My name is Brian Street.
I am a Van Vleck Assistant Professor at the University
of Wisconsin-Madison.
A copy of my CV can be found here.
I received my PhD from Princeton University in 2007, and my advisor was Eli Stein.
street
(at)
math.wisc.edu
Papers:
A. Nachman and B. Street, Reconstruction in the Calderón Problem with Partial Data, preprint, 17 pages -- arXiv
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 Green's functions for the Laplacian, and corresponding single layer potentials, which may be of independent interest.
B. Street, Multi-parameter Carnot-Carathéodory balls and the theorem of Frobenius, preprint, 82 pages -- New version on August 11, 2009 arXiv
Multi-parameter Carnot-Carathéodory balls are studied, 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, maximal functions associated
to certain multi-paramter families of Carnot-Carathéodory balls
are also studied.
B. Street, A Parametrix for Kohn's operator, preprint, 38 pages -- PDF
Kohn constructed examples of sums of squares of complex vector fields satisfying Hörmander's condition that lose derivatives, but are nevertheless hypoelliptic. He also demonstrated optimal L2 regularity. In
this paper, we construct parametricies for Kohn's 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. This is the paper version of my thesis, which was done under the
supervision of Eli Stein.
Recently, Nagel and Stein studied the ☐b heat equation, where ☐b is the Kohn Laplacian on the boundary of a weakly pseudoconvex domain of finite type in
C2. They showed that the Schwartz kernel of e-t☐b satisfies good "off-diagonal" estimates, while that of e-t☐b-π satisfies good "on-diagonal" estimates, where
π denotes the Szegö projection. We offer a simple proof of these results, which easily generalizes to
other, similar situations. Our methods involve adapting the well-known relationship between the heat equation
and the finite propagation speed of the wave equation to this situation. In addition, we apply these methods to
study multipliers of the form m(☐b). In particular, we show that m(☐b) is an NIS operator, where m satisfies an appropriate
Mihlin−Hörmander condition.
We present an intrinsically defined algebra of operators containing the
right and left invariant Calderón−Zygmund operators on a
stratified group. The operators in our algebra are pseudolocal and
bounded on Lp (1<p<∞). This algebra provides an example of an algebra of singular integrals that falls outside of the
classical Calderón−Zygmund theory.
