Symmetrization of distributions and its application

Abstract:

Let P be a polynomial such that k of the $n - 1$ principal curvatures are different from zero at each point of $N(P) = \{ s \in {R^n}:P(s) = 0\} ;N(P)$ is assumed to be nonempty, bounded, and $n - 1$ dimensional. If ${\text {Supp}}\;\varphi \subset {U^\delta } = \{ s \in {R^n}:|P(s)| < \delta \}$ with $\delta$ small and $\varphi \in C_c^\infty ({R^n})$, let ${\varphi ^\rho }$ be the integral of $\varphi$ over $N(P - q)$ if $q \in [ - \delta ,\delta ]$ and ${\varphi ^\sigma }(s) = {\varphi ^\rho }(P(s))$ on ${U^\delta }$ and $= 0$ outside ${U^\delta }$. Then ${\varphi ^\sigma } \in C_c^\infty ({R^n})$. We define the symmetrization ${v^\sigma }$ of a distribution v, with ${\text {Supp}}\;v \subset {U^\delta }$, in a natural way. Setting $u = {\mathcal {F}^{ - 1}}\{ v\}$ and ${u_0} = {\mathcal {F}^{ - 1}}\{ {v^\sigma }\}$, we prove that ${u_0}$ is the integral of the product of u with some function $w(,)$ which depends only on P. This result is used to prove a Liouville type theorem for entire solutions of $P( - i{D_x})u(x) = f(x)$, with $f \in C_c^\infty ({R^n})$.