Local Hardy spaces with variable exponents associated to non-negative self-adjoint operators satisfying Gaussian estimates

In this paper we introduce variable exponent local Hardy spaces associated with a non-negative self-adjoint operator L. We define them by using an area square integral involving the heat semigroup associated to L. A molecular characterization is established and as an aplication of the molecular characterization we prove that our local Hardy space coincides with the (global) variable exponent Hardy space associated to L, provided that 0 does not belong to the spectrum of L. Also, we show that it coincides with the global variable exponent Hardy space associated to L+I.