Criteria for the Boundedness of Potential Operators in Grand Lebesgue Spaces

Abstract

It is shown that that the fractional integral operators with the parameter $α$, $0<α<1$, are not bounded between the generalized grand Lebesgue spaces $L^{p), θ_1}$ and $L^{q), θ_2}$ for $θ_2 < (1+αq)θ_1$, where $1<p<1/α$ and $q=\frac{p}{1-αp}$. Besides this, it is proved that the one--weight inequality $$ \|I_α(fw^α)\|_{L_{w}^{q),θ(1+αq)}}\leq c\|f\|_{L_{w}^{p),θ}}, $$ where $I_α$ is the Riesz potential operator on the interval $[0,1]$, holds if and only if $w\in A_{1+q/p'}$.

15 pages