Trace inequalities for fractional integrals in grand Lebesgue spaces

Vakhtang Kokilashvili¹ ; Alexander Meskhi²

¹ A. Razmadze Mathematical Institute I. Javakhishvili Tbilisi State University 2, University St. 0186 Tbilisi, Georgia
² A. Razmadze Mathematical Institute I. Javakhishvili Tbilisi State University 2, University St. 0186 Tbilisi, Georgia and Department of Mathematics Faculty of Informatics and Control Systems Georgian Technical University 77, Kostava St. 0175 Tbilisi, Georgia

Studia Mathematica, Tome 210 (2012) no. 2, pp. 159-176

Résumé

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.

Détail

DOI : 10.4064/sm210-2-4

Keywords: rning boundedness fractional maximal potential operators defined quasi metric measure spaces theta theta trace inequality where infty theta satisfies doubling condition results even euclidean spaces example general results adams type necessary sufficient conditions guaranteeing trace inequality fractional maximal functions potentials defined so called s sets mathbb follow trace inequalities one sided potentials strong fractional maximal functions potentials product kernels fractional maximal functions potentials defined half space proved terms adams type criteria finally remark fefferman stein type inequality hardy littlewood maximal functions calder zygmund singular integrals holds grand lebesgue spaces