Let α
1, …, α
d' be an algebraic basis of rank
r in a Lie algebra \mathfrak{g} of a connected Lie group
G and let
Aι be the left differential operator in the direction α
i on the
Lp-spaces with respect to the left, or right, Haar measure, where
p∈[1, ∞]. We consider
m-th order operators
H=∑
cαAαwith complex variable bounded coefficients
cα which are subcoercive of step
r, i. e., for all
g∈
G the form obtained by fixing the
cα at
g is subcoercive of step
r and the ellipticity constant is bounded from below uniformly by a positive constant. If the principal coefficients are
m-times differentiate in
L∞ in the directions of α
1, …, α
d' we prove that the closure of
H generates a consistent interpolation semigroup
S which has a kernel. We show that
S is holomorphic on a non-empty
p-independent sector and if
H is formally self-adjoint then the holomorphy angle is π/2. We also derive ‘Gaussian’ type bounds for the kernel and its derivatives up to order
m−1.
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