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 Eurasian Math. J., 2017, Volume 8, Number 1, Pages 34–49 (Mi emj246)

Embedding relations between weighted complementary local Morrey-type spaces and weighted local Morrey-type spaces

A. Gogatishvilia, R. Mustafayevbc, T. Ünverc

a Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic
b Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan, B. Vahabzade St. 9, Baku, AZ 1141, Azerbaijan
c Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey

Abstract: In this paper embedding relations between weighted complementary local Morrey-type spaces $^cLM_{p\theta,\omega}(\mathbb{R}^n,v)$ and weighted local Morrey-type spaces $LM_{p\theta,\omega}(\mathbb{R}^n,v)$ are characterized. In particular, two-sided estimates of the optimal constant $c$ in the inequality
$$( \int_0^\infty( \int_{B(0,t)} f(x)^{p_2}v_2(x) dx )^{\frac{q_2}{p_2}}u_2(t) dt )^{\frac1{q_2}} \leqslant c (\int_0^\infty(\int_{^cB(0,t)}f(x)^{p_1}v_1(x) dx)^{\frac{q_1}{p_1}}u_1(t) dt)^{\frac1{q_1}},\quad f\geqslant0$$
are obtained, where $p_1$, $p_2$, $q_1$, $q_2\in(0,\infty)$, $p_2\leqslant q_2$ and $u_1$, $u_2$ and $v_1$, $v_2$ are weights on $(0,\infty)$ and $\mathbb{R}^n$, respectively. The proof is based on the combination of the duality techniques with estimates of optimal constants of the embedding relations between weighted local Morrey-type and complementary local Morrey-type spaces and weighted Lebesgue spaces, which allows to reduce the problem to using of the known Hardy-type inequalities.

Keywords and phrases: local Morrey-type spaces, embeddings, iterated Hardy inequalities.

 Funding Agency Grant Number Grantová Agentura České Republiky P201-13-14743SRVO 67985840 Shota Rustaveli National Science Foundation DI/9/5-100/13 The research of A. Gogatishvili was partially supported by the grant P201-13-14743S of the Grant Agency of the Czech Republic and RVO: 67985840 and by Shota Rustaveli National Science Foundation, grant no. DI/9/5-100/13 (Function spaces, weighted inequalities for integral operators and problems of summability of Fourier series).

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Bibliographic databases:
MSC: 46E30, 26D10
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Citation: A. Gogatishvili, R. Mustafayev, T. Ünver, “Embedding relations between weighted complementary local Morrey-type spaces and weighted local Morrey-type spaces”, Eurasian Math. J., 8:1 (2017), 34–49

Citation in format AMSBIB
\Bibitem{GogMusUnv17} \by A.~Gogatishvili, R.~Mustafayev, T.~\"Unver \paper Embedding relations between weighted complementary local Morrey-type spaces and weighted local Morrey-type spaces \jour Eurasian Math. J. \yr 2017 \vol 8 \issue 1 \pages 34--49 \mathnet{http://mi.mathnet.ru/emj246} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000411744800003}