Maximal Regularity and the Method of Fourier Multipliers

2014 ◽  
pp. 19-45
Author(s):  
Ravi P. Agarwal ◽  
Claudio Cuevas ◽  
Carlos Lizama
Keyword(s):  
Maximal Regularity ◽  
Fourier Multipliers

scholarly journals $L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Carlos Lizama ◽  
Marina Murillo-Arcila
Keyword(s):  
Acoustic Waves ◽  
Maximal Regularity ◽  
Bubble Radius ◽  
Fourier Multipliers ◽  
Cylindrical Domain ◽  
Lebesgue Spaces ◽  
Regularity Problem ◽  
Bubbly Liquids

Abstract We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden–Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden–Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers.

Hp-Maximal Regularity and Operator Valued Multipliers on Hardy Spaces

2007 ◽  
Vol 59 (6) ◽  
pp. 1207-1222 ◽  
Author(s):  
Shangquan Bu ◽  
Christian Le Merdy
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Hardy Spaces ◽  
Maximal Regularity ◽  
Type Theorem ◽  
Closed Operator ◽  
Fourier Multipliers ◽  
The Cauchy Problem

AbstractWe consider maximal regularity in the Hp sense for the Cauchy problem u′(t) + Au(t) = f(t) (t ∈ ℝ), where A is a closed operator on a Banach space X and f is an X-valued function defined on ℝ. We prove that if X is an AUMD Banach space, then A satisfies Hp-maximal regularity if and only if A is Rademacher sectorial of type < . Moreover we find an operator A with Hp-maximal regularity that does not have the classical Lp-maximal regularity. We prove a related Mikhlin type theorem for operator valued Fourier multipliers on Hardy spaces Hp(ℝ X), in the case when X is an AUMD Banach space.

scholarly journals Lp-Lq-Well Posedness for the Moore–Gibson–Thompson Equation with Two Temperatures on Cylindrical Domains

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1748
Author(s):  
Carlos Lizama ◽  
Marina Murillo-Arcila
Keyword(s):  
Cauchy Problem ◽  
Sound Propagation ◽  
Maximal Regularity ◽  
Fourier Multipliers ◽  
Well Posedness ◽  
Elastic Media ◽  
Regularity Estimates ◽  
The Cauchy Problem ◽  
Cylindrical Domains ◽  
Two Temperatures

We examine the Cauchy problem for a model of linear acoustics, called the Moore–Gibson–Thompson equation, describing a sound propagation in thermo-viscous elastic media with two temperatures on cylindrical domains. For an adequate combination of the parameters of the model we prove Lp-Lq-well-posedness, and we provide maximal regularity estimates which are optimal thanks to the theory of operator-valued Fourier multipliers.

scholarly journals Maximal Regularity for Flexible Structural Systems in Lebesgue Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Claudio Fernández ◽  
Carlos Lizama ◽  
Verónica Poblete
Keyword(s):  
Maximal Regularity ◽  
Flexible Structures ◽  
Fourier Multipliers ◽  
Smooth Domain ◽  
Structural Systems ◽  
Material Damping ◽  
Lebesgue Spaces ◽  
Dirichlet Laplacian ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

We study abstract equations of the formλu′′′(t)+u′′(t)=c2Au(t)+c2μAu′(t)+f(t),0<λ<μwhich is motivated by the study of vibrations of flexible structures possessing internal material damping. We introduce the notion of(α;β;γ)-regularized families, which is a particular case of(a;k)-regularized families, and characterize maximal regularity inLp-spaces based on the technique of Fourier multipliers. Finally, an application with the Dirichlet-Laplacian in a bounded smooth domain is given.

scholarly journals Fourier multipliers for Hölder continuous functions and maximal regularity

2004 ◽  
Vol 160 (1) ◽  
pp. 23-51 ◽  
Author(s):  
Wolfgang Arendt ◽  
Charles Batty ◽  
Shangquan Bu
Keyword(s):  
Maximal Regularity ◽  
Continuous Functions ◽  
Fourier Multipliers ◽  
Hölder Continuous ◽  
Hölder Continuous Functions

scholarly journals OPERATOR-VALUED FOURIER MULTIPLIERS ON PERIODIC BESOV SPACES AND APPLICATIONS

2004 ◽  
Vol 47 (1) ◽  
pp. 15-33 ◽  
Author(s):  
Wolfgang Arendt ◽  
Shangquan Bu
Keyword(s):  
Banach Space ◽  
Besov Space ◽  
Real Line ◽  
Besov Spaces ◽  
Fourier Multiplier ◽  
Maximal Regularity ◽  
Fourier Multipliers ◽  
Subject Classification ◽  
Cauchy Problems ◽  
Multiplier Theorem

AbstractLet $1\leq p,q\leq\infty$, $s\in\mathbb{R}$ and let $X$ be a Banach space. We show that the analogue of Marcinkiewicz’s Fourier multiplier theorem on $L^p(\mathbb{T})$ holds for the Besov space $B_{p,q}^s(\mathbb{T};X)$ if and only if $1\ltp\lt\infty$ and $X$ is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann’s result (Math. Nachr.186 (1997), 5–56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.AMS 2000 Mathematics subject classification: Primary 47D06; 42A45

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 919-936 ◽  
Author(s):  
Jiao Chen ◽  
Wei Ding ◽  
Guozhen Lu
Keyword(s):  
Hardy Spaces ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Multipliers ◽  
Regularity Of Solutions ◽  
Local Version ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Local Hardy Spaces ◽  
The One

AbstractAfter the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are {L^{p}({\mathbb{R}^{n}})} bounded for {1<p<\infty}, but only bounded on local Hardy spaces {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for {0<p\leq 1}. Though much work has been done on the {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {1<p<\infty} and Hardy {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of {0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

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