Heat kernel in the framework of zero order Mehler-Fock transform (Q2008155)

Let \(P_\xi\) denotes the Legendre function and \(\mathcal{M}\) the Mehler-Fock transform \[ \mathcal{M} f (\tau) := \int_1^\infty f(x) P_{i\tau-\frac12} (x) dx \] defining an isometric isomorphism between Hilbert spaces \(L^2((1,\infty),dx)\) and \(L^2((0,\infty),\tau \tan(\pi \tau) d\tau)\). The authors consider the associated heat kernel \[ h(t;x,y): = \int_0^\infty f(x) \exp\left( -t\tau^2 - \frac t4 \right) P_{i\tau-\frac12} (x) P_{i\tau-\frac12} (y) \tau \tan(\pi \tau) d\tau \] as well as the Weierstrass type integral transform \[ W_t f(x) :=\int_1^\infty f(y) h(t;x,y) dy \] and investigate some basic properties of \(\mathcal{M}\), \(h(t;x,y)\) and \(W_t \). They show in particular that \(h(t;x,y)\) is the fundamental solution for some generalized diffusion equation. Moreover, upper bounds for \(h(t;x,y)\) and a Heisenberg type inequality for \(\mathcal{M}\) are obtained. Boundedness and an inversion formula of \(W_t\) are also discussed.