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Michael Christ

Publications and source records attributed to Michael Christ.

At least 73 records · Page 4Linked to original sources
arXiv2005

Necessary conditions for vector-valued operator inequalities in harmonic analysis

Via a random construction we establish necessary conditions for $L^p(\ell^q)$ inequalities for certain families of operators arising in harmonic analysis. In particular we consider dilates of a convolution kernel with compactly supported Fourier transform, vector maximal functions acting on classes of entire functions of exponential type, and a characterization of Sobolev spaces by square functions and pointwise moduli of smoothness.

math.CA↗
arXiv2005

A remark on sums of squares of complex vector fields

This note is a comment on a recent paper by J. J. Kohn. We give an example of a second order partial differential operator, expressed as a sum of squares of complex vector fields satisfying the bracket condition, that is not hypoelliptic.

math.CV↗
arXiv2005

Nonuniqueness of weak solutions of the nonlinear Schroedinger equation

Generalized solutions of the Cauchy problem for the one-dimensional periodic nonlinear Schrödinger equation, with certain nonlinearities, are not unique. For any $s<0$ there exist nonzero generalized solutions varying continuously in the Sobolev space $H^s$, with identically vanishing initial data.

math.AP↗
arXiv2005

Power series solution of a nonlinear Schroedinger equation

A slightly modified variant of the cubic periodic one-dimensional nonlinear Schroedinger equation is shown to admit weak solutions for all initial data in certain function spaces wider than L^2. These solutions depend uniformly continuously on the initial data, in the norms considered. The solutions are constructed as sums of infinite series of multilinear operators applied to initial data; no fixed point argument or energy inequality are used. In a companion paper we have shown that weak solutions in these same function spaces are however not unique.

math.AP↗
arXiv2004

Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrödinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities. In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces,in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the $L^2$ topology, despite the existence of global weak solutions at this regularity. Finally, we investigate the KdV equation at the endpoint regularity $H^{-3/4}$, and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval $[-T,T]$ and a locally Lipschitz continuous map taking the initial data in $H^{-3/4}$ to a distributional solution $u \in C^0 ([-T,T]; $H^{-3/4})$ which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV.

math.AP↗
arXiv2003

Compactness in the d-bar Neumann problem, magnetic Schrodinger operators, and the Aharonov-Bohm effect

Compactness of the d-bar Neumann operator is studied for weakly pseudoconvex bounded Hartogs domains in two dimensions. A nonsmooth example is constructed in which condition (P) fails to hold, yet the Neumann operator is compact. The main result, in contrast, is that for smoothly bounded Hartogs domains, condition (P) of Catlin and Sibony is equivalent to compactness. The analyses of both compactness and condition (P) boil down to properties of the lowest eigenvalues of certain sequences of Schrodinger operators, with and without magnetic fields, parametrized by a Fourier variable resulting from the Hartogs symmetry. The nonsmooth counterexample is based on the Aharonov-Bohm phenomenon of quantum mechanics. For smooth domains, we prove that there always exists an exceptional sequence of Fourier variables for which the Aharonov-Bohm effect is quite weak. This sequence can be quite sparse, so that the failure of compactness is due to a rather subtle effect.

math.CV↗
arXiv2003

Instability of the periodic nonlinear Schrodinger equation

We study the periodic non-linear Schrodinger equations with odd integer power nonlinearities, for initial data which are assumed to be small in some negative order Sobolev space, but which may have large L^2 mass. These equations are known to be illposed in H^s for all negative s, in the sense that the solution map fails to be uniformly continuous from H^s to itself, even for short times and small norms. Here we show that these equations are even more unstable. For the cubic equation, the solution map is discontinuous from H^s to even the space of distributions. For the quintic and higher order nonlinearities, there exist pairs of solutions which are uniformly bounded in H^s, are arbitrarily close in any C^N norm at time zero, and fail to be close in the distribution topology at an arbitrarily small positive time.

math.AP↗
arXiv2003

On multilinear oscillatory integrals, nonsingular and singular

Basic questions concerning nonsingular multilinear operators with oscillatory factors are posed and partially answered. Lebesgue space norm inequalities are established for multilinear integral operators of Calderon-Zygmund type which incorporate oscillatory factors exp(iP), where P is a real-valued polynomial with large coefficients. A related problem concerning upper bounds for measures of sublevel sets is solved.

math.CA↗
arXiv2003

Ill-posedness for nonlinear Schrodinger and wave equations

The nonlinear wave and Schrodinger equations on Euclidean space of any dimension, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space of index s whenever the exponent s is lower than that predicted by scaling or Galilean invariances, or when the regularity is too low to support distributional solutions. This extends previous work of the authors, which treated the one-dimensional cubic nonlinear Schrodinger equation. In the defocusing case soliton or blowup examples are unavailable, and a proof of ill-posedness requires the construction of other solutions. In earlier work this was achieved using certain long-time asymptotic behavior which occurs only for low power nonlinearities. Here we analyze instead a class of solutions for which the zero-dispersion limit provides a good approximation.

math.AP↗
arXiv1999

Singular and maximal Radon transforms: Analysis and geometry

The purpose of this paper is to prove the L^p boundedness of singular Radon transforms and their maximal analogues. These operators differ from the traditional singular integrals and maximal functions in that their definition at any point x in R^n involves integration over a k-dimensional submanifold of R^n, depending on x, with k < n. The role of the underlying geometric data which determines the submanifolds and how they depend on x, makes the analysis of these operators quite different from their standard analogues. In fact, much of our work is involved in the examination of the resulting geometric situation, and the elucidation of an attached notion of curvature (a kind of ``finite-type'' condition) which is crucial for our analysis.

math.CA↗
arXiv1998

Hypoellipticity: Geometrization and Speculation

To any finite collection of smooth real vector fields $X_j$ in $\reals^n$ we associate a metric in the phase space $T^*\reals^n$. The relation between the asymptotic behavior of this metric and hypoellipticity of $\sum X_j^2$, in the smooth, real analytic, and Gevrey categories, is explored.

math.FA↗
arXiv1997

Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: some optimal results

The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic Schrödinger operators is preserved under all perturbations $V(x)$ satisfying $|V(x)|\leq C(1+x)^{-α}$, $α>\frac{1}{2}.$ This result is optimal in the power scale. More general classes of perturbing potentials which are not necessarily power decaying are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on new maximal function and norm estimates and almost everywhere convergence results for certain multilinear integral operators.

math.SP↗
arXiv1996

Intermediate Optimal Gevrey Exponents Occur

Hypoellipticity in Gevrey classes $G^s$ is characterized for a simple family of sums of squares of vector fields satisfying the bracket hypothesis, with analytic coefficients. It is shown that hypoellipticity holds if and only if $s$ is greater than or equal to an optimal exponent that may take on any rational value.

math.FA↗
arXiv1996

Analytic hypoellipticity in dimension two

A simple geometric condition is sufficient for analytic hypoellipticity of sums of squares of two vector fields in ${\mathbb R}^2$. This condition is proved to be necessary for generic vector fields and for various special cases, and to be both necessary and sufficient for a closely related family of operators.

math.FA↗
arXiv1995

Global Irregularity For Degenerate Elliptic Operators

Examples are given of degenerate elliptic operators on smooth, compact manifolds that are not globally regular in $C^\infty$. These operators degenerate only in a rather mild fashion. Certain weak regularity results are proved, and an interpretation of global irregularity in terms of the associated heat semigroup is given.

math.AP↗