Finite mean oscillation and the Beltrami equation

2006 ◽  
Vol 153 (1) ◽  
pp. 247-266 ◽  
Author(s):  
Vladimir Ryazanov ◽  
Uri Srebro ◽  
Eduard Yakubov
Keyword(s):  
Beltrami Equation ◽  
Mean Oscillation ◽  
Finite Mean Oscillation

scholarly journals On boundary extension of one class of mappings in terms of prime ends

2020 ◽  
Vol 53 (1) ◽  
pp. 29-40
Author(s):  
E.A. Sevost'yanov ◽  
S. A. Skvortsov ◽  
I.A. Sverchevska
Keyword(s):  
Metric Spaces ◽  
Quasiconformal Mappings ◽  
Boundary Extension ◽  
Finite Distortion ◽  
Mean Oscillation ◽  
Prime Ends ◽  
Finite Mean Oscillation

Here we consider the classes of mappings of metric spaces that distort the modulus of families of paths similarly to Poletsky inequality. For domains, which are not locally connected at the boundaries, we obtain results on the boundary extension of the indicated mappings. We also investigate the local and global behaviorof mappings in the context of the equicontinuity of their families. The main statements of the article are proved under the condition that the majorant responsible for the distortion of the modulus of the families of paths has a finite mean oscillation at the corresponding points. The results are applicable to well-known classes of conformal and quasiconformal mappings as well as mappings with a finite distortion.

Finite mean oscillation in upper regular metric spaces

2017 ◽  
Vol 38 (2) ◽  
pp. 206-212
Author(s):  
E. Afanas’eva ◽  
A. Golberg ◽  
R. Salimov
Keyword(s):  
Metric Spaces ◽  
Mean Oscillation ◽  
Finite Mean Oscillation

scholarly journals Some estimates for the commutators of multilinear maximal function on Morrey-type space

2021 ◽  
Vol 19 (1) ◽  
pp. 515-530
Author(s):  
Xiao Yu ◽  
Pu Zhang ◽  
Hongliang Li
Keyword(s):  
Maximal Function ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Bounded Mean Oscillation ◽  
Generalized Morrey Spaces ◽  
Type Space ◽  
Bmo Function ◽  
Mean Oscillation ◽  
Multilinear Maximal Function ◽  
Equivalent Conditions

Abstract In this paper, we study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.

Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz boundary

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Roland Duduchava
Keyword(s):  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Beltrami Equation ◽  
Bessel Potential ◽  
Convolution Operators ◽  
Classical Setting ◽  
Type Boundary ◽  
Mellin Convolution ◽  
Bessel Potential Spaces

AbstractThe purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 in the non-classical setting when solutions are sought in the Bessel potential spaces \mathbb{H}^{s}_{p}(\mathcal{C}), \frac{1}{p}<s<1+\frac{1}{p}, 1<p<\infty. Fredholm criteria and unique solvability criteria are found. By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain \Omega_{\alpha}\subset\mathbb{R}^{2} of magnitude 𝛼. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J.27 (2020), 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2\times 2 systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes \mathbb{R}^{+} in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys.60 (2013), 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl.443 (2016), 2, 707–731].

scholarly journals EXPLICIT INTERPOLATION BOUNDS BETWEEN HARDY SPACE AND

2013 ◽  
Vol 95 (2) ◽  
pp. 158-168
Author(s):  
H.-Q. BUI ◽  
R. S. LAUGESEN
Keyword(s):  
Growth Rate ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Mean Oscillation ◽  
Complex Interpolation ◽  
Bounded Linear ◽  
Exponential Bound ◽  
Mean Oscillation ◽  
Good Lambda Inequality ◽  
Interpolation Result

AbstractEvery bounded linear operator that maps ${H}^{1} $ to ${L}^{1} $ and ${L}^{2} $ to ${L}^{2} $ is bounded from ${L}^{p} $ to ${L}^{p} $ for each $p\in (1, 2)$, by a famous interpolation result of Fefferman and Stein. We prove ${L}^{p} $-norm bounds that grow like $O(1/ (p- 1))$ as $p\downarrow 1$. This growth rate is optimal, and improves significantly on the previously known exponential bound $O({2}^{1/ (p- 1)} )$. For $p\in (2, \infty )$, we prove explicit ${L}^{p} $ estimates on each bounded linear operator mapping ${L}^{\infty } $ to bounded mean oscillation ($\mathit{BMO}$) and ${L}^{2} $ to ${L}^{2} $. This $\mathit{BMO}$ interpolation result implies the ${H}^{1} $ result above, by duality. In addition, we obtain stronger results by working with dyadic ${H}^{1} $ and dyadic $\mathit{BMO}$. The proofs proceed by complex interpolation, after we develop an optimal dyadic ‘good lambda’ inequality for the dyadic $\sharp $-maximal operator.

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