Fractional, Maximal and Singular Operators in Variable Exponent Lorentz Spaces

TL;DR

This paper introduces variable exponent Lorentz spaces and proves the boundedness of singular, fractional, and ergodic operators within them, relaxing traditional local log-condition requirements on exponents.

Contribution

It demonstrates that operator boundedness in these spaces is achievable under decay conditions on exponents, advancing the theory beyond existing local log-condition constraints.

Findings

Boundedness of singular integral operators established in variable exponent Lorentz spaces.

Operators are bounded without the usual local log-condition on exponents.

Decay conditions on exponents suffice for boundedness, based on Hardy inequalities.

Abstract

We introduce the Lorentz space $\mathcal{L}^{p(\cdot), q(\cdot)}$ with variable exponents $p(t),q(t)$ and prove the boundedness of singular integral and fractional type operators, and corresponding ergodic operators in these spaces. The main goal of the paper is to show that the boundedness of these operators in the spaces $\mathcal{L}^{p(\cdot), q(\cdot)}$ is possible without the local log-condition on the exponents, typical for the variable exponent Lebesgue spaces; instead the exponents $p(s)$ and $q(s)$ should only satisfy decay conditions of log-type as $s\to 0$ and $s\to\infty$. To prove this, we base ourselves on the recent progress in the problem of the validity of Hardy inequalities in variable exponent Lebesgue spaces.