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We consider fractional Schr\"odinger operators with possibly singularpotentials and derive certain spatially averaged estimates for its complex-timeheat kernel. The main tool is a Phragm\'en-Lindel\"of theorem for polynomiallybounded functions on the right complex half-plane. The interpolation leads topossibly nonoptimal off-diagonal bounds.

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