Stability of Toeplitz Operators on Fractional Parabolic Hardy Spaces

We study Toeplitz operators on fractional parabolic Hardy spaces associated with the fundamental solutions of fractional heat operators. For a finite positive Borel measure on the upper half-space, we first recall the characterization of boundedness and compactness of Toeplitz operators in terms of Carleson measures. The main result of this paper establishes the stability of Toeplitz operators with respect to variations of the fractional parameter (0, 1]. Assuming that satisfies a uniform Carleson condition on a compact interval of parameters, we prove that the corresponding family of Toeplitz operators depends continuously on in the operator norm topology. As immediate consequences, we obtain the stability of boundedness and compactness under perturbations of the fractional parameter. The proofs rely on uniform kernel estimates for fractional heat kernels and on a Carleson embedding approach via Poisson extension operators. This is a preprint of a manuscript submitted to Integral Equations and Operator Theory.