Boundedness of Positive Integral Operators on Lorentz-Gamma Spaces

We characterize the boundedness of a positive integral operator $T_K$, with kernel $K\in M_{+}({\mathbb{R}}^{2n})$, between Lorentz-Gamma spaces ${\mathrm{\Gamma }}_{p,{\phi }_2}({\mathbb{R}}^n)$ and ${\mathrm{\Gamma }}_{q,{\phi }_1}({\mathbb{R}}^n)$, $1<p\le q<\mathrm{\infty }$. The key step reduces the $n$-dimensional problem to a one-dimensional weighted norm inequality for the composed operator $T_{L}S$, where $L=(K^{{\ast }_2}{)}^{{\ast }_1}$ is the iterated rearrangement of $K$ introduced by Blozinski~\cite{B} and $S$ is the Stieltjes transform. Explicit Muckenhoupt-type conditions are obtained for the case $L(t,s)=(t+s{)}^{-1}$, corresponding to the iterated Stieltjes operator $S^2$.