Fractional Gaussian estimates and holomorphy of semigroups

• Published in: Archiv der Mathematik Vol. 113; no. 6; pp. 629 - 647
• Main Authors: Keyantuo, Valentin, Seoanes, Fabian, Warma, Mahamadi
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-12-2019; Springer Nature B.V
• Subjects: Continuity (mathematics); Dirichlet problem; Mathematical analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); 47D06; 47D03; Fractional Gaussian estimates; Holomorphy; Fractional Laplace operator; 35R11; Fractional heat equation; Semigroup
• ISSN: 0003-889X; 1420-8938
• DOI: 10.1007/s00013-019-01381-y

Let
Ω
⊂
R
N
be an arbitrary open set,
0
<
s
<
1
and denote by
(
e
-
t
(
-
Δ
)
R
N
s
)
t
≥
0
the semigroup on
L
2
(
R
N
)
generated by the fractional Laplace operator. In the first part of the paper, 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
|
≤
M
e
-
b
t
(
-
Δ
)
R
N
s
|
f
|
,
0
≤
t
≤
1
,
f
∈
L
2
(
Ω
)
, for some constants
M
≥
1
and
b
≥
0
, then
T
defines a bounded holomorphic semigroup of angle
π
2
that interpolates on
L
p
(
Ω
)
,
1
≤
p
<
∞
. Using a duality argument, we prove that the same result also holds on the space of continuous functions. In the second part, we apply the above results to the realization of fractional order operators with the exterior Dirichlet conditions.