scholarly journals Weak estimates for the maximal and Riesz potential operators on non-homogeneous central Morrey type spaces in L^1 over metric measure spaces

2020 ◽  
Vol 45 (2) ◽  
pp. 1187-1207
Author(s):  
Katsuo Matsuoka ◽  
Yoshihiro Mizuta ◽  
Tetsu Shimomura
Keyword(s):  
Riesz Potential ◽  
Metric Measure Spaces ◽  
Potential Operators ◽  
Measure Spaces ◽  
Type Spaces

Commutators of sublinear operators in grand morrey spaces

2019 ◽  
Vol 56 (2) ◽  
pp. 211-232
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi ◽  
Humberto Rafeiro
Keyword(s):  
Integral Operators ◽  
Morrey Spaces ◽  
Doubling Measure ◽  
Maximal Operators ◽  
Metric Measure Spaces ◽  
Potential Operators ◽  
Sublinear Operators ◽  
Measure Spaces

Abstract In this paper we establish the boundedness of commutators of sublinear operators in weighted grand Morrey spaces. The sublinear operators under consideration contain integral operators such as Hardy-Littlewood and fractional maximal operators, Calderón-Zygmund operators, potential operators etc. The operators and spaces are defined on quasi-metric measure spaces with doubling measure.

scholarly journals Newtonian Spaces Based on Quasi-Banach Function Lattices

2016 ◽  
Vol 119 (1) ◽  
pp. 133 ◽  
Author(s):  
Lukáš Malý
Keyword(s):  
Absolute Continuity ◽  
General Setting ◽  
Sobolev Type ◽  
Metric Measure Spaces ◽  
First Order ◽  
Upper Gradient ◽  
Measure Spaces ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
Sobolev Capacity

In this paper, first-order Sobolev-type spaces on abstract metric measure spaces are defined using the notion of (weak) upper gradients, where the summability of a function and its upper gradient is measured by the "norm" of a quasi-Banach function lattice. This approach gives rise to so-called Newtonian spaces. Tools such as moduli of curve families and Sobolev capacity are developed, which allows us to study basic properties of these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are established.

scholarly journals SPACES OF LIPSCHITZ TYPE ON METRIC SPACES AND THEIR APPLICATIONS

2004 ◽  
Vol 47 (3) ◽  
pp. 709-752 ◽  
Author(s):  
Dachun Yang ◽  
Yong Lin
Keyword(s):  
Sobolev Spaces ◽  
Riemannian Manifolds ◽  
Metric Spaces ◽  
Sobolev Embedding ◽  
Metric Measure Spaces ◽  
Spaces Of Homogeneous Type ◽  
Sobolev Embedding Theorem ◽  
Measure Spaces ◽  
Type Spaces ◽  
Self Similar

AbstractNew spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.AMS 2000 Mathematics subject classification: Primary 42B35. Secondary 46E35; 58J35; 43A99

scholarly journals Lower estimates of heat kernels for non-local Dirichlet forms on metric measure spaces

2017 ◽  
Vol 272 (8) ◽  
pp. 3311-3346 ◽  
Author(s):  
Alexander Grigor'yan ◽  
Eryan Hu ◽  
Jiaxin Hu
Keyword(s):  
Dirichlet Forms ◽  
Heat Kernels ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Non Local

Trace inequalities for fractional integrals in mixed norm grand lebesgue spaces

2020 ◽  
Vol 23 (5) ◽  
pp. 1452-1471
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi
Keyword(s):  
Riesz Potential ◽  
Fractional Integrals ◽  
Mixed Norm ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Grand Lebesgue Spaces ◽  
Measure Spaces ◽  
Necessary And Sufficient ◽  
Bounded Sets ◽  
Trace Inequalities

Abstract D. Adams type trace inequalities for multiple fractional integral operators in grand Lebesgue spaces with mixed norms are established. Operators under consideration contain multiple fractional integrals defined on the product of quasi-metric measure spaces, and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously necessary and sufficient for appropriate trace inequalities. The derived results are new even for multiple Riesz potential operators defined on the product of Euclidean spaces.

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