$L^{p}$ behavior of certain second order partial differential operators

Abstract:

We give examples of bounded inverses of polynomials in ${{\textbf {R}}^n}$, $n > 1$, which are not Fourier multipliers of ${L^p} ({{\textbf {R}}^n})$ for any $p \ne 2$. Our main tool is the Kakeya set construction of C. Fefferman. Using these results, we relate the invertibility on ${L^p}$ of a linear second order constant coefficient differential operator D on ${{\textbf {R}}^n}$ to the geometric structure of quadratic surfaces associated to its symbol d. This work was motivated by multiplier conjectures of N. Rivière and R. Strichartz.