The Amalgam Spaces $W( L^{p( x) },\ell^{\{ p_{n}\} })$ and Boundedness of Hardy–Littlewood Maximal Operators

Abstract

Let $L^{q ( x ) } (\mathbb{R} )$ be variable exponent Lebesgue space and $\ell^{ \{ q_{n} \} }$ be discrete analog of this space. In this work we define the amalgam spaces W(L ^p(x),L ^q(x)) and $W ( L^{p ( x ) },\ell^{ \{ q_{n} \} } )$, and discuss some basic properties of these spaces. Since the global components $L^{q ( x ) } (\mathbb{R} )$ and $\ell^{ \{ q_{n} \} }$ are not translation invariant, these spaces are not a Wiener amalgam space. But we show that there are similar properties of these spaces to the Wiener amalgam spaces. We also show that there is a variable exponent q(x) such that the sequence space $\ell^{ \{ q_{n} \}}$ is the discrete space of $L^{q ( x ) } (\mathbb{R} )$. By using this result we prove that $W ( L^{p ( x ) },\ell ^{ \{ p_{n} \} } ) =L^{p ( x ) } (\mathbb{R} )$. We also study the frame expansion in $L^{p ( x ) } ( \mathbb{R} )$. At the end of this work we prove that the Hardy–Littlewood maximal operator from $W ( L^{s ( x ) },\ell^{ \{t_{n} \} } )$ into $W ( L^{u ( x ) },\ell^{ \{v_{n} \} } )$ is bounded under some assumptions.

This paper is in final form and no version of it will be submitted for publication elsewhere.