Criteria for the Boundedness of Potential Operators in Grand Lebesgue Spaces

TL;DR

This paper investigates the boundedness of fractional integral operators in grand Lebesgue spaces, establishing conditions under which these operators are bounded or unbounded, and characterizing the associated weight functions.

Contribution

It provides new criteria for the boundedness of potential operators in grand Lebesgue spaces and characterizes the weights for which one-weight inequalities hold.

Findings

Fractional integral operators are not bounded between certain grand Lebesgue spaces under specified conditions.

The one-weight inequality for Riesz potentials holds if and only if the weight belongs to the class A_{1+q/p'}.

Abstract

It is shown that that the fractional integral operators with the parameter $\alpha$, $0<\alpha<1$, are not bounded between the generalized grand Lebesgue spaces $L^{p), \theta_1}$ and $L^{q), \theta_2}$ for $\theta_2 < (1+\alpha q)\theta_1$, where $1<p<1/\alpha$ and $q=\frac{p}{1-\alpha p}$. Besides this, it is proved that the one--weight inequality $$ \|I_{\alpha}(fw^{\alpha})\|_{L_{w}^{q),\theta(1+\alpha q)}}\leq c\|f\|_{L_{w}^{p),\theta}}, $$ where $I_{\alpha}$ is the Riesz potential operator on the interval $[0,1]$, holds if and only if $w\in A_{1+q/p'}$.