, Volume 37, Issue 2, pp 381-393

A sharp weightedL 2-estimate for the solution to the time-dependent Schrödinger equation

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Abstract

For Ξ∈R n ,tR andfS(R n ) define \(\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)\) . We determine the optimal regularitys 0 such that $$\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$$ holds whereC is independent offS(R n ) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11].

Our theorems can be generalized to the case where the exp(it|ξ|2) is replaced by exp(it|ξ|a),a≠2.

The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL 2(R n ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity.

Part of this research was carried out in July 1994 when I enjoyed a stay at Johannes Kepler Universität, Linz. I would like to thank Jim Cooper, Eva Matoušková, Paul Müller, Charles Stegall, Michael Schmuckenschläger and Renata Mühlbachler for having created a stimulating and friendly atmosphere in the functional analysis group at Linz. I would also like to thank Professor Per Sjölin for valuable comments and for encouraging me to carry out the investigation presented here, Professor Krzysztof Stempak, Uniwersytet Wrocławski, Wrocław for help regarding improvements of the presentation and the referee for comments and criticism.