Fractional Gaussian estimates and holomorphy of semigroups

Abstract

Let Ω ⊂ RN be an arbitrary open set and denote by (e−t(−∆)sRN )t≥0 (where 0 < s < 1) the semigroup on L2 (RN) generated by the fractional Laplace operator. In the first part of the thesis we show that if T is a self-adjoint semigroup on L2(Ω) satisfying a fractional Gaussian estimate in the sense that |T(t)f| ≤ Me−bt(−∆)sRN |f|, 0 ≤ t ≤ 1, f ∈ L2(Ω), for some constants M ≥ 1 and b ≥ 0, then T defines a bounded holomorphic semigroup of angle π/2 that interpolates on Lp (Ω), 1 ≤ p < ∞. Additionally, if T0 is a semigroup on C0(Ω) such that T0(t)f = T(t)f for all f ∈ C0(Ω) ∩ L2 (Ω), we prove that the same result also holds on the space C0(Ω). If Ω is bounded then the same conclusion holds for C(Ω). Also, we apply the above results to the realization of fractional order operators with the exterior Dirichlet conditions.