A weighted estimate for two dimensional Schrodinger, matrix schrodinger and wave equations with resonance of first kind at zero energy

We study the two dimensional Schr\"odinger operator, $H=-\Delta+V$, in the weighted L^1(\R^2) \rightarrow L^{\infty}(\R^2) setting when there is a resonance of the first kind at zero energy. In particular, we show that if |V(x)|\les \la x \ra ^{-3-} and there is only s-wave resonance at zero of H, then \big\| w^{-1} \big( e^{itH}P_{ac} f - {\f 1 t } F f \big) \big\| _{\infty} \leq \frac {C} {|t| (\log|t|)^2 } \|wf\|_1 |t|>2, with w(x)=\log^2(2+|x|). Here Ff=c \psi\la f,\psi \ra, where \psi is an s-wave resonance function. We also extend this result to matrix Schr\"odinger and wave equations with potentials under similar conditions.