Flow monotonicity and Strichartz inequalities

We identify complete monotonicity properties underlying a variety of well-known sharp Strichartz inequalities in euclidean space.

Related papers:

title: Monotonicity and its analytic and geometric implications
abstract: In this expository article, we discuss various monotonicity formulas for parabolic and elliptic operators and explain how the analysis of function spaces and the geometry of the underlining spaces are intertwined. After briefly discussing some of the well-known analytical applications of monotonicity for parabolic operators, we turn to their elliptic counterparts, their geometric meaning, and some geometric consequences.

title: Heat-flow monotonicity of Strichartz norms
abstract: Most notably we prove that for $d=1,2$ the classical Strichartz norm $$\|e^{i s\Delta}f\|_{L^{2+4/d}_{s,x}(\mathbb{R}\times\mathbb{R}^d)}$$ associated to the free Schr\"{o}dinger equation is nondecreasing as the initial datum $f$ evolves under a certain quadratic heat-flow.

title: A sharp Strichartz estimate for the wave equation with data in the energy space
abstract: 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.

title: A Sharp Inequality for the Strichartz Norm
abstract: Let be the solution of the linear Schrodinger equationIn the first part of this paper, we obtain a sharp inequality for the Strichartz norm , where , and , that admits only Gaussian maximizers. As corollaries, we obtain sharp forms of the classical Strichartz inequalities in low dimensions (works of Foschi  and Hundertmark-Zharnitsky [6]) and also sharp forms of some Sobolev-Strichartz inequalities. In the second part of the paper, we express Foschi's [4] sharp inequalities for the Schrodinger and wave equations in the broader setting of sharp restriction/extension estimates for the paraboloid and the cone.

title: Maximizers for the Strichartz Inequality
abstract: We compute explicitly the best constants and, by solving some functional equations, we find all maximizers for homogeneous Strichartz estimates for the Schrodinger equation and for the wave equation in the cases when the Lebesgue exponent is an even integer

title: Hardy-Littlewood-Sobolev inequalities via fast diffusion flows
abstract: We give a simple proof of the λ = d - 2 cases of the sharp Hardy-Littlewood-Sobolev inequality for d≥3, and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for d = 2 via a monotone flow governed by the fast diffusion equation.

title: On sharp bilinear Strichartz estimates of Ozawa-Tsutsumi type
abstract: We provide a comprehensive analysis of sharp bilinear estimates of Ozawa-Tsutsumi type for solutions u of the free Schrodinger equation, which give sharp control on $|u|^2$ in classical Sobolev spaces. In particular, we provide a generalization of their estimates in such a way that provides a unification with some sharp bilinear estimates proved by Carneiro and Planchon-Vega, via entirely different methods, by seeing them all as special cases of a one parameter family of sharp estimates. We show that the extremal functions are solutions of the Maxwell-Boltzmann functional equation and provide a new proof that this equation admits only Gaussian solutions. We also make a connection to certain sharp estimates on $u^2$ involving certain dispersive Sobolev norms.

title: Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid
abstract: We study the problem of existence of extremizers for the L2 to Lp adjoint Fourier restriction inequalities for the hyperboloid in dimensions 3 and 4 in the case p is an even integer. We use the method developed by Foschi in  to show that extremizers do not exist.

title: Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion
abstract: In the euclidean space, Sobolev and Hardy-Littlewood-Sobolev inequalities can be related by duality. In this paper, we investigate how to relate these inequalities using the flow of a fast diffusion equation in dimension $d\ge3$. The main consequence is an improvement of Sobolev's inequality when $d\ge5$, which involves the various terms of the dual Hardy-Littlewood-Sobolev inequality. In dimension $d=2$, Onofri's inequality plays the role of Sobolev's inequality and can also be related to its dual inequality, the logarithmic Hardy-Littlewood-Sobolev inequality, by a super-fast diffusion equation.

title: A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions
abstract: We prove a sharp bilinear inequality for the Klein-Gordon equation on $\sr^{d+1}$, for any $d \geq 2$. This extends work of Ozawa-Rogers and Quilodr\'an for the Klein-Gordon equation and generalises work of Bez-Rogers for the wave equation. As a consequence we obtain a sharp Strichartz estimate for the solution of the Klein-Gordon equation in five spatial dimensions for data belonging to $H^1$. We show that maximisers for this estimate do not exist and that any maximising sequence of initial data concentrates at spatial infinity.

title: Uniqueness of Extremizers for an Endpoint Inequality of the $$k$$k-Plane Transform
abstract: The $$k$$k-plane transform is a bounded operator from $$L^p(\mathbb {R}^n)$$Lp(Rn) to $$L^q$$Lq of the Grassmann manifold of all affine $$k$$k-planes in $$\mathbb {R}^n$$Rn for certain exponents depending on $$k$$k and $$n$$n. In the endpoint case $$q=n+1$$q=n+1, we identify all extremizers of the associated inequality for the general $$k$$k-plane transform.

title: Sharp Strichartz estimates on nontrapping asymptotically conic manifolds
abstract: <abstract abstract-type="TeX"><p>We obtain the Strichartz inequalities ║u║<sub xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L<sup>q</sup><sub class="stack">t</sub>L<sup>r</sup><sub class="stack">x</sub></i> ([0,1]×<i>M</i>)</sub> ≥ C║<i>u</i>(0)<sub xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L<sup>2</sup></i>║(<i>M</i>)</sub> for any smooth <i>n</i>-dimensional Riemannian manifold <i>M</i> which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and nontrapping, where <i>u</i> is a solution to the Schrödinger equation <i>iu</i><sub xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>t</i></sub> + 1/2 Δ<i><sub>M</sub>u</i> = 0, and 2 < <i>q, r</i> ≥ ∞ are admissible Strichartz exponents (2/<i>q</i> + <i>n/r</i> = <i>n</i>/2). This corresponds with the estimates available for Euclidean space (except for the endpoint (<i>q, r</i>) = (2, 2<i>n/n</i>-2) when <i>n</i> > 2). These estimates imply existence theorems for semi-linear Schrödinger equations on <i>M</i>, by adapting arguments from Cazenave and Weissler and Kato.</p><p>This result improves on our previous result, which was an <i>L</i><sup>4</sup><sub class="stack" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t,x</sub> Strichartz estimate in three dimensions. It is closely related to results of Staffilani-Tataru, Burq, Robbiano-Zuily and Tataru, who consider the case of asymptotically flat manifolds.

title: The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation
abstract: We consider the cubic nonlinear Schrodinger (NLS) equation set on a two dimensional box of size $L$ with periodic boundary conditions. By taking the large box limit $L \to \infty$ in the weakly nonlinear regime (characterized by smallness in the critical space), we derive a new equation set on $\R^2$ that approximates the dynamics of the frequency modes. This nonlinear equation turns out to be Hamiltonian and enjoys interesting symmetries, such as its invariance under Fourier transform, as well as several families of explicit solutions. A large part of this work is devoted to a rigorous approximation result that allows to project the long-time dynamics of the limit equation into that of the cubic NLS equation on a box of finite size.

title: Global Maximizers for the Sphere Adjoint Fourier Restriction Inequality
abstract: We show that constant functions are global maximizers for the adjoint Fourier restriction inequality for the sphere.

title: ENDPOINT STRICHARTZ ESTIMATES By MARKUS KEEL and TERENCE TAO
abstract: We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrödinger equation (in dimension n 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation.

title: On the Strichartz Estimates for the Kinetic Transport Equation
abstract: We show that the endpoint Strichartz estimate for the kinetic transport equation is false in all dimensions. We also present an alternative approach to proving the non-endpoint cases using multilinear analysis.

title: Best constant and value of extremizers for a k-plane transform inequality
abstract: The k-plane transformRk acting on test functions onR satisfies a dilationinvariantL → L inequality for some exponents p, q. We will explicit some extremizers and the value of the best constant for any value of k andd, solving the endpoint case of a conjecture from Baernstein and Loss. This extends their own res ult for k = 2 and Christ’s result for k = d− 1.

title: Extremizers of a Radon transform inequality
abstract: The Radon transform is a bounded operator from L^p of Euclidean space R^d to L^q of the Grassmann manifold of all affine hyperplanes in R^d, for certain exponents. We identify all extremizers of the associated inequality for the endpoint case p=(d+1)/d and q=d+1.

title: On the multilinear restriction and Kakeya conjectures
abstract: We prove d-linear analogues of the classical restriction and Kakeya conjectures in Rd. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called “joints” problem, as well as presenting some n-linear analogues for n < d.