Uniform Sobolev Inequalities Research Articles

From spectral cluster to uniform resolvent estimates on compact manifolds

It is well known that uniform resolvent estimates imply spectral cluster estimates. We show that the converse is also true in some cases. In particular, Sogge's universal spectral cluster estimates for the Laplace–Beltrami operator on closed Riemannian manifolds directly imply uniform resolvent estimates outside a parabolic region, without any reference to parametrices. The method is purely functional analytic and takes full advantage of the known spectral cluster bounds. This yields new resolvent estimates for manifolds with boundary or with low-regularity metrics, among other examples. Moreover, we show that the resolvent estimates are stable under perturbations and use this to establish uniform Sobolev and spectral cluster inequalities for Schrödinger operators with singular potentials.

The 𝑊^{𝑠,𝑝}-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential

In this paper, we investigate the W s , p W^{s,p} -boundedness for stationary wave operators of the Schrödinger operator with inverse-square potential L a = − Δ + a | x | 2 , a ≥ − ( d − 2 ) 2 4 , \begin{equation*} \mathcal L_a=-\Delta +\tfrac {a}{|x|^2}, \quad a\geq -\tfrac {(d-2)^2}{4}, \end{equation*} in dimension d ≥ 2 d\geq 2 . We construct the stationary wave operators in terms of integrals of Bessel functions and spherical harmonics, and prove that they are W s , p W^{s,p} -bounded for certain p p and s s which depend on a a . As corollaries, we solve some open problems associated with the operator L a \mathcal L_a , which include the dispersive estimates and the local smoothing estimates in dimension d ≥ 2 d\geq 2 . We also generalize some known results such as the uniform Sobolev inequalities, the equivalence of Sobolev norms and the Mikhlin multiplier theorem, to a larger range of indices. These results are important in the description of linear and nonlinear dynamics for dispersive equations with inverse-square potential.

Strichartz and uniform Sobolev inequalities for the elastic wave equation

We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are deduced. For the purpose we diagonalize the symbols of the Lamé operator and its semigroup, which also gives an alternative and simpler proofs of the previous results on perturbed elastic wave equations. Furthermore, we obtain uniform Sobolev inequalities for the elastic wave operator.

A Uniform Sobolev Inequality for Ancient Ricci Flows with Bounded Nash Entropy

Abstract This note is a continuation of [7]. We shall show that an ancient Ricci flow with uniformly bounded Nash entropy also has uniformly bounded $\nu $-functional. Consequently, on such an ancient solution, there are uniform logarithmic Sobolev and Sobolev inequalities. We emphasize that the main theorem in this paper is true so long as the theory in [3] is valid, and in particular, when the underlying manifold is closed.

Uniform Sobolev estimates on compact manifolds involving singular potentials

We obtain generalizations of the uniform Sobolev inequalities of Kenig, Ruiz and the fourth author (1986) for Euclidean spaces and Dos Santos Ferreira, Kenig and Salo (2014) for compact Riemannian manifolds involving critically singular potentials V\in L^{n/2} . We also obtain the analogous improved quasimode estimates of the first, third and fourth author (2021), Hassell and Tacy (2015), the first and fourth author (2019), and Hickman (2020), as well as analogues of the improved uniform Sobolev estimates of Bourgain, Shao, the fourth author and Yao (2015), and Hickman (2020), involving such potentials. Additionally, on S^n , we obtain sharp uniform Sobolev inequalities involving such potentials for the optimal range of exponents, which extend the results of S. Huang and the fourth author (2014). For general Riemannian manifolds, we improve the earlier results in of the first, third and fourth authors (2021) by obtaining quasimode estimates for a larger (and optimal) range of exponents under the weaker assumption that V\in L^{n/2} .

A Conservative Scheme with Optimal Error Estimates for a Multidimensional Space–Fractional Gross–Pitaevskii Equation

Abstract The present work departs from an extended form of the classical multi-dimensional Gross–Pitaevskii equation, which considers fractional derivatives of the Riesz type in space, a generalized potential function and angular momentum rotation. It is well known that the classical system possesses functionals which are preserved throughout time. It is easy to check that the generalized fractional model considered in this work also possesses conserved quantities, whence the development of conservative and efficient numerical schemes is pragmatically justified. Motivated by these facts, we propose a finite-difference method based on weighted-shifted Grünwald differences to approximate the solutions of the generalized Gross–Pitaevskii system. We provide here a discrete extension of the uniform Sobolev inequality to multiple dimensions, and show that the proposed method is capable of preserving discrete forms of the mass and the energy of the model. Moreover, we establish thoroughly the stability and the convergence of the technique, and provide some illustrative simulations to show that the method is capable of preserving the total mass and the total energy of the generalized system.

Endpoint uniform Sobolev inequalities for elliptic operators with applications

In this paper, we obtain sharp restricted weak-type uniform Sobolev inequalities for a class of elliptic operators P(D). In particular, we improve a recent result of Sikora, Yan and Yao [24] to the endpoint case. The sharpness is proved by constructing explicit counterexamples. As applications, we establish Stein-Tomas type inequalities for H=P(D)+V with certain class of potentials V.

Eigenvalue bounds for non-self-adjoint Schrödinger operators with the inverse-square potential

The purpose of this paper is to study spectral properties of the non-self-adjoint Schrödinger operator -\Delta-\frac{(n-2)^2}{4|x|^{2}}+V on \mathbb R^n with complex-valued potential V\in L^{p,\infty} , p > n/2 . We prove Keller type inequalities which measure the radius of a disc containing the discrete spectrum, in terms of the L^{p,\infty} norm of V . Similar inequalities also hold if the inverse-square potential is replaced by a large class of subcritical potentials with critical singularities at the origin. The main new ingredient in the proof is the uniform Sobolev inequality of Kenig–Ruiz–Sogge type for Schrödinger operators with strongly singular potentials, which may be of independent interest.

An endpoint version of uniform Sobolev inequalities

Abstract We prove an endpoint version of the uniform Sobolev inequalities in [C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 1987, 329–347]. It was known that strong type inequalities no longer hold at the endpoints; however, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient in our proof is a type of interpolation first introduced by Bourgain [J. Bourgain, Esitmations de certaines functions maximales, C. R. Acad. Sci. Paris 310 1985, 499–502]. We also prove restricted weak type Stein–Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold.

Uniform resolvent and Strichartz estimates for Schrödinger equations with critical singularities

This paper deals with global dispersive properties of Schrödinger equations with real-valued potentials exhibiting critical singularities, where our class of potentials is more general than inverse-square type potentials and includes several anisotropic potentials. We first prove weighted resolvent estimates, which are uniform with respect to the energy, with a large class of weight functions in Morrey–Campanato spaces. Uniform Sobolev inequalities in Lorentz spaces are also studied. The proof employs the iterated resolvent identity and a classical multiplier technique. As an application, the full set of global-in-time Strichartz estimates including the endpoint case, is derived. In the proof of Strichartz estimates, we develop a general criterion on perturbations ensuring that both homogeneous and inhomogeneous endpoint estimates can be recovered from resolvent estimates. Finally, we also investigate uniform resolvent estimates for long range repulsive potentials with critical singularities by using an elementary version of the Mourre theory.

Spectral Multipliers, Bochner–Riesz Means and Uniform Sobolev Inequalities for Elliptic Operators

This article comprises two parts. In the first, we study |$L^p$| to |$L^q$| bounds for spectral multipliers and Bochner–Riesz means with negative index in the general setting of abstract self-adjoint operators. In the second, we obtain the uniform Sobolev estimates for constant coefficients higher order elliptic operators |$P(D)-z$| and all |$z\in {\mathbb C}\backslash [0, \infty)$|⁠, which give an extension of the second order results of Kenig–Ruiz–Sogge [42]. Next we use perturbation techniques to prove the uniform Sobolev estimates for Schrödingier operators |$P(D)+V$| with small integrable potentials |$V$|⁠. Finally, we deduce spectral multiplier applications for all these operators, including sharp Bochner–Riesz summability results.

Uniform Sobolev inequalities for second order non-elliptic differential operators

We study uniform Sobolev inequalities for the second order differential operators P(D) of non-elliptic type. For d≥3 we prove that the Sobolev type estimate ‖u‖Lq(Rd)≤C‖P(D)u‖Lp(Rd) holds with C independent of the first order and the constant terms of P(D) if and only if 1/p−1/q=2/d and 2d(d−1)d2+2d−4<p<2(d−1)d. We also obtain restricted weak type endpoint estimates for the critical (p,q)=(2(d−1)d,2d(d−1)(d−2)2), (2d(d−1)d2+2d−4,2(d−1)d−2). As a consequence, the result extends the class of functions for which the unique continuation for the inequality |P(D)u|≤|Vu| holds.

Convergence of the Generalized Kähler-Ricci Flow

In this paper, we consider the convergence of the generalized Kahler-Ricci flow with semi-positive twisted form \(\theta \) on Kahler manifold \(M\). We give detailed proofs of the uniform Sobolev inequality and some uniform estimates for the metric potential and the generalized Ricci potential along the flow. Then assuming that there exists a generalized Kahler-Einstein metric, if the twisting form \(\theta \) is strictly positive at a point or \(M\) admits no nontrivial Hamiltonian holomorphic vector field, we prove that the generalized Kahler-Ricci flow must converge in \(C^\infty \) topology to a generalized Kahler-Einstein metric exponentially fast, where we get the exponential decay without using the Futaki invariant.

Sobolev-type inequalities and heat kernel bounds along the geometric flow

This paper is concerned with Sobolev-type inequalities and upper bound for the fundamental solution to the heat-type equation defined on compact manifold whose metric evolves by the generalized geometric flow. It turns out that the pointwise estimates obtained in this paper depend on the constants in the uniform Sobolev inequalities for the flow or the best constants in the euclidean Sobolev embedding. We give various illustrations to show that our results are valid in many contexts of geometric flow, where we may not need explicit curvature constraint. Moreover, our approach here also demonstrates equivalence of Sobolev inequalities, log-Sobolev inequalities, ultracontractive estimates and heat kernel upper bounds.

The Logarithmic Sobolev and Sobolev Inequalities Along the Ricci Flow

Based on Perelman’s entropy monotonicity, uniform logarithmic Sobolev inequalities along the Ricci flow are derived. Then uniform Sobolev inequalities along the Ricci flow are derived via harmonic analysis of the integral transform of the relevant heat operator. These inequalities are fundamental analytic properties of the Ricci flow. They are also extended to the volume-normalized Ricci flow and the Kahler–Ricci flow.

The Logarithmic Sobolev Inequality Along the Ricci Flow: The Case $$\lambda _0(g_0)=0$$ λ 0 ( g 0 ) = 0

A uniform logarithmic Sobolev inequality, a uniform Sobolev inequality and a uniform \(\kappa \)-noncollapsing estimate along the Ricci flow are established in the situation that a certain smallest eigenvalue for the initial metric is zero.

Uniform Sobolev Inequality along the Sasaki–Ricci Flow

In this paper we prove a uniform Sobolev inequality along the Sasaki–Ricci flow. In the process, we develop the theory of basic Lebesgue and Sobolev function spaces, and prove some general results about the decomposition of the heat kernel for a class of elliptic operators on a Sasaki manifold.

Eigenvalue bounds for Schrödinger operators with complex potentials

We show that the absolute values of non-positive eigenvalues of Schr\"odinger operators with complex potentials can be bounded in terms of L_p-norms of the potential. This extends an inequality of Abramov, Aslanyan, and Davies to higher dimensions and proves a conjecture by Laptev and Safronov. Our main ingredient are the uniform Sobolev inequalities of Kenig, Ruiz, and Sogge.

Erratum to: A Uniform Sobolev Inequality Under Ricci Flow

In the paper [3], the constants in Theorem 1.1 (a) should depend on time in general. Here, we provide the corrected statement and proof. “A uniform Sobolev inequality under Ricci flow”, the constants A and B of Theorem 1.1 (a) are correct only for finite time. For large time t, they should depend on time t via the controlling constants in the log Sobolev inequality of the metric t g(0), which is finite only sometimes. Step 2 of the proof should follow the method outlined at the end of the introduction instead. Here is the corrected statement for that theorem.

Addendum to: A Uniform Sobolev Inequality under Ricci Flow

Addendum to: A Uniform Sobolev Inequality under Ricci Flow