Schur-type Banach modules of integral kernels acting on mixed-norm Lebesgue spaces

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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New Generalizations and Results in Shift-Invariant Subspaces of Mixed-Norm Lebesgue Spaces \({L_{\vec{p}}(\mathbb{R}^d)}\)

Mathematics ◽

10.3390/math9030227 ◽

2021 ◽

Vol 9 (3) ◽

pp. 227 ◽

Cited By ~ 1

Author(s):

Junjian Zhao ◽

Wei-Shih Du ◽

Yasong Chen

Keyword(s):

Invariant Subspace ◽

Type Inequality ◽

Invariant Subspaces ◽

Minkowski Inequality ◽

Hölder Inequality ◽

Stability Theorem ◽

Mixed Norm ◽

Lebesgue Spaces

In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd). We obtain a mixed-norm Hölder inequality, a mixed-norm Minkowski inequality, a mixed-norm convolution inequality, a convolution-Hölder type inequality and a stability theorem to mixed-norm case in the setting of shift-invariant subspace of Lp→(Rd). Our new results unify and refine the existing results in the literature.

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Non-Linear Potential Theory in Lebesgue Spaces with Mixed Norm

Potential Theory ◽

10.1007/978-1-4613-0981-9_40 ◽

1988 ◽

pp. 325-331 ◽

Cited By ~ 1

Author(s):

Tord Sjödin

Keyword(s):

Potential Theory ◽

Mixed Norm ◽

Linear Potential ◽

Lebesgue Spaces ◽

Non Linear ◽

Linear Potential Theory

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Extension of multilinear fractional integral operators to linear operators on mixed-norm Lebesgue spaces

Mathematische Annalen ◽

10.1007/s00208-020-02105-2 ◽

2020 ◽

Author(s):

Ting Chen ◽

Wenchang Sun

Keyword(s):

Fractional Integral ◽

Integral Operators ◽

Linear Operators ◽

Mixed Norm ◽

Lebesgue Spaces ◽

Fractional Integral Operators ◽

Multilinear Fractional Integral

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Convolution Algebraic Structures Defined by Hardy-Type Operators

Journal of Function Spaces and Applications ◽

10.1155/2013/212465 ◽

2013 ◽

Vol 2013 ◽

pp. 1-13

Author(s):

Pedro J. Miana ◽

Juan J. Royo ◽

Luis Sánchez-Lajusticia

Keyword(s):

Sobolev Spaces ◽

Banach Algebras ◽

Weighted Sobolev Spaces ◽

Integral Kernel ◽

Lebesgue Spaces ◽

Hardy Type ◽

Weighted Lebesgue Spaces ◽

Banach Modules ◽

Hardy Type Operator ◽

Fractional Derivation

The main aim of this paper is to show that certain Banach spaces, defined via integral kernel operators, are Banach modules (with respect to some known Banach algebras and convolution products onℝ+). To do this, we consider some suitable kernels such that the Hardy-type operator is bounded in weighted Lebesgue spacesLωpℝ+forp≥1. We also show new inequalities in these weighted Lebesgue spaces. These results are applied to several concrete function spaces, for example, weighted Sobolev spaces and fractional Sobolev spaces defined by Weyl fractional derivation.

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