Yen Do

theory for outer measures and two themes of Lennart Carleson united

We develop a theory of Lp spaces based on outer measures rather than measures. This theory includes the classical Lp theory on measure spaces as special case. It also covers parts of potential theory and Carleson embedding theorems. The theory turns out to be an elegant language to describe aspects of classical singular integral theory such as paraproduct estimates and T(1) theorems, and it is particularly useful for generalizations of singular integral theory in time-frequency analysis. We formulate and prove a generalized Carleson embedding theorem and give a relatively short reduction of basic estimates for the bilinear Hilbert transform to this new Carleson embedding theorem.

Variational estimates for paraproducts

We generalize a family of variation norm estimates of Lepingle with endpoint estimates of Bourgain and Pisier-Xu to a family of variational estimates for paraproducts, both in the discrete and the continuous setting. This expands on work of Friz and Victoir, our focus being on the continuous case and an expanded range of variation exponents.

Weighted bounds for variational Fourier series

For 1<p<infty and for weight w in A_p, we show that the r-variation of the Fourier sums of any function in L^p(w) is finite a.e. for r larger than a finite constant depending on w and p. The fact that the variation exponent depends on w is necessary. This strengthens previous work of Hunt-Young and is a weighted extension of a variational Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses weighted adaptation of phase plane analysis and a weighted extension of a variational inequality of Lepingle.

On the convergence of lacunary Walsh–Fourier series

We study the Walsh-Fourier series of S_{n_j}f, along a lacunary subsequence of integers {n_j}. Under a suitable integrability condition, we show that the sequence converges to f a.e. Integral condition is only slightly larger than what the sharp integrability condition would be, by a result of Konyagin. The condition is: f is in L loglog L (logloglog L). The method of proof uses four ingredients, (1) analysis on the Walsh Phase Plane, (2) the new multi-frequency Calderon-Zygmund Decomposition of Nazarov-Oberlin-Thiele, (3) a classical inequality of Zygmund, giving an improvement in the Hausdorff-Young inequality for lacunary subsequences of integers, and (4) the extrapolation method of Carro-Martin, which generalizes the work of Antonov and Arias-de-Reyna.

THE SPECTRUM OF RANDOM KERNEL MATRICES: UNIVERSALITY RESULTS FOR ROUGH AND VARYING KERNELS

We consider random matrices whose entries are f( ) or f(||Xi-Xj||^2) for iid vectors Xi in R^p with normalized distribution. Assuming that f is sufficiently smooth and the distribution of Xis is sufficiently nice, El Karoui [17] showed that the spectral distributions of these matrices behave as if f is linear in the Marchenko--Pastur limit. When Xis are Gaussian vectors, variants of this phenomenon were recently proved for varying kernels, i.e. when f may depend on p, by Cheng and Singer [13]. Two results are shown in this paper: first it is shown that for a large class of distributions the regularity assumptions on f in El Karouis results can be reduced to minimal; and secondly it is shown that the Gaussian assumptions in Cheng--Singers result can be removed, answering a question posed in [13] about the universality of the limiting spectral distribution.

Real roots of random polynomials: expectation and repulsion

Let

Positive sparse domination of variational Carleson operators

P_{n}(x)= \sum_{i=0}^n \xi_i x^i

Variational bounds for a dyadic model of the bilinear Hilbert transform

be a Kac random polynomial where the coefficients

Weighted Bounds for Variational Walsh–Fourier Series

\xi_i

A nonlinear stationary phase method for oscillatory Riemann-Hilbert problems

are iid copies of a given random variable