Resumen de A Hörmander-type spectral multiplier theorem for operators without heat kernel
Hörmander’s famous Fourier multiplier theorem ensures the L_p-boundedness of F(-\Delta _{\mathbb{R}} D) whenever F\in \mathcal{H}(s) for some s>\frac{D}{2}, where we denote by \mathcal{H} (s) the set of functions satisfying the Hörmander condition for s derivatives. Spectral multiplier theorems are extensions of this result to more general operators A \ge 0 and yield the L_p-boundedness of F(A) provided F\in \mathcal{H}(s) for some s sufficiently large. The harmonic oscillator A=-\Delta _{\mathbb{R}}+x^2 shows that in general s> \frac{D}{2} is not sufficient even if A has a heat kernel satisfying gaussian estimates. In this paper, we prove the L_p-boundedness of F(A) whenever F\in \mathcal{H}(s) for some s>\frac{D+1}{2}, provided A satisfies generalized gaussian estimates. This assumption allows to treat even operators A without heat kernel (e.g. operators of higher order and operators with complex or unbounded coefficients) which was impossible for all known spectral multiplier results.
