Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations

Abstract

Let $u(x,t)$ be the solution of the Schrödinger or wave equation with $L_2$ initial data. We provide counterexamples to plausible conjectures involving the decay in $t$ of the $\BMO$ norm of $u(t,\cdot )$. The proofs make use of random methods, in particular, Brownian motion. (Since this paper was written, the unsolved problem remaining in this paper has been solved by Keel and Tao.)

Also available at http://www.math.missouri.edu/~stephen/preprints/

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