Surjectivity Of Convolution Operators Research Articles

Surjectivity of Convolution Operators on Harmonic NA Groups

Let μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} be a radial compactly supported distribution on a harmonic NA group. We prove that the right convolution operator cμ:f↦f∗μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_{\\mu }:f \\mapsto f* \\mu $$\\end{document} maps the space of smooth v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {v}$$\\end{document}-radial functions onto itself if and only if the spherical Fourier transform μ~(λ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{\\mu }(\\lambda )$$\\end{document}, λ∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\in \\mathbb {C}$$\\end{document}, is slowly decreasing. As an application, we prove that certain averages over spheres are surjective on the space of smooth v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {v}$$\\end{document}-radial functions.

Asymptotic Expansions of Finite Hankel Transforms and the Surjectivity of Convolution Operators

A compactly supported distribution is called invertible in the sense of Ehrenpreis-Hörmander if the convolution with it induces a surjection from $\mathcal{C}^{\infty}(\mathbb{R}^{n})$ to itself. We give sufficient conditions for radial functions to be invertible. Our analysis is based on the asymptotic expansions of finite Hankel transforms. The dominant term may be the contribution from the origin or from the boundary of the support of the function. For the proof, we propose a new method to calculate the asymptotic expansions of finite Hankel transforms of functions with singularities at a point other than the origin.

Surjectivity of convolution operators on noncompact symmetric spaces

Let μ be a K-invariant compactly supported distribution on a noncompact Riemannian symmetric space X=G/K. If the spherical Fourier transform μ˜(λ) is slowly decreasing, it is known that the right convolution operator cμ:f↦f⁎μ maps E(X) onto E(X). In this paper, we prove the converse of this result. We also prove that cμ has a fundamental solution if and only if μ˜(λ) is slowly decreasing.

Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type

Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type

Existence of Fundamental Solutions and Surjectivity of Convolution Operators on Classes of Ultra-Differentiable Functions

Existence of Fundamental Solutions and Surjectivity of Convolution Operators on Classes of Ultra-Differentiable Functions