$L^ p$ continuity of pseudo-differential operators with discontinuous symbol

Abstract:

A set $\Gamma \subset {{\mathbf {R}}^n} \times {{\mathbf {R}}^n}$ is called conic if $\left ( {x,\xi } \right ) \in \Gamma$ implies $\left ( {x,t\xi } \right ) \in \Gamma$, for all $t > 0$. In localizing a partial differential equation, the following question arose: Let us consider a pseudo-differential operator $L$ whose symbol is the characteristic function of a conic subset of ${{\mathbf {R}}^n} \times {{\mathbf {R}}^n}$. Will $L$ be a bounded operator in ${L^2}\left ( {{{\mathbf {R}}^n}} \right )$? We present as a counterexample a pseudo-differential operator $L$ whose symbol is the characteristic function of a conic subset of ${{\mathbf {R}}^2} \times {{\mathbf {R}}^2}$, which is unbounded on ${L^p}\left ( {{{\mathbf {R}}^2}} \right )$ for $1 \leq p \leq 2$.