Criteria for the Boundedness of Potential Operators in Grand Lebesgue Spaces

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 $$\Vert I_{\alpha }(f{w}^{\alpha }){\Vert }_{L_w^{q),\theta (1+\alpha q)}}\le c\Vert f{\Vert }_{L_w^{p),\theta }},$$ where $I_{\alpha }$ is the Riesz potential operator on the interval $[0,1]$, holds if and only if $w\in A_{1+q/p^{\prime }}$.

15 pages