Boundedness of Positive Integral Operators on Lorentz-Gamma Spaces

We characterize the boundedness of a positive integral operator $T_K$, with kernel $K\in M_+(\R^{2n})$, between Lorentz-Gamma spaces $Γ_{p,ϕ_2}(\R^n)$ and $Γ_{q,ϕ_1}(\R^n)$, $1<p\le q<\infty$. The key step reduces the $n$-dimensional problem to a one-dimensional weighted norm inequality for the composed operator $T_LS$, where $L=(K^{*_2})^{*_1}$ 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 $S^2$.