REMARKS ON THE STRONG MAXIMUM PRINCIPLE FOR NONLOCAL OPERATORS

In this note, we study the existence of a strong maximum principle for the nonlocal operator M[u](x) := integral(G) J(g)u(x * g(-1))d mu(g) - u(x), where G is a topological group acting continuously on a Hausdorff space X and u is an element of C(X). First we investigate the general situation and derive a pre-maximum principle. Then we restrict our analysis to the case of homogeneous spaces (i.e., X = G/H). For such Hausdorff spaces, depending on the topology, we give a condition on J such that a strong maximum principle holds for M. We also revisit the classical case of the convolution operator (i.e. G - (R-n, +), X = R-n, d mu = dy).