ACTA issues

The duals of $A^p$ and $S^p$ the spaces of absolutely and strongly convergent Fourier series of index $p>1$

I. Szalay , N. Tanović-Miller

Acta Sci. Math. (Szeged) 60:3-4(1995), 637-657

Abstract. Let $A^p$ and $S^p$, $p\ge1$, denote the spaces of functions whose Fourier series are absolutely, respectively strongly convergent a.e. of index $p\ge1$. They are Banach spaces; $A^p\subset S^p\subset\cap _{r\ge1}L^p$ for $p>1$, and $A^1={\cal A}$ is the classical space of absolutely convergent trigonometric series. It is well known that $\tilde{\cal A}=\ell ^1$ so that the dual space $({\cal A})^*$ can be identified with the space of Radon measures. We extend this statement to the function spaces $A^p$, $p>1$ by showing that $(A^p)^*$, the dual of $A^p$, can be identified with $A^q$, $1/p+1/q=1$. Hence, the spaces $A^p$, $p\ge1$ are reflexive. We prove this by observing that $A^p$, $p\ge1$ can be characterized in terms of the sequences of Fourier coefficients and applying the sequence space approach. Similarly, giving various description of the spaces $\widehat{S^p}$, we examine the duals of $S^p$ for $p>1$ and show that these spaces are not reflexive. The dual space $(S^p)^*$ can be identified with the space $\{g\in L^1:\sum_{j=0}^\infty\|d^jg\|_{S^q}< \infty\} $, where $d^jg$ denotes the $j$-th dyadic section of $g$. All of these spaces are examples of the so called mixed norm spaces.

AMS Subject Classification (1991): 42A16, 42A20

Received September 28, 1994. (Registered under 5656/2009.)