Local Hardy Spaces with Variable Exponents Associated with Non-negative Self-Adjoint Operators Satisfying Gaussian Estimates.

In this paper we introduce variable exponent local Hardy spaces h L p (·) (R n) associated with a non-negative self-adjoint operator L. We assume that, for every t > 0 , the operator e - t L has an integral representation whose kernel satisfies a Gaussian upper bound. We define h L p (·) (R n) by using an area square integral involving the semigroup { e - t L } t > 0 . A molecular characterization of h L p (·) (R n) is established. As an application of the molecular characterization, we prove that h L p (·) (R n) coincides with the (global) Hardy space H L p (·) (R n) provided that 0 does not belong to the spectrum of L. Also, we show that h L p (·) (R n) = H L + I p (·) (R n) . [ABSTRACT FROM AUTHOR]