Weighted norm inequalities in a bounded domain by the sparse domination method

Emma-Karoliina Kurki,Antti V Vähäkangas

doi:10.1007/s13163-020-00358-8

Emma-Karoliina Kurki, Antti V Vähäkangas

Open Access

https://doi.org/10.1007/s13163-020-00358-8

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Abstract

We prove a local two-weight Poincaré inequality for cubes using the sparse domination method that has been influential in harmonic analysis. The proof involves a localized version of the Fefferman–Stein inequality for the sharp maximal function. By establishing a local-to-global result in a bounded domain satisfying a Boman chain condition, we show a two-weight p-Poincaré inequality in such domains. As an application we show that certain nonnegative supersolutions of the p-Laplace equation and distance weights are p-admissible in a bounded domain, in the sense that they support versions of the p-Poincaré inequality.

Highlights

Poincaré inequalities are useful tools in analysis, especially so in the theory of partial differential equations (PDEs)
Vähäkangas on a bounded domain only. This setting is natural from the viewpoint of analysis of PDEs and calls for a localized argument, which we borrow from harmonic analysis
The method has been very influential in harmonic analysis, in which several weighted inequalities for singular integrals and potential operators have been established by using the idea of pointwise dyadic domination, including the celebrated A2 theorem

Summary

Introduction

Poincaré inequalities are useful tools in analysis, especially so in the theory of partial differential equations (PDEs). A key new feature of Theorem 5.4 is that the assumptions concerning the weights v and w are localized to the fixed cube Q0, which is necessary in order to establish the local version of (3) in Theorem 7.7 This distinguishes our main result (2) (Theorem 7.1) from other two-weight inequalities in the same vein. In order to prove (1), we proceed in two stages Both are based on the idea of sparse domination, in which one first provides a pointwise inequality in terms of a sparse dyadic operator. The method has been very influential in harmonic analysis, in which several weighted inequalities for singular integrals and potential operators have been established by using the idea of pointwise dyadic domination, including the celebrated A2 theorem. Compared to state-of-the-art instances of the sparse domination argument, our version is vastly simpler, yet perfectly sufficient for its purpose and with the further advantage of being localized

Setup A cube in Rn is a half-open set of the form

Sparse domination I

Local Fefferman–Stein inequality

Sparse domination II

From local to global