We consider generalized Morrey type spaces \( {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) \) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets \( \Omega \subset {\mathbb{R}^n} \), we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem \( {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) \) for the potential type operator I α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.
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Translated from Problems in Mathematical Analysis 50, September 2010, pp. 3–20
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Guliyev, V.S., Hasanov, J.J. & Samko, S.G. Boundedness of maximal, potential type, and singular integral operators in the generalized variable exponent Morrey type spaces. J Math Sci 170, 423–443 (2010). https://doi.org/10.1007/s10958-010-0095-7
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DOI: https://doi.org/10.1007/s10958-010-0095-7