Almost-everywhere convergence of Fourier series for functions in Sobolev spaces

TL;DR

This paper proves that Fourier series of functions in certain Sobolev spaces converge almost everywhere, extending known results for Fourier integrals to a broader class of functions on multi-dimensional tori.

Contribution

It extends the transplantation technique from $L_p$ spaces to Sobolev spaces, enabling almost-everywhere convergence results for Fourier series in these spaces.

Findings

Almost-everywhere convergence for Sobolev space functions

Extension of transplantation technique to Sobolev spaces

Generalization of Carbery and Soria's Fourier integral results

Abstract

Let $S_\lambda F(x)$ be the spherical partial sums of the multiple Fourier series of function $F\in L_2(\mathbb{T}^N)$. We prove almost-everywhere convergence $S_\lambda F(x)\rightarrow F(x)$ for functions in Sobolev spaces $H_p^a(\mathbb{T}^N)$ provided $1< p \leq 2$ and $a> (N-1)(\frac{1}{p}-\frac{1}{2})$. For multiple Fourier integrals this is well known result of Carbery and Soria (1988). To prove our result, we first extend the transplantation technic of Kenig and Tomas (1980) from $L_p$ spaces to $H_p^a$ spaces, then apply it to the Carbery and Soria result.