Criteria for the Boundedness of Potential Operators in Grand Lebesgue Spaces

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