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

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.