Pseudo-Differential Operators in Algebras of Generalized Functions and Global Hypoellipticity

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.

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$$L^p$$-Bounds for Pseudo-differential Operators on Graded Lie Groups

Journal of Geometric Analysis ◽

10.1007/s12220-021-00694-1 ◽

2021 ◽

Author(s):

Duván Cardona ◽

Julio Delgado ◽

Michael Ruzhansky

Keyword(s):

Lie Groups ◽

Differential Operators ◽

Pseudo Differential Operators

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Minimal and maximal extensions of M-hypoelliptic proper uniform pseudo-differential operators in $$L^p$$-spaces on non-compact manifolds

Journal of Pseudo-Differential Operators and Applications ◽

10.1007/s11868-021-00393-z ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Ognjen Milatovic

Keyword(s):

Differential Operators ◽

Compact Manifolds ◽

Pseudo Differential Operators

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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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