,) such that for allt >0, |retΔf|LpC t1/2kfkLp.(Gp) If the Riesz transform|rΔ1/2|is bounded inLp, by analyticity of the heat semigroup, then (Gp Therefore,) holds.q˜+is greater than the supremum on the exponentspfor which the Riesz transform is bounded onLp particular. Inq+2 by [CD]. AsaconsequenceofourmainresultsandintheabsenceofPoincare´inequalitieswe obtain the following (see Corollary 4.5 below for the precise statement):
Theorem 1.1.LetMbe complete non-compact connected Riemannian manifold sat-isfying the doubling volume property and(U E). Given1p <we setp= n p/(np)if1p < nandp=otherwise. (a)Givenm1(mis taken large enough when1< p < n), letStm=I(IetΔ)m and1< q < p. Then, for any smooth function with compact supportfwe have ZB|fStmBf|q1/qCkX1φ(k)r(σkB)Zσ|Δ1/2f|p1/p, kB whereφ(k) =σk θandθdepends onm,nandp. (b)For anyp((q˜+)0,)[2,), any1< q < pand any smooth function with compact supportfwe have ZB|fetBΔf|q1/qCkX1ec σkr(σkB)ZσkB|rf|p1/p. In this resultσis a large constant depending on the doubling condition (see below).
The plan of the paper is as follows. In Section 2 we give some preliminaries and definitions. The main result and its different extensions are in Section 3. Applications are considered in Section 4. In particular, we devote Sections 4.1 and 4.2 to study variousPoincar´etypeinequalitiesingeneralspacesofhomogeneoustype.Inthe former we start from an estimate whose right hand side is localized to the given ball B, in the latter we take into account the lack of localization of the approximation of the identity or the semigroup and the right hand side contains a series of terms as in the applications to manifolds stated above. As a consequence, in Section 4.1 we obtainglobalpseudo-Poincar´einequalities.InSection4.3weconsidertheapplication