Small Divisor problems and $A_p$ weights with an application

TL;DR

This paper links Muckenhoupt A_p weights to small divisor problems, providing a quantitative version of the Ehrenpreis-Malgrange theorem for PDEs with applications to Sobolev space estimates.

Contribution

It introduces a novel connection between A_p weights and small divisor issues, enabling a quantitative analysis of local solvability for constant coefficient PDEs.

Findings

Established a link between A_p weights and small divisor problems.

Developed a quantitative version of the Ehrenpreis-Malgrange theorem.

Applied the results to Sobolev space estimates and provided examples.

Abstract

We establish a link between Muckenhoupt $A_p$ weights and a means to address small divisor problems. We use this link to obtain a quantitative version of the Ehrenpreis-Malgrange theorem of local solvability for constant coefficient PDE. We give an example as to how our theorem applies. In our quantitative version of the Ehrenpreis-Malgrange theorem, the loss of derivatives in the solvability estimate is measured in the scale of Sobolev spaces via the use of Muckenhoupt A_p weights. A part of our results are global in nature.