We establish the optimal
interior regularity of solutions to
where
denotes
the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition
on
, namely the
group convolution
is
, where
is the fundamental
solution of
.
The
regularity is understood in the sense of Folland and Stein. In the classical Euclidean
setting, the first seeds of the above problem were already present in the 1991 paper of
Sakai and are also related to quadrature domains. As a special instance of our results,
when
is nonnegative and satisfies the above equation, we recover the
regularity of solutions to the obstacle problem in stratified groups, which
was previously established by Danielli, Garofalo and Salsa. Our regularity
result is sharp: it can be seen as the subelliptic counterpart of the
regularity result due to Andersson, Lindgren and Shahgholian.
Keywords
sub-Laplacian, obstacle problem, subelliptic equations,
stratified groups