Poincaré inequalities and the sharp maximal inequalities with L φ -norms for differential forms

Yueming Lu,Gang Bao

doi:10.1186/1029-242x-2013-400

Abstract

This paper is concerned with the Poincare inequalities and the sharp maximal inequalities for differential forms with -norm, where φ satisfies nonstandard growth conditions. These results can be used to estimate the norms of classical operators and analyze integral properties of differential forms.

Highlights

In this paper, we consider the functionalI(, v) = φ |dv| dx, ( )where φ : [, ∞) → [, ∞) is a Young function satisfying the nonstandard growth condition pφ(t) ≤ tφ (t) ≤ qφ(t), < p ≤ q < ∞.This condition was first used in [ ]; the author used this condition to get the higher integrability of the gradient of minimizers
Where φ : [, ∞) → [, ∞) is a Young function satisfying the nonstandard growth condition pφ(t) ≤ tφ (t) ≤ qφ(t), < p ≤ q < ∞
Lp(, l) is the space of l-forms u(x) = I uI(x) dxI with uI∈ Lp(, R) for all ordered l-tuples I, it is a Banach space endowed with the norm u p, = ( |u|p dx) /p

Summary

Introduction

Where φ : [ , ∞) → [ , ∞) is a Young function satisfying the nonstandard growth condition pφ(t) ≤ tφ (t) ≤ qφ(t), < p ≤ q < ∞. A differential l-form ω on is a Schwartz distribution on with values in l(Rn), and it can be denoted by u(x) = uI (x) dxI =. Lp( , l) is the space of l-forms u(x) = I uI(x) dxI with uI∈ Lp( , R) for all ordered l-tuples I, it is a Banach space endowed with the norm u p, = ( |u|p dx) /p. W ,p( , l) are those differential l-forms on , whose coefficients are in W ,p( , R), and it is a Banach space endowed with the norm u ,p = u p + u p. In Section , Poincaré inequalities for differential forms with Orlicz norm are obtained. In Section , using the methods that appeared in Section and Section , we get some estimates of classical operators

Poincaré inequalities

Application