Trace inequalities for fractional integrals in grand Lebesgue spaces
Volume 210 / 2012
Abstract
rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from $L^{p), \theta} (X, \mu)$ to $L^{q), q\theta/p}(X, \nu)$ (trace inequality), where $1< p< q< \infty$, $\theta> 0$ and $\mu$ satisfies the doubling condition in $X$. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called $s$-sets in ${\mathbb{R}}^n$ follow. Trace inequalities for one-sided potentials, strong fractional maximal functions and potentials with product kernels, fractional maximal functions and potentials defined on the half-space are also proved in terms of Adams-type criteria. Finally, we remark that a Fefferman–Stein-type inequality for Hardy–Littlewood maximal functions and Calderón–Zygmund singular integrals holds in grand Lebesgue spaces.
