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 $$ |I_{\alpha}(fw^{\alpha})|{L{w}^{q),\theta(1+\alpha q)}}\leq c|f|{L{w}^{p),\theta}}, $$ where $I_{\alpha}$ is the Riesz potential operator on the interval $[0,1]$, holds if and only if $w\in A_{1+q/p'}$.