Ciprian Demeter
We present a slightly simpler proof of the multilinear refined Strichartz estimate, and prove a slightly more general linear refined Strichartz estimate. Our arguments seek to clarify the connection between these estimates, refined decoupling and tube incidences.
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Some analogues of the Schrödinger refined Strichartz inequalities (Du, Guth, Li and Zhang) are obtained for the wave equation. These are used to improve the best known $L^2$ fractal Strichartz inequalities for the wave equation in dimensions $d \geq 4$.
There are two parts for this paper. In the first part, we extend some results in a recent paper by Du, Guth, Li and Zhang to a more general class of phase functions. The main methods are Bourgain-Demeter's $l^2$ decoupling theorem and induction on scales. In the second part, we prove some positive results for the maximal extension operator for hypersurfaces with positive principal curvatures. The main methods are sharp $L^2$ estimates by Du and Zhang, and the bilinear method by Wolff and Tao.
We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schrödinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the $L^2$-critical setting due to an enormous group of symmetries, while on the other hand, the spacetime Fourier analysis employed by the existing approaches to the constant-coefficient equation are not adapted to nontranslation-invariant situations, especially with potentials …
We obtain partial improvement toward the pointwise convergence problem of Schrödinger solutions, in the general setting of fractal measure. In particular, we show that, for $n\geq 3$, $\lim_{t \to 0} e^{itΔ}f(x) = f(x)$ almost everywhere with respect to Lebesgue measure for all $f \in H^s (\mathbb{R}^n)$ provided that $s>(n+1)/2(n+2)$. The proof uses linear refined Strichartz estimates. We also prove a multilinear refined Strichartz using decoupling and multilinear Kakeya.
A sharpened energy-Strichartz inequality for the wave equation
本研究针对Bez和Rogers提出的1+5维能量空间波动方程精确Strichartz估计，借鉴经典精细化Sobolev估计思路，添加与极值元集距离成正比的项实现了估计的改进。
We prove Strichartz estimates over large time scales for the Schrodinger equation set on irrational tori. They are optimal for Lebesgue exponents $p > 6$.
We prove a sharp, global-in-time Strichartz estimate for the Schrödinger equation on the cylinder $\mathbb{R}\times\mathbb{T}$.
In this article we study the Schrödinger 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 of the spectral projections associated harmonic oscillator proved in \cite{KT}. Our Strichartz type estimates are sharp in sense of regularity of initial data.
In this note we study the eigenvalue problem for a quadratic form associated with Strichartz estimates for the Schrödinger 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.
We show that the endpoint Strichartz estimate for the kinetic transport equation is false in all dimensions. We also present a new approach to proving the non-endpoint cases using multilinear analysis.
We consider Strichartz estimates for the wave equation with respect to general measures which satisfy certain growth condition. In $\mathbb R^{3+1}$ we obtain the sharp estimate and in higher dimensions improve the previous results.
We calculate the the sharp constant and characterise the extremal initial data in $\dot{H}^{\frac{3}{4}}\times\dot{H}^{-\frac{1}{4}}$ for the $L^4$ Sobolev--Strichartz estimate for the wave equation in four space dimensions.
We prove scale-invariant Strichartz inequalities for the Schrodinger equation on rectangular tori (rational or irrational) in all dimensions. We use these estimates to give a unified and simpler treatment of local well-posedness of the energy-critical nonlinear Schrodinger equation in dimensions three and four.
Strong-type inhomogeneous Strichartz estimates are shown to be false for the wave equation outside the so-called acceptable region. On a critical line where the acceptability condition marginally fails, we prove substitute estimates with a weak-type norm in the temporal variable. We achieve this by establishing such weak-type inhomogeneous Strichartz estimates in an abstract setting. The application to the wave equation rests on a slightly stronger form of the standard dispersive estimate in terms of certain Besov spaces.
In this paper we prove standard and reversed Strichartz estimates for the Klein--Gordon equation in $\mathbb{R}^{3+1}$. Instead of the Fourier theory, our analysis is based on fundamental solutions of the free equations and fractional integrations. In the final part of this paper, we apply Strichartz estimates in the study of a semilinear Klein--Gordon equation.
We prove a sharp bilinear estimate for the wave equation from which we obtain the sharp constant in the Strichartz estimate which controls the $L^4_{t,x}(\R^{5+1})$ norm of the solution in terms of the energy. We also characterise the maximisers.
We prove generalized Strichartz estimates with weaker angular integrability for the Schrödinger equation. Our estimates are sharp except some endpoints. Then we apply these new estimates to prove the scattering for the 3D Zakharov system with small data in the energy space with low angular regularity. Our results improve the results obtained recently in \cite{GLNW}.
We obtain scattering for the 3D Zakharov system with non-radial small data in the energy space with angular regularity of degree one. The main ingredient is a generalized Strichartz estimate for the Schrödinger equation in the space of $L^2$ angular integrability.
In this paper we study Strichartz estimates for dispersive equations which are defined by radially symmetric pseudo-differential operators, and of which initial data belongs to spaces of Sobolev type defined in spherical coordinates. We obtain the space time estimates on the best possible range including the endpoint cases.
On sharp Strichartz estimate for hyperbolic Schrödinger equation on $\mathbb{T}^3$
本文用关联几何方法证明了三维环面上双曲薛定谔方程的尖锐Strichartz估计，并由此得到非线性方程的最优局部适定性。
Sharp Strichartz estimates on the one-dimensional torus
本文证明了一维环面上的尖锐Strichartz估计，给出了精确的常数与条件。
Refinements of Strichartz estimates on tori and applications
本文在环面上证明依赖频率时间区间的三线性与新线性 Strichartz 估计，并将其用于 Airy 传播子，得到修正 KdV 方程解存在性的最优 Sobolev 正则性。
Using the div-curl inequalities of Bourgain-Brezis [?MR2057026] and van Schaftingen [?MR2078071], we prove an improved Strichartz estimate for systems of inhomogeneous wave and Schrodinger equations, for which the inhomogeneity is a divergence-free vector field at each given time. The novelty of the result is that one can allow $L^1_x$ norms of the inhomogeneity in the right hand side of the estimate.
We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcricital case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.
We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schrödinger 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 using stable Lévy process in $\R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^\infty$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates.
We prove global-in-time Strichartz-type estimates for the Schrödinger equation on manifolds of the form $\mathbb{R}^{n}\times \mathbb{T}^{d}$, where $\mathbb{T}^{d}$ is a $d$-dimensional torus. Our results generalize and improve a global space-time estimate for the Schrödinger equation on $\mathbb{R} \times \mathbb{T}^{2}$ due to Z. Hani and B. Pausader.
Strichartz estimates for quasi-periodic functions and applications
本文利用$\ell^2$-解耦建立拟周期函数的Strichartz估计，并通过多线性精细化证明非线性薛定谔方程的低正则局部适定性，对三次情形得到尖锐结果。
A note on Strichartz estimates for the wave equation with orthonormal initial data
本文关注波动方程及初值数据正交族的 Strichartz 估计，综述了已知结果，针对未解决案例提出合理猜想，并在空间极大边界案例中提供了新结果。
Boundary Strichartz estimates and pointwise convergence for orthonormal systems
本文研究了费米子系统的极大估计，建立了空间变量的边界多体 Strichartz 估计，并证明了新的时间极大估计，显著扩展了关于费米子系统 Carleson 逐点收敛问题的已有成果。
We generalize the Strichartz estimates for Schrödinger operators on compact manifolds of Burq, Gérard and Tzvetkov [10] by allowing critically singular potentials $V$. Specifically, we show that their $1/p$--loss $L^p_tL^q_x(I\times M)$-Strichartz estimates hold for $e^{-itH_V}$ when $H_V=-Δ_g+V(x)$ with $V\in L^{n/2}(M)$ if $n\ge3$ or $V\in L^{1+δ}(M)$, $δ>0$, if $n=2$, with $(p,q)$ being as in the Keel-Tao theorem and $I\subset {\mathbb R}$ a bounded interval. We do this by formulating and proving new "quasimode" estimates for scaled dyadic unperturbed Schrödinger operators and taking advant…
We prove a bilinear $L^2(\R^d) \times L^2(\R^d) \to L^2(\R^{d+1})$ estimate for a pair of oscillatory integral operators with different asymptotic parameters and phase functions satisfying a transversality condition. This is then used to prove a bilinear refinement to Strichartz estimates on closed manifolds, similar to that on $\R^d$, but at a relevant semi-classical scale. These estimates will be employed elsewhere to prove global well-posedness below $H^1$ for the cubic nonlinear Schrödinger equation on closed surfaces.
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 Schrödinger equation. As applications, we prove the Strauss' conjecture with a kind of mild rough data for $2\le n\le 4$, and a result of global well-posedness with small data for some nonlinear Schrödinger equation with $L^2$-subcritical nonlinearity.
Strichartz estimates for quasi-periodic functions on long time intervals: An arithmetic approach
该研究针对带准周期初值的一维薛定谔方程，结合代数频率比的特性，利用Roth型丢番图不等式等工具改进了长时Strichartz估计并证明了端点L⁴估计。
We propose a conjecture for long time Strichartz estimates on generic (non-rectangular) flat tori. We proceed to partially prove it in dimension 2. Our arguments involve on the one hand Weyl bounds; and on the other hands bounds on the number of solutions of Diophantine problems.
This note presents a new proof of the well-known Strichartz estimates for the Schrödinger equation in $2+1$ dimensions, building on ideas from our recent work \cite{MO}.
Endpoint orthonormal Strichartz estimates
本文在Keel–Tao框架下建立正交Strichartz估计，解决了d≥2时薛定谔方程的端点猜想，同时给出自由传输方程对应猜想的肯定答案，并改进了单个函数的Strichartz估计。
Lossless Strichartz and spectral projection estimates on unbounded manifolds
本文研究了在非紧流形上无损斯特里茨茨和谱投影估计的改进方法，并通过新的残差估计获得了相关结果。
The Strichartz estimates for Schrödinger equations can be improved when the data is spread out in either physical or frequency space. In this paper we give refinements of the 2-dimensional homogeneous Strichartz estimate on the maximum size of a single wave packet. Different approaches are used in the proofs, including arithmetic approaches, polynomial partitioning, and the $l^2$ Decoupling Theorem, for different cases. We also give examples to show that the refinements we obtain cannot be further improved when $2 \leq p \leq 4$ and $p = 6$.
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