Fractional gaussian estimates and holomorphy of semigroups
Date
2019-10Author
Seoanes Correa, Fabian
Keyantuo, Valentín (Advisor)
Warma, Mahamadi (Advisor)
El Mennaoui, Omar (Reader)
Li, Liangquin (Reader)
Shan, Lin (Reader)
Metadata
Show full item recordAbstract
Let Ω ⊂ RN be an arbitrary open set and denote by (e−t(−∆)sRN )t≥0 (where 0 < s < 1) the semigroup on L 2 (R N ) generated by the fractional Laplace operator.
In the first part of the thesis we show that if T is a self-adjoint semigroup on L
2(Ω) satisfying a fractional Gaussian estimate in the sense that |T(t)f| ≤ Me−bt(−∆)s
RN |f|,0 ≤ t ≤ 1, f ∈ L2(Ω), for some constants M ≥ 1 and b ≥ 0, then T defines a
bounded holomorphic semigroup of angle π2 that interpolates on L p
(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) ∩ L 2(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.