On an Interesting Class of Variable Exponents

Abstract

Let $\mathcal{M}(\mathbb{R}^n)$ be the class of functions $p:\mathbb{R}^n\to[1,\infty]$ bounded away from one and infinity and such that the Hardy-Littlewood maximal function is bounded on the variable Lebesgue space $L^{p(\cdot)}(\mathbb{R}^n)$. We denote by $\mathcal{M}^*(\mathbb{R}^n)$ the class of variable exponents $p\in\mathcal{M}(\mathbb{R}^n)$ for which $1/p(x)=\theta/p_0+(1-\theta)/p_1(x)$ with some $p_0\in(1,\infty)$, $\theta\in(0,1)$, and $p_1\in\mathcal{M}(\mathbb{R}^n)$. Rabinovich and Samko \cite{RS08} observed that each globally log-H\"older continuous exponent belongs to $\mathcal{M}^*(\mathbb{R}^n)$. We show that the class $\mathcal{M}^*(\mathbb{R}^n)$ contains many interesting exponents beyond the class of globally log-H\"older continuous exponents.