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Series: Analysis Seminar

This talk concerns recent joint work with E. M. Stein on the extension to higher dimension of Calder\'on's andCoifman-McIntosh-Meyer's seminal results about the Cauchy integral for a Lipschitz planar curve (interpreted as the boundary of a Lipschitz domain $D\subset\mathbb C$). From the point of view of complex analysis, a fundamental feature of the 1-dimensional Cauchy kernel:\vskip-1.0em$$H(w, z) = \frac{1}{2\pi i}\frac{dw}{w-z}$$\smallskip\vskip-0.7em\noindent is that it is holomorphic (that is, analytic) as a function of $z\in D$. In great contrast with the one-dimensional theory, in higher dimension there is no obvious holomorphic analogueof $H(w, z)$. This is because of geometric obstructions (the Levi problem) that in dimension 1 are irrelevant. A good candidate kernel for the higher dimensional setting was first identified by Jean Lerayin the context of a $C^\infty$-smooth, convex domain $D$: while these conditions on $D$ can be relaxed a bit, if the domain is less than $C^2$-smooth (much less Lipschitz!) Leray's construction becomes conceptually problematic.In this talk I will present {\em(a)}, the construction of theCauchy-Leray kernel and {\em(b)}, the $L^p(bD)$-boundedness of the induced singular integral operator under the weakest currently known assumptions on the domain's regularity -- in the case of a planar domain these are akin to Lipschitz boundary, but in our higher-dimensional context the assumptions we make are in fact optimal. The proofs rely in a fundamental way on a suitably adapted version of the so-called ``\,$T(1)$-theorem technique'' from real harmonic analysis.Time permitting, I will describe applications of this work to complex function theory -- specifically, to the Szeg\H o and Bergman projections (that is, the orthogonal projections of $L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic functions).

Series: Analysis Seminar

Consistent reconstruction is a method for estimating a signal from a
collection of noisy linear measurements that are corrupted by uniform
noise. This problem arises, for example, in analog-to-digital
conversion under the uniform noise model for memoryless scalar
quantization. We shall give an overview of consistent reconstruction
and prove optimal mean squared error bounds for the quality of
approximation. We shall also discuss an iterative alternative (due to
Rangan and Goyal) to consistent reconstruction which is also able to
achieve optimal mean squared error; this is closely related to the
classical Kaczmarz algorithm and provides a simple example of the power
of randomization in numerical algorithms.

Series: Analysis Seminar

For $\mathbb B^n$ the unit ball of $\mathbb C^n$, we consider Bergman-Orlicz spaces of holomorphic functions in $L_\alpha^\Phi(\mathbb B^n)$, which are generalizations of classical Bergman spaces. Weobtain their atomic decomposition and then prove weak factorization theorems involving the Bloch space and Bergman-Orlicz space and also weak factorization involving two Bergman-Orlicz spaces. This talk is based on joint work with D. Bekolle and A. Bonami.

Series: Analysis Seminar

In this talk we will discuss the connection between functions
with bounded mean oscillation (BMO) and commutators of Calderon-Zygmund
operators. In particular, we will discuss how to characterize certain BMO
spaces related to second order differential operators in terms of Riesz
transforms adapted to the operator and how to characterize commutators when
acting on weighted Lebesgue spaces.

Series: Analysis Seminar

We generalize the Calderon commutator to the higher-dimensional multicommutator with more input functions in higher dimensions. For this new multilinear operator, we establish the strong boundedness of it in all possible open points by a new multilinear multiplier theorem utilizing a new type of Sobolev spaces.

Series: Analysis Seminar

The asymptotic distribution of the zeros of two families of multiple
orthogonal polynomials will be given, namely the Jacobi-Pineiro polynomials (which
are an extension of the Jacobi polynomials) and the multiple Laguerre polynomials
of the first kind (which are an extension of the Laguerre polynomials). We use the
nearest neighbor recurrence relations for these polynomials and a recent result on
the ratio asymptotics of multiple orthogonal polynomials. We show how these
asymptotic zero distributions are related to the Fuss-Catalan distribution.

Series: Analysis Seminar

Compressed sensing illustrates the possibility of acquiring and reconstructing sparse signals via underdetermined (linear) systems. It is believed that iid Gaussian measurement vectors give near optimal results, with the necessary number of measurements on the order of $s \log(n/s)$ - $n$ is ambient dimension and $s$ is the sparsity threshold. The recovery algorithm used above relies on a certain quasi-isometry property of the measurement matrix. A surprising result is that the same order of measurements gives an analogous quasi-isometry in the extreme quantization of one-bit sensing. Bylik and Lacey deliver this result as a consequence of a certain stochastic process on the sphere. We will discuss an alternative method that relies heavily on the VC-dimension of a class of subsets on the sphere.

Series: Analysis Seminar

In this work we prove that the space of two parameter, matrix-valued BMO functions can be
characterized by considering iterated commutators with the Hilbert transform. Specifically, we
prove that the norm in the BMO space is equivalent to the norm of the
commutator of the BMO function with the Hilbert transform, as an
operator on L^2.
The upper bound estimate relies on a representation of the Hilbert transform as an average of dyadic
shifts, and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and
Lacey's wavelet proof for the scalar case.

Series: Analysis Seminar

We will describe sufficient conditions on a set $\Lambda \subset [0,2\pi) $ so that the maximal operator below is bound on $\ell^2(Z)$. $$\sup _{\lambda \in \Lambda} \Big| \sum_{n\neq 0} e^{i \lambda n^2} f(x-n)/n\Big|$$ The integral version of this result is an influential result to E.M. Stein. Of course one should be able to take $\Lambda = [0,2\pi) $, but such a proof would have to go far beyond the already complicated one we will describe. Joint work with Ben Krause.

Series: Analysis Seminar

A signal is a high dimensional vector x, and a measurement is the inner product . A one-bit measurement is the sign of . These are basic objects, as will be explained in the talk, with the help of some videos of photons. The import of this talk is that one bit measurements can be as effective as the measurements themselves, in that the same number of measurements in linear and one bit cases ensure the RIP property. This is explained by a connection with variants of classical spherical cap discrepancy. Joint work with Dimtriy Bilyk.