A lower estimate for weak-type Fourier multipliers

AbstractWe study sharp weak-type inequalities for a wide class of Fourier multipliers resulting from modulation of the jumps of Lévy processes. In particular, we obtain optimal estimates for second-order Riesz transforms, which lead to interesting a priori bounds for smooth functions on ℝd. The proofs rest on probabilistic methods: we deduce the above inequalities from the corresponding estimates for martingales. To obtain the lower bounds, we exploit the properties of laminates, important probability measures on the space of matrices of dimension 2×2, and some transference-type arguments.

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On restricted weak-type constants of Fourier multipliers

Publicacions Matemàtiques ◽

10.5565/publmat_58214_21 ◽

2014 ◽

Vol 58 ◽

pp. 415-443 ◽

Cited By ~ 1

Author(s):

A. Osękowski

Keyword(s):

Weak Type ◽

Fourier Multipliers

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Restriction results for multilinear multipliers in weighted settings

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210513000164 ◽

2015 ◽

Vol 145 (2) ◽

pp. 391-409

Author(s):

Salvador Rodríguez-López

Keyword(s):

Weak Type ◽

Fourier Multipliers ◽

Maximal Operators ◽

Lebesgue Spaces ◽

Weighted Lebesgue Spaces

We obtain restriction results of de Leeuw’s type for maximal operators defined through multilinear Fourier multipliers of either strong or weak type acting on weighted Lebesgue spaces. We give some applications of our development. In particular, we obtain periodic weighted results for Coifman–Meyer-, Hörmander- and Hörmander–Mihlin-type multilinear multipliers.

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Sharp Inequalities for the Haar System and Fourier Multipliers

Journal of Function Spaces and Applications ◽

10.1155/2013/646012 ◽

2013 ◽

Vol 2013 ◽

pp. 1-14

Author(s):

Adam Osȩkowski

Keyword(s):

Continuous Time ◽

Classical Result ◽

Weak Type ◽

Haar System ◽

Borel Subset ◽

General Setting ◽

Universal Constant ◽

Fourier Multipliers ◽

Arbitrary Subset ◽

Best Constant

A classical result of Paley and Marcinkiewicz asserts that the Haar systemh=hkk≥0on0,1forms an unconditional basis ofLp0,1provided1<p<∞. That is, if𝒫Jdenotes the projection onto the subspace generated byhjj∈J(Jis an arbitrary subset ofℕ), then𝒫JLp0,1→Lp0,1≤βpfor some universal constantβpdepending only onp. The purpose of this paper is to study related restricted weak-type bounds for the projections𝒫J. Specifically, for any1≤p<∞we identify the best constantCpsuch that𝒫JχALp,∞0,1≤CpχALp0,1for everyJ⊆ℕand any Borel subsetAof0,1. In fact, we prove this result in the more general setting of continuous-time martingales. As an application, a related estimate for a large class of Fourier multipliers is established.

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

Advances in Difference Equations ◽

10.1186/s13662-020-03054-5 ◽

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.

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

Forum Mathematicum ◽

10.1515/forum-2019-0319 ◽

2020 ◽

Vol 32 (4) ◽

pp. 919-936 ◽

Cited By ~ 2

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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Sharp Weak-Type (p, p) Estimates ($$1

Mediterranean Journal of Mathematics ◽

10.1007/s00009-020-01655-8 ◽

2021 ◽

Vol 18 (2) ◽

Author(s):

Adam Osękowski

Keyword(s):

Weak Type

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