Isometries of $L_p$-spaces of solutions of homogeneous partial differential equations

Abstract

Let $ n\geq 2, A=(a_{ij})_{i,j=1}^{n}$ be a real symmetric matrix, $a=(a_i)_{i=1}^{n}\in \Bbb R^n.$ Consider the differential operator $D_A = \sum_{i,j=1}^n a_{ij}{\partial^2 \over \partial x_i \partial x_j}+ \sum_{i=1}^n a_i{\partial \over \partial x_i}.$ Let $E$ be a bounded domain in $\Bbb R^n,$ $p>0.$ Denote by $L_{D_A}^p(E)$ the space of solutions of the equation $D_A f=0$ in the domain $E$ provided with the $L_p$-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 $L_{D_A}^p(E)$ is isometric to a subspace of $L_{D_B}^p(F)$ 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 $L_p$-isometries which reduces the problem to the question of which weighted composition operators carry solutions of the equation $D_A f=0$ in $E$ to solutions of the equation $D_B f=0$ in $F.$