Regularity of Fractional Heat Semigroup Associated with Schrodinger Operators

Let L = -Delta + V be a Schrodinger operator, where the potential V belongs to the reverse Holder class. By the subordinative formula, we introduce the fractional heat semigroup {e(-tL alpha)}(t>0), 0 < alpha < 1, associated with L. By the aid of the fundamental solution of the heat equation: partial derivative(t)u + Lu = partial derivative(t)u - Delta u + Vu = 0, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel k(alpha,t)(L) (., .), respectively. This method is independent of the Fourier transform, and can be applied to the second-order differential operators whose heat kernels satisfy the Gaussian upper bounds. As an application, we establish a Carleson measure characterization of the Campanato-type space BMOL gamma(R-n) via the fractional heat semigroup {e(-tL alpha)}(t>0).

Automated summary

This study introduces the fractional heat semigroup and estimates the gradient and time-fractional derivatives of the fractional heat kernel using the fundamental solution of the heat equation. The method is independent of the Fourier transform and applicable to second-order differential operators with heat kernels satisfying Gaussian upper bounds. An application of the method involves establishing a Carleson measure characterization of a specific type of space using the fractional heat semigroup.