Abstract
Consider the differential operator
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.
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The authors thank the referee(s) for meticulously reading our manuscript and giving us several valuable suggestions which improved the clarity of the paper. The authors also thank the handling editor for the help during the editorial process.
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Patra, P.S., Bais, S.R. & Venku Naidu, D. Application of Bargmann transform in the study of affine heat kernel transform. J. Pseudo-Differ. Oper. Appl. 15, 38 (2024). https://doi.org/10.1007/s11868-024-00603-4
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DOI: https://doi.org/10.1007/s11868-024-00603-4
Keywords
- Image characterization
- Holomorphic function spaces
- Affine heat kernel transform
- Translation-invariant spaces of analytic functions