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Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations. (English) Zbl 0955.35012

The author considers the \(L^p\)-\(L^q\)-estimate of solutions of wave equations and the Schrödinger equation. In particular is given a counterexample to the following estimate \[ \Biggl(\int^\infty_\infty\|B_t f\|^2_{\text{BMO}(\mathbb{R}^3)}dt\Biggr)^{{1\over 2}}\leq c\|f\|_{L^2(\mathbb{R}^3)}, \] where \(B_tf\) is a solution of a wave equation satisfying the initial conditions \(u(0)= 0\), \(u_t(0)= f\). Moreover, the author asserts the following two estimates do not hold: \[ \Biggl(\int^\infty_\infty\|A_t f\|^2_{\text{BMO}(\mathbb{R})}dt\Biggr)^{{1\over 2}}\leq c\|f\|_{L^2(\mathbb{R}^2)},\quad \Biggl(\int^\infty_\infty\|A_t f\|^q_{L^\infty(\mathbb{R}^2)}dt\Biggr)^{{1\over 2}}\leq c\|f\|_{L^2(\mathbb{R}^2)}, \] where \(A_t f\) is a solution of the Schrödinger equation with the initial data \(f\).

MSC:

35B45 A priori estimates in context of PDEs
35G10 Initial value problems for linear higher-order PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35L05 Wave equation
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References:

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