Boundedness of Positive Integral Operators on Lorentz-Gamma Spaces

TL;DR

This paper characterizes when positive integral operators are bounded between Lorentz-Gamma spaces, reducing the problem to a one-dimensional weighted inequality and providing explicit Muckenhoupt-type conditions.

Contribution

It introduces a reduction technique for boundedness problems to one-dimensional inequalities and derives explicit conditions for specific kernels in Lorentz-Gamma spaces.

Findings

Reduced n-dimensional boundedness to one-dimensional weighted inequalities.

Derived explicit Muckenhoupt-type conditions for specific kernels.

Characterized boundedness of integral operators on Lorentz-Gamma spaces.

Abstract

We characterize the boundedness of a positive integral operator $T_K$, with kernel $K\in M_+(\R^{2n})$, between Lorentz-Gamma spaces $\Gamma_{p,\phi_2}(\R^n)$ and $\Gamma_{q,\phi_1}(\R^n)$, $1<p\le q<\infty$. 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 $S^2$.