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In this note we study the eigenvalue problem for a quadratic form associated with Strichartz estimates for the Schr{o}dinger equation, proving in particular a sharp Strichartz inequality for the case of odd initial data. We also describe an alternative method that is applicable to a wider class of matrix problems.
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Let $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally log-Holder continuous condition and $L$ a one to one operator of type $omega$ in $L^2({mathbb R}^n)$, with $omegain[0,,pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W!H_L^{p(cdot)}(mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space $W!T^{p(cdot)}(mathbb R^n)$ which is also obtained in this article. In particular, when $L$ is non-negative and self-adjoint, the authors obtain the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$. As an application of the molecular characterization, when $L$ is the second-order divergence form elliptic operator with complex bounded measurable coefficient, the authors prove that the associated Riesz transform $ abla L^{-1/2}$ is bounded from $W!H_L^{p(cdot)}(mathbb R^n)$ to the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$. Moreover, when $L$ is non-negative and self-adjoint with the kernels of ${e^{-tL}}_{t>0}$ satisfying the Gauss upper bound estimates, the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$ is further used to characterize the space via non-tangential maximal functions.
We prove a number of textit{a priori} estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value problems of Dirichlet and Neumann type in various topologies. We work in classes of solutions which include the energy solutions. For those solutions, we use a description using the first order systems satisfied by their conormal gradients and the theory of Hardy spaces associated with such systems but the method also allows us to design solutions which are not necessarily energy solutions. We obtain precise comparisons between square functions, non-tangential maximal functions and norms of boundary trace. The main thesis is that the range of exponents for such results is related to when those Hardy spaces (which could be abstract spaces) are identified to concrete spaces of tempered distributions. We consider some adapted non-tangential sharp functions and prove comparisons with square functions. We obtain boundedness results for layer potentials, boundary behavior, in particular strong limits, which is new, and jump relations. One application is an extrapolation for solvability a la {v{S}}ne{ui}berg. Another one is stability of solvability in perturbing the coefficients in $L^infty$ without further assumptions. We stress that our results do not require De Giorgi-Nash assumptions, and we improve the available ones when we do so.
Using a new local smoothing estimate of the first and third authors, we prove local-in-time Strichartz and smoothing estimates without a loss exterior to a large class of polygonal obstacles with arbitrary boundary conditions and global-in-time Strichartz estimates without a loss exterior to a large class of polygonal obstacles with Dirichlet boundary conditions. In addition, we prove a global-in-time local smoothing estimate in exterior wedge domains with Dirichlet boundary conditions and discuss some nonlinear applications.
In this paper, we establish an optimal dual version of trace estimate involving angular regularity. Based on this estimate, we get the generalized Morawetz estimates and weighted Strichartz estimates for the solutions to a large class of evolution equations, including the wave and Schr{o}dinger equation. As applications, we prove the Strauss conjecture with a kind of mild rough data for $2le nle 4$, and a result of global well-posedness with small data for some nonlinear Schr{o}dinger equation with $L^2$-subcritical nonlinearity.
We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of earlier work of Metcalfe-Tataru, in order to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.
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