Some non-standard biparametric Poincar\'e type inequalities through harmonic analysis

TL;DR

This paper establishes new biparametric Poincaré inequalities with weights, using harmonic analysis techniques like maximal functions, fractional integrals, and extrapolation to improve and sharpen existing estimates.

Contribution

It introduces non-standard biparametric Poincaré inequalities with weights and refines these results through extrapolation methods, advancing harmonic analysis tools.

Findings

Derived new biparametric Poincaré inequalities with weights

Utilized maximal functions and fractional integrals in proofs

Achieved sharper estimates via extrapolation techniques

Abstract

We show some non-standard Poincar\'e type estimates in the biparametric setting with appropriate weights. We will derive these results using variants from classical estimates exploiting the interplay between maximal functions and fractional integrals. We also provide a sharper result by using extrapolation techniques.