ISOMETRIES OF Lp-SPACES OF SOLUTIONS OF HOMOGENEOUS PARTIAL DIFFERENTIAL EQUATIONS

Let n ≥ 2, A = (aij) n i,j=1 be a real symmetric matrix, a = (ai) n i=1 ∈ R . Consider the differential operator DA = ∑n i,j=1 aij ∂ ∂xi∂xj + ∑n i=1 ai ∂ ∂xi . Let E be a bounded domain in R, p > 0. Denote by L DA (E) the space of solutions of the equation DAf = 0 in the domain E provided with the Lp-norm. We prove that, for matrices A, B, vectors a, b, bounded domains E, F, and every p > 0 which is not an even integer, the space LpDA (E) is isometric to a subspace of L p DB (F ) if and only if the matrices A and B have equal signatures, and the domains E and F coincide up to a natural mapping which in the most cases is affine. We use the extension method for Lp-isometries which reduces the problem to the question of which weighted composition operators carry solutions of the equation DAf = 0 in E to solutions of the equation DBf = 0 in F.