Richard Oberlin

A variation norm Carleson theorem

By a standard approximation argument it follows that S[f ] may be meaningfully defined as a continuous function in ξ for almost every x whenever f ∈ L and the a priori bound of the theorem continues to hold for such functions. Theorem 1.1 is intimately related to almost everywhere convergence of partial Fourier sums for functions in L[0, 1]. Via a transference principle [12], it is indeed equivalent to the celebrated theorem by Carleson [2] for p = 2 and the extension of Carleson’s theorem by Hunt [9] for 1 < p < ∞; see also [7],[15], and [8]. The main purpose of this paper is to sharpen Theorem 1.1 towards control of the variation norm in the parameter ξ. Thus we consider mixed L and V r norms of the type:

THE KAKEYA SET AND MAXIMAL CONJECTURES FOR ALGEBRAIC VARIETIES OVER FINITE FIELDS

Using the polynomial method of Dvir [On the size of Kakeya sets in finite fields. Preprint ], we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties W over finite fields F . For instance, given an ( n −1)-dimensional projective variety W ⊂¶ n ( F ), we establish the Kakeya maximal estimate for all functions f : F n → R and d ≥1, where for each w ∈ W , the supremum is over all irreducible algebraic curves in F n of degree at most d that pass through w but do not lie in W , and with C n , W , d depending only on n , d and the degree of W ; the special case when W is the hyperplane at infinity in particular establishes the Kakeya maximal function conjecture in finite fields, which in turn strengthens the results of Dvir.

Sampling on the Sierpinski Gasket

We study regular and irregular sampling for functions defined on the Sierpinski Gasket (SG), where we interpret “bandlimited” to mean the function has a finite expansion in the first dm Dirichlet eigenfunctions of the Laplacian as defined by Kigami, and dm is the cardinality of the sampling set. In the regular case, we take the sampling set to be the nonboundary vertices of the level m graph approximating SG. We prove that regular sampling is always possible, and we give an algorithm to compute the sampling functions, based on an extension of the spectral decimation method of Fukushima and Shima to include inner products. We give experimental evidence that the sampling functions decay rapidly away from the sampling point, in striking contrast to the classical theory on the line where the sinc function exhibits excruciatingly slow decay. Similar behavior appears to hold for certain Dirichlet kernels. We show by example that the sampling formula provides an appealing method of approximating functions that are not necessarily bandlimited, and so might be useful for numerical analysis. We give experimental evidence that reasonable perturbations of one of the regular sampling sets remains a sampling set. In contrast to what happens on the unit interval, it is not true that all sets of the correct cardinality are sampling sets.

Estimates for the X-ray transform restricted to 2-manifolds

We prove almost sharp mixed-norm estimates for the X-ray transform restricted to lines whose directions lie on certain well-curved two dimensional manifolds.

Estimates for Compositions of Maximal Operators with Singular Integrals

We prove weak-type (1,1) estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator

A Marcinkiewicz maximal-multiplier theorem

\Delta^*\Psi

Bounds on the Walsh model for

where

M^{q,*}

\Delta^*

Carleson and related operators

is Bourgains maximal multiplier operator and

SPHERICAL MEANS AND PINNED DISTANCE SETS

\Psi