首页/论文/2005.04113

arXiv 2005.041132020-05-08math.FAmath.RT

Surjectivity of Convolution Operators on Noncompact Symmetric Spaces

Fulton Gonzalez, Tomoyuki Kakehi, Jue Wang

英文摘要：

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

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Surjectivity of Mean Value Operators on Noncompact Symmetric Spaces

Let $X=G/K$ be a symmetric space of the non-compact type. We prove that the mean value operator over translated $K$-orbits of a fixed point is surjective on the space of smooth functions on $X$ if $X$ is either complex or of rank one. For higher rank spaces it is shown that the same statement is true for points in an appropriate Weyl subchamber.

2016-08-05math.FA

调和NA群上卷积算子的满射性

Surjectivity of convolution operators on harmonic $NA$ groups

本文刻画了调和NA群上径向紧支撑分布诱导的右卷积算子在光滑径向函数空间上满射的充要条件，即其球面傅里叶变换缓慢递减，并由此证明了球面平均算子的满射性。

2024-10-19math.FA

有限Hankel变换的渐近展开与卷积算子的满射性

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

本文研究Ehrenpreis-Hörmander意义下卷积算子的满射性，通过有限Hankel变换的渐近展开给出径向函数可逆的充分条件，并提出处理非原点奇点的新方法。

2024-05-27math.FA

强制卷积满射性的算子

Operators that coerce the surjectivity of convolution

考虑的是运营商，离开一套不可逆的（在这个意义上的Escherapeis）分布稳定。他们同时概括了操作卷积的分布与紧凑的支持和操作的乘法由一个真正的解析函数;他们在这里被称为伪卷积，因为他们也概括了伪微分算子。(It证明了椭圆实解析伪微分算子使不可逆分布和可逆分布都保持不变。但是当放松实解析性的条件时，这样的算子可以将一个不可逆分布映射到一个可逆分布--假定两种情况下的可逆性都涉及同一个函数空间。然而，通过改变空间，人们可以测量非解析扰动可能引入的“非可逆性损失”。这种现象在这里研究使用Beurling类的功能和测量的规律性运营商符号在Denjoy-Carleman意义上; Gevrey的情况变得特别简单。

2013-12-17math.FA

Noncompactness of Fourier Convolution Operators on Banach Function Spaces

Let $X(\mathbb{R})$ be a separable Banach function space such that the Hardy-Littlewood maximal operator $M$ is bounded on $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$. Suppose $a$ is a Fourier multiplier on the space $X(\mathbb{R})$. We show that the Fourier convolution operator $W^0(a)$ with symbol $a$ is compact on the space $X(\mathbb{R})$ if and only if $a=0$. This result implies that nontrivial Fourier convolution operators on Lebesgue spaces with Muckenhoupt weights are never compact.

2019-09-30math.FA

The smoothness of orbital measures on noncompact symmetric spaces

Let $G/K$ be an irreducible symmetric space where $G$ is a non-compact, connected Lie group and $K$ is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $% L^{2}(G)$ and hence is an absolutely continuous measure with respect to Haar measure. The number $r$ is approximately the rank of $G/K$. For the special case of the orbital measures, $ν_{a_{i}}$, supported on the double cosets $Ka_{i}K$ where $a_{i}$ belongs to the dense set of regular elements, we p…

2020-09-29math.FA

更多相关论文收起更多论文

On perturbation of a surjective convolution operator

Let $μ\in {\cal E}'({\mathbb R}^n)$ be a compactly supported distribution such that its support is a convex set with non-empty interior. Let $X_2$ be a convex domain in ${\mathbb R}^n$, $X_1 = X_2 + supp \ μ$. Assuming that a convolution operator $A: {\cal E}(X_1) \to {\cal E}(X_2)$ acting by the rule $(Af)(x) = (μ* f)(x)$ is surjective we provide a condition on a linear continuous operator $B: {\cal E}(X_1) \to {\cal E}(X_2)$ that guarantees surjectivity of the operator $A+B$.

2016-12-16math.FA

Generalized spherical mean value operators on Euclidean space

We consider the Neumann version of the spherical mean value operator and its variants in the space of smooth functions, distributions and compactly supported ones. Surjectivity and range characterization issues are addressed from the viewpoint of convolution equations.

2020-03-22math.FA

Convolution kernels versus spectral multipliers for sub-Laplacians on groups of polynomial growth

Let $\mathcal{L}$ be a sub-Laplacian on a connected Lie group $G$ of polynomial growth. It is well known that, if $F : \mathbb{R} \to \mathbb{C}$ is in the Schwartz class $\mathcal{S}(\mathbb{R})$, then the convolution kernel $\mathcal{K}_{F(\mathcal{L})}$ of the operator $F(\mathcal{L})$ is in the Schwartz class $\mathcal{S}(G)$. Here we prove a sort of converse implication for a class of groups $G$ including all solvable noncompact groups of polynomial growth. We also discuss the problem whether integrability of $\mathcal{K}_{F(\mathcal{L})}$ implies continuity of $F$.

2018-05-10math.CAmath.FA

凸分析在数学中的应用

Applications of Convex Analysis within Mathematics

本文研究了凸分析及其理论应用。我们首先将凸分析的重要工具应用于优化和分析。然后，我们展示了凸分析，特别是无穷卷积在单调算子理论中的各种深入应用。其中，我们在希尔伯特空间中重新获得了Minty满射性定理，并给出了自反空间中和定理的一个新证明。更技术性地，我们还讨论了极大单调算子的自共轭表示。最后，我们考虑数学分析中的各种其他应用。

2013-07-21math.FAmath.OC

对称空间上的Lp-分布

Lp-distributions on symmetric spaces

在非紧型黎曼对称空间上引入了$L^p$ -分布的概念，并建立了它们的主要性质.我们用Laplace-Beltrami算子来描述检验函数空间的拓扑结构。该技术是基于椭圆算子的先验估计。我们表明，结构定理，类似于$\Rn$，持有对称空间。我们给出卷积的估计。

2006-11-19math.DGmath.FA