Brian Street

Reconstruction in the Calderon Problem with Partial Data

We consider the problem of recovering the coefficient σ(x) of the elliptic equation ▿·(σ▿u) = 0 in a body from measurements of the Cauchy data on possibly very small subsets of its surface. We give a constructive proof of a uniqueness result by Kenig, Sjöstrand, and Uhlmann. We construct a uniquely specified family of solutions such that their traces on the boundary can be calculated by solving an integral equation which involves only the given partial Cauchy data. The construction entails a new family of Greens functions for the Laplacian, and corresponding single layer potentials, which may be of independent interest.

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.

Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius

We study multi-parameter Carnot-Caratheodory balls, generalizing results due to Nagel, Stein, and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, we study maximal functions associated to certain multi-parameter families of Carnot-Caratheodory balls.

Multi-parameter singular radon transforms I: The L 2 theory

The purpose of this paper is to study the L2 boundedness of operators of the form f ↦ ψ(x) ∫ f (γt(x))K(t)dt, where γt(x) is a C∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝN × ℝn, satisfying γ0(x) ≡ x, ψ is a C∞ cut-off function supported on a small neighborhood of 0 ∈ ℝn, and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝN. The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L2. The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderón- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of Lp boundedness, while the third paper deals with the special case when γ is real analytic.

Multi-parameter singular integrals

This book develops a new theory of multi-parameter singular integrals associated with Carnot-Caratheodory balls. Brian Street first details the classical theory of Calderon-Zygmund singular integrals and applications to linear partial differential equations. He then outlines the theory of multi-parameter Carnot-Caratheodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. Street then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. Multi-parameter Singular Integrals will interest graduate students and researchers working in singular integrals and related fields.

A parametrix for Kohn's operator

Abstract Recently, Kohn constructed examples of sums of squares of complex vector fields satisfying Hörmanders condition that lose derivatives, but are nevertheless hypoelliptic. He also demonstrated optimal L 2 regularity. In this paper, we construct parametricies for Kohns operators, which lead to the corresponding Lp (1 < p < ∞) and Lipschitz regularity. In fact, our parametrix construction generalizes to a somewhat larger class of operators, yielding some new examples of operators which are hypoelliptic, but lose derivatives.

WHAT ELSE about...Hypoellipticity

Let P = ∑|α|≤m aα(x)∂ x be a linear partial differential operator of degree m on Rn with smooth coefficients aα ∈ C∞(Rn). Consider the equation Pu = f (where u and f are distributions). A basic problem is: given f what can one say about u? Even the simplest P have nontrivial nullspaces, so one cannot hope to recover u completely from f. One often resorts to understanding what sort of properties u inherits from f. For example, if f is smooth, does it follow that u is smooth?