Multilinear Stockwell transforms

2018 ◽  
Vol 13 (4) ◽  
pp. 35
Author(s):  
Viorel Catană
Keyword(s):  
Weight Function ◽  
Uncertainty Principle ◽  
Differential Operators ◽  
Tempered Distributions ◽  
Pseudo Differential Operators ◽  
Schwartz Class ◽  
Convolution Products ◽  
Decreasing Functions

The main aim of this paper is to introduce multilinear versions of the Stockwell transforms (also named S-transforms) by using the fact that S-transforms can be written as convolution products. Further on we extend the multilinear S-transforms from the Schwartz class of rapidly decreasing functions to the space of tempered distributions. In the sequel we give a relation between multilinear S-transforms and multilinear pseudo-differential operators. We also state and prove some boundedness results regarding multilinear S-transforms on the Lebegue’s spaces Lp(Rn) and also on the Hörmander’s spaces Bp,k(Rn), where p ≥ 1 and k is a temperate weight function. In the end, a weak uncertainty principle for multilinear S-transforms and for its adjoint is also given.

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

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].

scholarly journals On the Weyl transform for rotationally invariant symbols

2019 ◽  
Vol 10 (4) ◽  
pp. 769-791 ◽  
Author(s):  
Norbert Ortner ◽  
Peter Wagner
Keyword(s):  
Asymptotic Behavior ◽  
Differential Operators ◽  
Symmetric Distributions ◽  
Yield Criteria ◽  
Weyl Transform ◽  
Pseudo Differential Operators ◽  
Weyl Transforms ◽  
Rotationally Invariant

Abstract Several formulas for the eigenvalues $$\lambda _j$$ λ j of the Weyl transforms $$W_\sigma $$ W σ of symbols $$\sigma $$ σ given by radially symmetric distributions are derived. These yield criteria for the boundedness and the compactness, respectively, of the pseudo-differential operators $$W_\sigma .$$ W σ . We investigate some examples by analyzing the asymptotic behavior of $$\lambda _j$$ λ j for $$j\rightarrow \infty $$ j → ∞ .

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