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Niveau: Supérieur, Doctorat, Bac+8

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 <∞.

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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Published by | mijec |

Reads | 8 |

Language | English |

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