Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey spaces
We show the boundedness of the Riesz-type potential operator of variable order $\beta (x)$ from the variable exponent Herz – Morrey spaces $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_1 ,q_1 (\cdot )}(\mathbb{R}^n)$ into the weighted space $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_2 ,q_2 (\cdot )}(\mathb...
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| Date: | 2017 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2017
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1769 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | We show the boundedness of the Riesz-type potential operator of variable order $\beta (x)$ from the variable exponent Herz –
Morrey spaces $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_1 ,q_1 (\cdot )}(\mathbb{R}^n)$ into the weighted space $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_2 ,q_2 (\cdot )}(\mathbb{R}^n, \omega )$, where
$\alpha (x) \in L^{\infty} (\mathbb{R}^n) is log-Holder continuous both at the origin and at infinity, $\omega = (1+| x| ) \gamma (x)$ with some $\gamma (x) > 0$, and $1/q_1 (x) 1/q_2 (x) = \beta (x)/n$
when $q_1 (x)$ is not necessarily constant at infinity. It is assumed that the exponent $q_1 (x)$ satisfies the logarithmic continuity
condition both locally and at infinity and $1 < (q_1)_{\infty} \leq q_1(x) \leq (q_1)_+ < \infty, \;x \in \mathbb{R}$. |
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