arXiv subjects
Publications and source records attributed to Michael Christ.
The Riesz-Sobolev inequality relates the convolution of nonnegative functions on Euclidean space to the convolution of their symmetric nonincreasing rearrangements. We show that for dimension one, for indicator functions of sets, if the inequality is sufficiently close to an equality then the sets in question must nearly coincide with intervals.
Those functions which nearly extremize Young's convolution inequality are characterized for discrete groups which have no nontrivial finite subgroups. Near-extremizers of the Hausdorff-Young inequality are characterized for Z^d.
It is a well known general principle that the Fourier transform of a random measure is small, except at the zero frequency, in various senses for appropriate notions of randomness. In this note we develop analogues of this principle for two classes of random multilinear operators. Two ergodic theoretic applications, involving correlations over randomly generated sparse subsequences, are obtained as corollaries of the main results.
Examples are constructed of sparse subsequences of the integers for which the associated maximal averages operator is of weak type (1,1). A consequence, by transference, is that an almost everywhere L^1 -- type ergodic theorem holds for corresponding subsequences of iterates of general measure-preserving transformations. These examples can be constructed so that n_k has growth rate k^m for any prescribed integer power m greater than or equal to 2. Urban and Zienkiewicz have established the same conclusion for other subsequences, which have growth rate k^m for noninteger exponents m sufficiently close to 1; the first novelty here is that the exponent can be arbitrarily large. In contrast, Buczolich and Mauldin have shown that the corresponding conclusion fails to hold if n_k is exactly k^2. The rather simple analysis relies on certain exponential sum bounds of Weil, together with a decomposition of Calderon-Zygmund type in which the exceptional set is defined in terms of the subsequence in question. The subsequences used are closely related to those employed by Rudin in a 1960 paper in which examples of Lambda(p) sets were constructed.
We examine a certain class of trilinear integral operators which incorporate oscillatory factors e^{iP}, where P is a real-valued polynomial, and prove smallness of such integrals in the presence of rapid oscillations.
In 2005, Li, Tao, Thiele and the author raised a general question concerning upper bounds for a class of multilinear oscillatory integral operators, and established such bounds in a few cases. Most cases remain open. The present paper is concerned with sublevel set bounds, which would be a consequence of the oscillatory integral bounds, if valid. These sublevel set bounds are established here in a weak form but in nearly full generality, subject only to a rationality hypothesis. The proof relies on an extension of Szemeredi's theorem due to Furstenberg and Katznelson.
Christ, Li, Tao, and Thiele have established multilinear oscillatory integral operator inequalities under severe dimensional restrictions. These restrictions are relaxed substantially in the present paper, but are by no means eliminated. The simple method used here has the advantage of yielding power type bounds, but cannot be expected to extend to the general case.
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.
Convolution with an appropriate surface measure on a paraboloid is known to define a bounded operator T from L^p(R^d) to L^q(R^d) for certain exponents p,q. By a quasiextremal for the associated inequality, we mean a function f for which the norm of Tf is at least a constant c times the norm of f. Our main result characterizes all quasiextremals, with some quantitative control in terms of c. Several related results are also discussed. This is the first in a series of at least four articles about a circle of questions concerning the inverse problem of deducing information about f from information about the ratio of the norm of Tf to the norm of f.
Convolution with an appropriate surface measure on a paraboloid in R^d defines a bounded operator T from L^p(R^d) to L^q(R^d) for certain exponents p,q. In this article it is proved that there exist functions which extremize the associated inequality.
Let $d\ge 2$ and $T$ be the convolution operator $Tf(x)=\int_{\reals^{d-1}} f(x'-t,x_d-|t|^2)\,dt$, which is is bounded from $L^{(d+1)/d}(\reals^d)$ to $L^{d+1}(\reals^d)$. We show that any critical point $f\in L^{(d+1)/d}$ of the functional $\norm{Tf}_{d+1}/\norm{f}_{(d+1)/d}$ is infinitely differentiable, and that $|x|^δf\in L^{(d+1)/d}$ for some $δ>0$. In particular, this holds for all extremizers of the associated inequality. This is done by exploiting a generalized Euler-Lagrange equation, and certain weighted norm inequalities for $T$.
It was proved independently by Foschi and Hundertmark, Zharnitsky that Gaussians extremize the adjoint Fourier restriction inequality for L^2 functions on the paraboloid in the two lowest-dimesional cases. Here we prove that Gaussians are critical points for the L^p to L^q adjoint Fourier restriction inequalities if and only if p=2. Also, Gaussians are critial points for the L^2 to L^r_t L^q_x Strichartz inequalities for all admissible pairs (r,q) in (1,infinity)^2.
The adjoint Fourier restriction inequality for the sphere $S^2$ states that if $f\in\lt(S^2,σ)$ then $\widehat{fσ}\in L^4(\reals^3)$. We prove that all critical points $f$ of the functional $\norm{\widehat{fσ}}_{L^4}/\norm{f}_{\lt}$ are smooth; that any complex-valued extremizer for the inequality is a nonnegative extremizer multiplied by the character $e^{ix\cdotξ}$ for some $ξ$; and that complex-valued extremizing sequences for the inequality are precompact modulo multiplication by characters.
The adjoint Fourier restriction inequality of Tomas and Stein states that the mapping $f\mapsto \widehat{fσ}$ is bounded from $\lt(S^2)$ to $L^4(\reals^3)$. We prove that there exist functions which extremize this inequality, and that any extremizing sequence of nonnegative functions has a subsequence which converges to an extremizer.
Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrödinger equation on the real line are studied in Sobolev spaces $H^s$, for $s$ negative but close to 0. For smooth solutions there is an {\em a priori} upper bound for the $H^s$ norm of the solution, in terms of the $H^s$ norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrary initial data in $H^s$.
We consider the Brascamp--Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse the inequalities from a structural perspective. We obtain two new proofs of Lieb's fundamental theorem concerning exhaustion by gaussians. Our techniques are partly based upon monotonicity formulas for positive solutions to heat equations in linear and multilinear settings.
In this paper we investigate the mapping properties in Lebesgue-type spaces of certain generalized Radon transforms defined by integration over curves.
A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalities of Holder, Young, and Loomis-Whitney. This is a companion to a recent paper by the same authors, containing a different proof and discussing certain variants.