We study convexity properties of the average integral operators naturally
associated with divergence-form second-order subelliptic operators
with nonnegative
characteristic form. When
is the classical Laplace operator, these average operators are the usual average
integrals over Euclidean spheres. In our subelliptic setting, the average operators are
(weighted) integrals over the level sets
of the fundamental solution
of
. We shall obtain characterizations
of the
-subharmonic
functions
(that is, the
weak solutions to
)
in terms of the convexity (w.r.t. a power of
) of the average
of
over
, as a function
of the radius
.
Solid average operators will be considered as well. Our main tools are representation
formulae of the (weak) derivatives of the average operators w.r.t. the radius. As
applications, we shall obtain Poisson–Jensen and Bôcher type results for
.
Keywords
subharmonic functions, hypoelliptic operator, convex
functions, average integral operator, divergence-form
operator.