Michael Goldberg (Cincinnati), Analysis Seminar

Title:  Pointwise bounds for the 3-dimensional wave equation and spectral multipliers

Abstract: The sine propagator for the wave equation in three dimensions, $\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}$, has an integral kernel $K(t,x,y)$ with the property $\int_{\mathbb R} |K(t, x, y)|dt = (2\pi|x-y|)^{-1}$.  Finiteness comes from the sharp Huygens principle and power-law decay comes from dispersion.  Estimates of this type are useful for proving “reversed Strichartz” inequalities that bound a solution in $L^p_x L^q_t$ for admissible pairs $(p,q)$.  We examine the propagator $\frac{\sin(t\sqrt{H})}{\sqrt{H}}P_{ac}(H)$ for operators $H = -\Delta + V$ with the potential $V$ belonging to the Kato-norm closure of test functions.  Assuming zero is not an eigenvalue or resonance, the bound $\int_{\mathbb R} |K(t,x,y) \leq C|x-y|^{-1}$ continues to be true.

Combined with a Huygens principle for the perturbed wave equation, this estimate suggests pointwise bounds for spectral multipliers of fractional integral or Hörmander-Mikhlin type.

This is joint work with Marius Beceanu (SUNY – Albany).