Uniform resolvent estimates for Schr\"odinger operator with an inverse-square potential

We study the uniform resolvent estimates for Schr\"odinger operator with a Hardy-type singular potential. Let $\mathcal{L}_V=-\Delta+V(x)$ where $\Delta$ is the usual Laplacian on $\mathbb{R}^n$ and $V(x)=V_0(\theta) r^{-2}$ where $r=|x|, \theta=x/|x|$ and $V_0(\theta)\in\mathcal{C}^1(\mathbb{S}^{n-1})$ is a real function such that the operator $-\Delta_\theta+V_0(\theta)+(n-2)^2/4$ is a strictly positive operator on $L^2(\mathbb{S}^{n-1})$. We prove some new uniform weighted resolvent estimates and also obtain some uniform Sobolev estimates associated with the operator $\mathcal{L}_V$.Comment: Comments are welcome.To appear in Journal of Functional Analysi

An improved maximal inequality for 2D fractional order Schr\"{o}dinger operators

The local maximal inequality for the Schr\"{o}dinger operators of order \a>1 is shown to be bounded from $H^s(\R^2)$ to $L^2$ for any $s>\frac38$. This improves the previous result of Sj\"{o}lin on the regularity of solutions to fractional order Schr\"{o}dinger equations. Our method is inspired by Bourgain's argument in case of \a=2. The extension from \a=2 to general \a>1 confronts three essential obstacles: the lack of Lee's reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable Galilean transformation in the deduction of the main theorem. We get around these difficulties by establishing a new reduction lemma at our disposal and analyzing all the possibilities in using the separateness of the segments to obtain the analogous bilinear $L^2-$estimates. To compensate the absence of Galilean invariance, we resort to Taylor's expansion for the phase function. The Bourgain-Guth inequality in \cite{ref Bourgain Guth} is also rebuilt to dominate the solution of fractional order Schr\"{o}dinger equations.Comment: Pages47, 3figures. To appear in Studia Mathematic