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ЖУРНАЛЫ // Eurasian Mathematical Journal // Архив

Eurasian Math. J., 2017, том 8, номер 1, страницы 34–49 (Mi emj246)

Эта публикация цитируется в 2 статьях

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

Аннотация: 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
$$
\left( \int_0^\infty\left( \int_{B(0,t)} f(x)^{p_2}v_2(x)\,dx \right)^{\frac{q_2}{p_2}}u_2(t)\,dt \right)^{\frac1{q_2}} \leqslant c \left(\int_0^\infty\left(\int_{^cB(0,t)}f(x)^{p_1}v_1(x)\,dx\right)^{\frac{q_1}{p_1}}u_1(t)\,dt\right)^{\frac1{q_1}},\quad f\geqslant0
$$
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.

Ключевые слова и фразы: local Morrey-type spaces, embeddings, iterated Hardy inequalities.

MSC: 46E30, 26D10

Поступила в редакцию: 06.12.2016

Язык публикации: английский



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