Related papers: Sharp Strichartz estimates for the Schr\"odinger e…
In this article we study the Schr\"odinger equation associated with Harmonic oscillator in the form of Strichartz type inequality. We give simple proofs for Strichartz type inequalities using purely the $L^2 \to L^p$ operator norm estimates…
We prove almost Strichartz estimates found after adding regularity in the spherical coordinates for Schr\"odinger-like equations. The estimates are sharp up to endpoints. The proof relies on estimates involving spherical averages. Sharpness…
In the first part of the paper we continue the study of solutions to Schr\"odinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schr\"odinger operator involves a…
In this note we obtain some Strichartz estimates for the Schr\"odinger equation associated to the twisted Laplacian on $\mathbb{C}^{n}\cong \mathbb{R}^{2n}$. The initial data will be considered in suitable Sobolev spaces associated to the…
In this paper we obtain some Strichartz estimates for the Schr\"odinger equation associated to the harmonic oscillator and the Laplacian. Our main tool will be some embeddings between Lebesgue spaces and suitable Triebel-Lizorkin spaces.
We obtain optimal space-time estimates in $L^q_{t,x}$ spaces for all $q\ge 2$ for solutions to the Schr\"odinger equation on Zoll manifolds, including, in particular, the standard round sphere $S^d$. The proof relies on the arithmetic…
We study the nonlinear Schr\"odinger equation associated with the sublaplacian L on the unit sphere $S^{2n+1}$ in $C^{n+1}$ equipped with its natural CR structure. We first prove Strichartz estimates with fractional loss of derivatives for…
We consider $L^p$-$L^q$ estimates for the spherical harmonic projection operators and obtain sharp bounds on a certain range of $p$, $q$. As an application, we provide a proof of off-diagonal Carleman estimates for the Laplacian, which…
In this paper, we obtain sharp Strichartz estimates for solutions of the wave equation $\square_\gg\phi=0$ where $\gg$ is a rough Lorentzian metric on a 4 dimensional space-time $\MM$. This is the last step of the proof of the bounded $L^2$…
In this article we study global-in-time Strichartz estimates for the Schr\"odinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article of the third author, where…
We prove Strichartz estimates with a loss of derivatives for the Schr\"odinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon…
In this article we establish optimal estimates for the first eigenvalue of Schr\"odinger operators on the d-dimensional unit sphere. These estimates depend on Lebsgue's norms of the potential, or of its inverse, and are equivalent to…
We prove generalized Strichartz estimates with weaker angular integrability for the Schr\"odinger equation. Our estimates are sharp except some endpoints. Then we apply these new estimates to prove the scattering for the 3D Zakharov system…
In this paper we study sharp estimates for the Schr\"odinger operator via the framework of orthogonal polynomials. We use spherical harmonics and Gegenbauer polynomials to prove a new weighted inequality for the Schr\"odinger equation that…
We prove Strichartz estimates on any compact connected simple Lie group. In the diagonal case of Bourgain's exponents $p=q,$ we provide a new regularity order $s_{0}^{R}(p)$ in the sense that our (reverse) Strichartz estimates are valid…
We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schr\"odinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^\infty$ estimates fail at the critical regularity in high dimensions by…
We prove Strichartz estimates for the Schr\"odinger equation which are scale-invariant up to an $\varepsilon$-loss on products of odd-dimensional spheres. Namely, for any product of odd-dimensional spheres…
Consider the wave equation associated with the Kohn Laplacian on groups of Heisenberg type. We construct parametrices using oscillatory integral representations and use them to prove sharp $L^p$ and Hardy space regularity results.
In this paper we discuss a priori estimates derived from the energy method to the initial value problem for the cubic nonlinear Schr\"odinger on the sphere $S^2$. Exploring suitable a priori estimates, we prove the existence of solution for…
We prove dispersive and Strichartz estimates for Schr\"{o}dinger equations on normal real form symmetric spaces. These estimates apply to the well-posedness and scattering for the nonlinear Schr\"{o}dinger equations.