Littlewood - Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces

Abstract

We point out that when the Hardy - Littlewood maximal operator is bounded on the space $L^{p(t)}(\mathbb{R}^n),\quad 1 < a \leq p(t) \leq b < \infty,\quad t \in \mathbb{R}$, the well-known characterization of spaces $L^{p(t)}(\mathbb{R}^n),\quad 1 < p < \infty$, by the Littlewood - Paley theory extends to the space $L^{p(t)}(\mathbb{R}^n).$ We show that if $n > 1,$ the Littlewood -Paley operator is bounded on $L^{p(t)}(\mathbb{R}^n),\quad 1 < a \leq p(t) \leq b < \infty,\quad t \in \mathbb{R},$ if and only if $p(t) =$ const.