Stephen Montgomery-Smith, Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations. Duke Math J. 91 (1998), 393-408. 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. (tex, dvi, ps, pdf.) Since this paper was written, the unsolved problem remaining in this paper has been solved by Keel and Tao. You may find a copy of their paper at either of their web sites. (Keel, Tao).

 

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