New Calderon Zygmund decomposition for Sobolev functions

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New Calderon-Zygmund decomposition for Sobolev functions N. Badr Institut Camille Jordan Universite Claude Bernard Lyon 1 43 boulevard du 11 Novembre 1918 F-69622 Villeurbanne Cedex F. Bernicot Universite de Paris-Sud F-91405 Orsay Cedex April 21, 2010 Abstract We state a new Calderon-Zygmund decomposition for Sobolev spaces on a dou- bling Riemannian manifold. Our hypotheses are weaker than those of the already known decomposition which used classical Poincare inequalities. Key-words : Calderon-Zygmund decomposition, Sobolev spaces, Poincare inequalities. MSC : 42B20, 46E35. Contents 1 Introduction 2 2 Preliminaries 3 2.1 The doubling property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Classical Poincare inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Estimates for the heat kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 The K-method of real interpolation . . . . . . . . . . .

measure space

qi ≤

lipshitz function

poincare inequalities

hardy spaces

poincare inequality

calderon- zygmund decomposition

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NewCalder´on-ZygmunddecompositionforSobolev functions

N. Badr Institut Camille Jordan Universit´eClaudeBernardLyon1 43 boulevard du 11 Novembre 1918 F-69622 Villeurbanne Cedex badr@math.univ-lyon1.fr F. Bernicot Universite´deParis-Sud F-91405 Orsay Cedex frederic.bernicot@math.u-psud.fr April 21, 2010

Abstract

WestateanewCalder´on-ZygmunddecompositionforSobolevspacesonadou-bling Riemannian manifold. Our hypotheses are weaker than those of the already knowndecompositionwhichusedclassicalPoincare´inequalities.

Key-words :s.ieitbolen,Soces,vspaac´roPniuqlaieenitiomposdecomundZ-gy´rnolaedC MSC :42B20, 46E35.

Contents 1 Introduction 2 2 Preliminaries 3 2.1 The doubling property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2ClassicalPoincare´inequality..........................4 2.3 Estimates for the heat kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 TheK-method of real interpolation. . . . . . . . . . . . . . . . . . . . . . 6 3New“Calder´on-Zygmund”decompositionsforSobolevfunctions.7 3.1 Decomposition using abstract “oscillation operators” . . . . . . . . . . . . 7 3.2 Application to real Interpolation of Sobolev spaces. . . . . . . . . . . . . . 12 3.3 Homogeneous version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1

4Pseudo-Poincare´inequalitiesandApplications15 4.1Theparticularcaseof“Pseudo-Poincar´eInequalities”............15 4.2 Application to Reverse Riesz transform inequalities. . . . . . . . . . . . . . 21 4.3 Application to Gagliardo-Nirenberg inequalities. . . . . . . . . . . . . . . . 21

1 Introduction ThepurposeofthisarticleistoweakenassumptionsofthealreadyknownCaldero´n-Zygmund decomposition for Sobolev functions. This well-known tool was ﬁrst stated by P.Auscherin[2].ItexactlycorrespondstotheCalder´on-Zygmunddecompositionina context of Sobolev spaces. Let us brieﬂy recall the ideas of such decomposition. In [34], E. Stein stated this decom-position for Lebesgue spaces as following. Let (X, d, µ) be a space of homogeneous type andp≥1. Given a functionf∈Lp(X), the decomposition gives a precise way of parti-tioningX oneinto two subsets: wherefis essentially small (bounded inL∞norm); the other a countable collection of cubes wherefis essentially large, but where some control of the function is obtained inL1hTsiro.mtstoeldanyZ-no´rednumgcisoasheldCaedat decomposition off, wherefis written as the sum of “good” and “bad” functions, using the above subsets. This decomposition is a basic tool in Harmonic analysis and the study of singular integrals. One of the applications is the following : anL2deunbo-´redlaCdumgyZ-noeratndoporis of weak type (1,1) and soLpbounded for everyp∈(1,∞). In [2], P. Auscher extended these ideas for Sobolev spaces. His decomposition is the following: Theorem 1.1Letn≥1,p∈[1,∞)andf∈ D0(Rn)be such thatkrfkLp<∞. Let α >0 one can ﬁnd a collection of cubes. Then,(Qi)i, functionsgandbisuch that f=g+Xbi i

and the following properties hold: