scholarly journals Calderón Operator on Local Morrey Spaces with Variable Exponents

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2977
Author(s):  
Kwok-Pun Ho
Keyword(s):  
Hölder Continuity ◽  
Morrey Spaces ◽  
Variable Exponents ◽  
Holder Continuity ◽  
Averaging Operators ◽  
Special Cases ◽  
Continuity Assumption ◽  
Hardy’S Inequalities ◽  
Extrapolation Theory ◽  
Local Morrey Spaces

In this paper, we establish the boundedness of the Calderón operator on local Morrey spaces with variable exponents. We obtain our result by extending the extrapolation theory of Rubio de Francia to the local Morrey spaces with variable exponents. The exponent functions of the local Morrey spaces with the exponent functions are only required to satisfy the log-Hölder continuity assumption at the origin and infinity only. As special cases of the main result, we have Hardy’s inequalities, the Hilbert inequalities and the boundedness of the Riemann–Liouville and Weyl averaging operators on local Morrey spaces with variable exponents.

scholarly journals Global higher integrability for minimisers of convex functionals with (p,q)-growth

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Lukas Koch
Keyword(s):  
Hölder Continuity ◽  
Growth Conditions ◽  
Holder Continuity ◽  
Higher Integrability ◽  
Convex Functionals ◽  
Global Higher Integrability ◽  
Continuity Assumption

AbstractWe prove global $$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) -regularity for minimisers of convex functionals of the form $${\mathscr {F}}(u)=\int _{\varOmega } F(x,Du)\,{\mathrm{d}}x$$ F ( u ) = ∫ Ω F ( x , D u ) d x .$$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform $$\alpha $$ α -Hölder continuity assumption in x and controlled (p, q)-growth conditions in z with $$q<\frac{(n+\alpha )p}{n}$$ q < ( n + α ) p n .

scholarly journals Global Hölder continuity of solutions to quasilinear equations with Morrey data

2021 ◽  
pp. 2150062
Author(s):  
Sun-Sig Byun ◽  
Dian K. Palagachev ◽  
Pilsoo Shin
Keyword(s):  
Hölder Continuity ◽  
Morrey Spaces ◽  
Regular Boundary ◽  
Holder Continuity ◽  
Density Condition ◽  
Classical Formula ◽  
Quasilinear Equations ◽  
Growth Structure ◽  
Carathéodory Functions ◽  
Global Boundedness

We deal with general quasilinear divergence-form coercive operators whose prototype is the [Formula: see text]-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property. We prove global boundedness and Hölder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical [Formula: see text]-result of Ladyzhenskaya and Ural’tseva to the settings of the Morrey spaces.

scholarly journals Hölder continuity of singular parabolic equations with variable nonlinearity

2020 ◽  
Vol 28 (3) ◽  
pp. 51-82
Author(s):  
Hamid El Bahja
Keyword(s):  
Iterative Method ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Hölder Continuity ◽  
Regularity Result ◽  
Hölder Regularity ◽  
Variable Exponents ◽  
Holder Continuity ◽  
Singular Parabolic Equations ◽  
Holder Regularity

AbstractIn this paper we obtain the local Hölder regularity of the weak solutions for singular parabolic equations with variable exponents. The proof is based on DiBenedetto’s technique called intrinsic scaling; by choosing an appropriate geometry one can deduce energy and logarithmic estimates from which one can implement an iterative method to obtain the regularity result.

Extrapolation to weighted Morrey spaces with variable exponents and applications

2021 ◽  
pp. 1-26
Author(s):  
Kwok-Pun Ho
Keyword(s):  
Differential Operators ◽  
Integral Operators ◽  
Fourier Integral ◽  
Morrey Spaces ◽  
Fourier Integral Operators ◽  
Variable Exponents ◽  
Weighted Morrey Spaces ◽  
Pseudo Differential Operators ◽  
Extrapolation Theory

Abstract This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

scholarly journals Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces

2016 ◽  
Vol 14 (1) ◽  
pp. 300-323 ◽  
Author(s):  
Ferit Gurbuz
Keyword(s):  
Large Class ◽  
Morrey Spaces ◽  
Rough Kernel ◽  
Campanato Space ◽  
Marcinkiewicz Operator ◽  
Sublinear Operators ◽  
Special Cases ◽  
Local Morrey Spaces

AbstractIn this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of a large class of parabolic rough operators on them. The author also establishes the parabolic local Campanato space estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respectively. At last, parabolic Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.

scholarly journals Extrapolation to mixed norm spaces and applications

2021 ◽  
Vol 25 (2) ◽  
pp. 281-296
Author(s):  
Kwok-Pun Ho
Keyword(s):  
Lorentz Spaces ◽  
Maximal Functions ◽  
Mixed Norm ◽  
Variable Exponents ◽  
Lebesgue Spaces ◽  
Mixed Norm Spaces ◽  
Special Cases ◽  
Extrapolation Theory

This paper establishes extrapolation theory to mixed norm spaces. By applying this extrapolation theory, we obtain the mapping properties of the Rubio de Francia Littlewood-Paley functions and the geometrical maximal functions on mixed norm spaces. As special cases of these results, we have the mapping properties on the mixed norm Lebesgue spaces with variable exponents and the mixed norm Lorentz spaces.

On nonhomeomorphic mappings with the inverse Poletsky inequality

2020 ◽  
Vol 17 (3) ◽  
pp. 414-436
Author(s):  
Evgeny Sevost'yanov ◽  
Serhii Skvortsov ◽  
Oleksandr Dovhopiatyi
Keyword(s):  
Boundary Behavior ◽  
Hölder Continuity ◽  
Continuous Extension ◽  
Quasiconformal Mappings ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion ◽  
Lipschitz Continuous ◽  
Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

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