On complex-time heat kernels of fractional Schr\"odinger operators via Phragm\'en-Lindel\"of principle

We consider fractional Schrodinger operators with possibly singular potentials and derive certain spatially averaged estimates for its complex-time heat kernel. The main tool is a Phragmen-Lindelof theorem for polynomially bounded functions on the right complex half-plane. The interpolation leads to possibly nonoptimal off-diagonal bounds.