Density of Analytic Polynomials in Abstract Hardy Spaces

TL;DR

This paper proves that analytic polynomials are dense in abstract Hardy spaces built on certain Banach function spaces, under conditions involving the boundedness of the Hardy-Littlewood maximal operator, with specific results for variable Lebesgue spaces.

Contribution

It establishes a general density result for analytic polynomials in abstract Hardy spaces based on the boundedness of the Hardy-Littlewood maximal operator.

Findings

Density of analytic polynomials in $H[X]$ under maximal operator boundedness

Application to variable Lebesgue spaces

Extension of classical Hardy space results

Abstract

Let $X$ be a separable Banach function space on the unit circle $\mathbb{T}$ and $H[X]$ be the abstract Hardy space built upon $X$. We show that the set of analytic polynomials is dense in $H[X]$ if the Hardy-Littlewood maximal operator is bounded on the associate space $X'$. This result is specified to the case of variable Lebesgue spaces.