Strichartz Inequalities Research Articles

Global in time Strichartz inequalities on asymptotically flat manifolds with temperate trapping

Global in time Strichartz inequalities on asymptotically flat manifolds with temperate trapping

Restriction Theorems and Strichartz Inequalities for the Laguerre Operator Involving Orthonormal Functions

Restriction Theorems and Strichartz Inequalities for the Laguerre Operator Involving Orthonormal Functions

The endpoint Stein–Tomas inequality: old and new

AbstractThe Stein–Tomas inequality from 1975 is a cornerstone of Fourier restriction theory. Despite its respectable age, it is a fertile ground for current research. This note is centered around three classical applications – to Strichartz inequalities, Salem sets and Roth’s theorem in the primes – and three recent improvements: the sharp endpoint Stein–Tomas inequality in three space dimensions, maximal and variational refinements, and the symmetric Stein–Tomas inequality with applications.

Strichartz inequalities with white noise potential on compact surfaces

Strichartz inequalities with white noise potential on compact surfaces

Restriction Theorem for the Fourier–Dunkl Transform and Its Applications to Strichartz Inequalities

Restriction Theorem for the Fourier–Dunkl Transform and Its Applications to Strichartz Inequalities

Extremals for $$\alpha $$-Strichartz Inequalities

A necessary and sufficient condition on the precompactness of extremal sequences for one-dimensional $$\alpha $$ -Strichartz inequalities, equivalently $$\alpha $$ -Fourier extension estimates, is established based on the profile decomposition arguments. One of our main tools is an operator-convergence dislocation property consequence which comes from the van der Corput Lemma. Our result is valid in asymmetric cases as well. In addition, we obtain the existence of extremals for non-endpoint $$\alpha $$ -Strichartz inequalities.

Local solution to an energy critical 2-D stochastic wave equation with exponential nonlinearity in a bounded domain

We prove the existence and the uniqueness of a local maximal solution to an H1-critical stochastic wave equation with multiplicative noise on a smooth bounded domain D⊂R2 with exponential nonlinearity. First, we derive the appropriate deterministic and stochastic Strichartz inequalities in suitable spaces and, then use them in arguments based on fixed point method to show the local well-posedness result. We also present an explosion result for the constructed unique local maximal solution.

Gaussians never extremize Strichartz inequalities for hyperbolic paraboloids

For ξ = ( ξ 1 , ξ 2 , … , ξ d ) ∈ R d \xi = (\xi _1, \xi _2, \ldots , \xi _d) \in \mathbb {R}^d let Q ( ξ ) ≔ ∑ j = 1 d σ j ξ j 2 Q(\xi ) \colonequals \sum _{j=1}^d \sigma _j \xi _j^2 be a quadratic form with signs σ j ∈ { ± 1 } \sigma _j \in \{\pm 1\} not all equal. Let S ⊂ R d + 1 S \subset \mathbb {R}^{d+1} be the hyperbolic paraboloid given by S = { ( ξ , τ ) ∈ R d × R : τ = Q ( ξ ) } S = \big \{(\xi , \tau ) \in \mathbb {R}^{d}\times \mathbb {R}\ : \ \tau = Q(\xi )\big \} . In this note we prove that Gaussians never extremize an L p ( R d ) → L q ( R d + 1 ) L^p(\mathbb {R}^d) \to L^{q}(\mathbb {R}^{d+1}) Fourier extension inequality associated to this surface.

Maximisers for Strichartz inequalities on the torus

We study the existence of maximisers for a one-parameter family of Strichartz inequalities on the torus. In general, maximising sequences can fail to be precompact in , and maximisers can fail to exist. We provide a sufficient condition for precompactness of maximising sequences (after translation in Fourier space), and verify the existence of maximisers for a range of values of the parameter. Maximisers for the Strichartz inequalities correspond to stable, periodic (in space and time) solutions of a model equation for optical pulses in a dispersion-managed fiber.

Wave equation on certain noncompact symmetric spaces

In this paper, we prove sharp pointwise kernel estimates and dispersive properties for the linear wave equation on noncompact Riemannian symmetric spaces G/K of any rank with G complex. As a consequence, we deduce Strichartz inequalities for a large family of admissible pairs and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on hyperbolic spaces.

Sharp Strichartz inequalities for fractional and higher-order Schrödinger equations

We investigate a class of sharp Fourier extension inequalities on the planar curves $s=|y|^p$, $p>1$. We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that extremizers exist if $1<p<p_0$, for some $p_0>4$. In particular, this resolves the dichotomy of Jiang, Pausader & Shao concerning the existence of extremizers for the Strichartz inequality for the fourth order Schr\"odinger equation in one spatial dimension. One of our tools is a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$, developed in a companion paper. We further show that any extremizer exhibits fast $L^2$-decay in physical space, and so its Fourier transform can be extended to an entire function on the whole complex plane. Finally, we investigate the extent to which our methods apply to the case of the planar curves $s=y|y|^{p-1}$, $p>1$.

Note on Strichartz inequalities for the wave equation with potential

We obtain Strichartz inequalities for the wave equation with potentials which behave like the inverse square potential $|x|^{-2}$ but might be not a radially symmetric function.

Critical Points of Strichartz Functional

We study a pair of infinite dimensional dynamical systems naturally associated with the study of minimizing/maximizing functions for the Strichartz inequalities for the Schrödinger equation. One system is of gradient type and the other one is a Hamiltonian system. For both systems, the corresponding sets of critical points, their stability, and the relation between the two are investigated. By a combination of numerical and analytical methods we argue that the Gaussian is a maximizer in a class of Strichartz inequalities for dimensions one, two, and three. The argument reduces to verification of an apparently new combinatorial inequality involving binomial coefficients.

Extremizers for the Airy–Strichartz inequality

We identify the compactness threshold for optimizing sequences of the Airy-- Strichartz inequality as an explicit multiple of the sharp constant in the Strichartz inequality. In particular, if the sharp constant in the Airy--Strichartz inequality is strictly smaller than this multiple of the sharp constant in the Strichartz inequality, then there is an optimizer for the former inequality. Our result is valid for the full range of Airy--Strichartz inequalities (except the endpoints) both in the diagonal and off-diagonal cases.

Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates

We generalize the theorems of Stein-Tomas and Strichartz about surface restrictions of Fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions. We deduce the corresponding Strichartz bounds for solutions to Schr\odinger equations up to the endpoint, thereby solving an open problem of Frank, Lewin, Lieb, and Seiringer. We also prove uniform Sobolev estimates in Schatten spaces, extending the results of Kenig, Ruiz, and Sogge. We finally provide applications of these results to a Limiting Absorption Principle in Schatten spaces, to the well-posedness of the Hartree equation in Schatten spaces, to Lieb-Thirring bounds for eigenvalues of Schr\odinger operators with complex potentials, and to Schatten properties of the scattering matrix.

Decay estimates for fractional wave equations on H-type groups

The aim of this paper is to establish the decay estimate for the fractional wave equation semigroup on H-type groups given by $e^{it\Delta^{\alpha}}$ , $0<\alpha<1$ . Combining the dispersive estimate and a standard duality argument, we also derive the corresponding Strichartz inequalities.

Local smoothing for the quantum Liouville equation

We analyze in this work the regularity properties of the density operator solution to the quantum Liouville equation. As was recently done for the Strichartz inequalities, we extend to systems of orthonormal functions the local smoothing estimates satisfied by the solutions to the Schrödinger equation. We show in particular that the local density associated to the solution to the free, linear, quantum Liouville equation admits locally fractional derivatives of given order provided the data belong to some Schatten spaces.

On one dimensional quantum Zakharov system

In this paper, we discuss the properties of one dimensional quantum Zakharov system which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The system with initial data $(E(0),n(0),\partial_t n(0))\in H^k\bigoplus H^l\bigoplus H^{l-2}$ is local well posedness in low regularity spaces. Especially, the low regularity result for $k$ satisfies $-3/4<k\leq -1/4$ is obtained by using the key observation that the convoluted phase function is convex and careful bilinear analysis. The result can not be obtained by using only Strichartz inequalities for "Schr\"{o}dinger" waves.

Scale-invariant Strichartz estimates on tori and applications

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.

Applications of Bourgain–Brézis inequalities to fluid mechanics and magnetism

As a consequence of inequalities due to Bourgain–Brézis, we obtain local-in-time well-posedness for the two-dimensional Navier–Stokes equation with velocity bounded in spacetime and initial vorticity in bounded variation. We also obtain spacetime estimates for the magnetic field vector through improved Strichartz inequalities.