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## Trace inequalities for fractional integrals in grand Lebesgue spaces

### Tom 210 / 2012

Studia Mathematica 210 (2012), 159-176 MSC: Primary 46E30; Secondary 31E05, 42B25. DOI: 10.4064/sm210-2-4

#### Streszczenie

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.

#### Autorzy

• Vakhtang KokilashviliA. Razmadze Mathematical Institute
I. Javakhishvili Tbilisi State University
2, University St.
0186 Tbilisi, Georgia
e-mail
• Alexander MeskhiA. 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
e-mail

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