scholarly journals Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Yoshihiro Sawano ◽  
Tetsu Shimomura
Keyword(s):  
Variable Exponent ◽  
Morrey Spaces ◽  
Doubling Measure ◽  
Riesz Potentials ◽  
Sobolev’S Inequality ◽  
Generalized Morrey Spaces ◽  
Measure Spaces

Sobolev’s Inequality for Riesz Potentials of Functions in Musielak–Orlicz–Morrey Spaces Over Non-doubling Metric Measure Spaces

2019 ◽  
Vol 63 (2) ◽  
pp. 287-303
Author(s):  
Takao Ohno ◽  
Tetsu Shimomura
Keyword(s):  
Morrey Spaces ◽  
Variable Exponents ◽  
Metric Measure Spaces ◽  
Riesz Potentials ◽  
Sobolev’S Inequality ◽  
Double Phase ◽  
Measure Spaces

AbstractOur aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $I_{\unicode[STIX]{x1D6FC}(\,\cdot \,),\unicode[STIX]{x1D70F}}f$ of order $\unicode[STIX]{x1D6FC}(\,\cdot \,)$ with $f\in L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705},\unicode[STIX]{x1D703}}(X)$ over bounded non-doubling metric measure spaces. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

SOBOLEV’S INEQUALITY FOR MUSIELAK–ORLICZ–MORREY SPACES OVER METRIC MEASURE SPACES

2019 ◽  
pp. 1-15
Author(s):  
TAKAO OHNO ◽  
TETSU SHIMOMURA
Keyword(s):  
Morrey Spaces ◽  
Variable Exponents ◽  
Metric Measure Spaces ◽  
Riesz Potentials ◽  
Sobolev’S Inequality ◽  
Double Phase ◽  
Measure Spaces

Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $J_{\unicode[STIX]{x1D6FC}(\cdot )}^{\unicode[STIX]{x1D70E}}f$ of functions $f$ in Musielak–Orlicz–Morrey spaces $L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705}}(X)$ . As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

Generalized Riesz potentials of functions in Morrey spaces L(1,ϕ;κ)(G) over non-doubling measure spaces

2020 ◽  
Vol 32 (2) ◽  
pp. 339-359 ◽  
Author(s):  
Yoshihiro Sawano ◽  
Masaki Shigematsu ◽  
Tetsu Shimomura
Keyword(s):  
Measure Space ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Doubling Measure ◽  
Doubling Condition ◽  
Riesz Potentials ◽  
General Measure ◽  
Open Set ◽  
Measure Spaces

AbstractThis paper proves the boundedness of the generalized Riesz potentials {I_{\rho,\mu,\tau}f} of functions in the Morrey space {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is {X)}, as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on {x\in X}. An example in the end of this article convincingly explains why the modification parameter τ must be introduced.

Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces

2021 ◽  
Vol 24 (6) ◽  
pp. 1643-1669
Author(s):  
Natasha Samko
Keyword(s):  
Upper Bound ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Target Space ◽  
Metric Measure Spaces ◽  
Generalized Morrey Spaces ◽  
Hardy Type ◽  
Measure Spaces ◽  
Hardy Operators ◽  
Hardy Type Operators

Abstract We study commutators of weighted fractional Hardy-type operators within the frameworks of local generalized Morrey spaces over quasi-metric measure spaces for a certain class of “radial” weights. Quasi-metric measure spaces may include, in particular, sets of fractional dimentsions. We prove theorems on the boundedness of commutators with CMO coefficients of these operators. Given a domain Morrey space 𝓛 p,φ (X) for the fractional Hardy operators or their commutators, we pay a special attention to the study of the range of the exponent q of the target space 𝓛 q,ψ (X). In particular, in the case of classical Morrey spaces, we provide the upper bound of this range which is greater than the known Adams exponent.

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