Application of Bargmann transform in the study of affine heat kernel transform

IF 0.9 3区 数学 Q2 MATHEMATICS
Partha Sarathi Patra, Shubham R. Bais, D. Venku Naidu
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引用次数: 0

Abstract

Consider the differential operator

$$\begin{aligned} \Delta _{a,b} = \Big (\frac{d^2}{dt^2} + \frac{4\pi ia}{b}t\frac{d}{dt} - \frac{4\pi ^2a^2t^2}{b^2} + \frac{2\pi ia}{b}I\Big ), \ t>0,\ a,b\in {\mathbb {R}}, \end{aligned}$$

where I is the identity operator. The operator \(\Delta _{a,b}\) is known as affine Laplacian. We consider the heat equation associated to the operator \(\Delta _{a,b}\) with initial condition f from \(L^2({\mathbb {R}}^n)\). Its solution is denoted by \(e^{t\Delta _{a,b}}f\). The transform \(f \mapsto e^{t\Delta _{a,b}}f\) is called affine heat kernel transform (or A-heat kernel transform). In this article, we consider (analytically extended) affine heat kernel transform and characterize the image of \(\displaystyle L^2({\mathbb {R}})\) under it as a weighted Bergman space of analytic functions on \({\mathbb {C}}\) with nonnegative weight. Consequently, we study \(L^p\)-boundedness of affine heat kernel transform, \(L^p\)-boundedness of affine Bargmann projection and related duality results. Moreover, we define affine Weyl translations and characterize the maximal and minimal spaces of analytic functions on \({\mathbb {C}}\) which are invariant under the affine Weyl translations.

巴格曼变换在仿射热核变换研究中的应用
考虑微分算子 $$begin{aligned}\Δ_{a,b} = \Big (\frac{d^2}{dt^2}+ \frac{4\pi ia}{b}t\frac{d}{dt} - \frac{4\pi ^2a^2t^2}{b^2}+ \frac{2\pi ia}{b}I\Big ), \t>0,\a,b\in {\mathbb {R}}, \end{aligned}$ 其中 I 是标识算子。算子 \(\Delta _{a,b}\) 被称为仿射拉普拉斯。我们考虑与初始条件 f 来自 \(L^2({\mathbb {R}}^n)\) 的算子 \(\Delta _{a,b}\) 相关的热方程。它的解用 \(e^{t\Delta _{a,b}}f\ 表示。)变换 \(f \mapsto e^{t\Delta _{a,b}}f\) 被称为仿射热核变换(或 A 热核变换)。在本文中,我们考虑(分析扩展的)仿射热核变换,并将其下的\(\displaystyle L^2({\mathbb {R}})\的图像描述为\({\mathbb {C}}\)上分析函数的加权伯格曼空间,其权重为非负。因此,我们研究了仿射热核变换的 \(L^p\)-boundedness 、仿射巴格曼投影的 \(L^p\)-boundedness 以及相关的对偶性结果。此外,我们定义了仿射韦尔平移,并描述了在\({\mathbb {C}}\) 上在仿射韦尔平移下不变的解析函数的最大和最小空间。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.20
自引率
9.10%
发文量
59
期刊介绍: The Journal of Pseudo-Differential Operators and Applications is a forum for high quality papers in the mathematics, applications and numerical analysis of pseudo-differential operators. Pseudo-differential operators are understood in a very broad sense embracing but not limited to harmonic analysis, functional analysis, operator theory and algebras, partial differential equations, geometry, mathematical physics and novel applications in engineering, geophysics and medical sciences.
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