Abstract. We develop a theory of L p spaces based on outer measures generated through coverings by distinguished sets. The theory includes as special case the classical L p theory on Euclidean spaces as well as some previously considered generalizations. The theory is a framework to describe aspects of singular integral theory such as Carleson embedding theorems, paraproduct estimates and T (1) theorems. It is particularly useful for generalizations of singular integral theory in time-frequency analysis, the latter originating in Carleson's investigation of convergence of Fourier series. We formulate and prove a generalized Carleson embedding theorem and give a relatively short reduction of the most basic L p estimates for the bilinear Hilbert transform to this new Carleson embedding theorem.

In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coefficients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.

Abstract. We show that for function f on [0, 1], with |f|(log log + |f|)(log log log + |f|) dx < ∞, and lacunary subsequence of integers {n j }, it holds that S n j f −→ f a.e., where S m f is the m-th Walsh-Fourier partial sum of f. According to a result of Konyagin, the sharp integrability condition would not have the triple-log term in it. The method of proof uses four ingredients, (1) analysis on the Walsh Phase Plane, (2) the new multi-frequency Calderón-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-Martín, which generalizes the work of Antonov and Arias-de-Reyna.

Let Pn(x) = n i=0 ξix i be a Kac random polynomial where the coefficients ξi are i.i.d. copies of a given random variable ξ. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a real double root.As an application, we consider the problem of estimating the number of real roots of Pn, which has a long history and in particular was the main subject of a celebrated series of papers by Littlewood and Offord from the 1940s. We show, for a large and natural family of atom variables ξ, that the expected number of real roots of Pn(x) is exactly (2/π) log n + C + o(1), where C is an absolute constant depending on the atom variable ξ. Prior to this paper, such a result was known only for the case when ξ is Gaussian.

Abstract. We consider random matrices whose entries are f (X T i X j ) or f ( X i − X j 2 ) for iid vectors X i ∈ R p with normalized distribution. Assuming that f is sufficiently smooth and the distribution of X i 's is sufficiently nice, El Karoui [17] showed that the spectral distributions of these matrices behave as if f is linear in the Marčhenko-Pastur limit. When X i 's are Gaussian vectors, variants of this phenomenon were recently proved for varying kernels, i.e. when f may depend on p, by Cheng-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 Karoui's results can be reduced to minimal; and secondly it is shown that the Gaussian assumptions in Cheng-Singer's result can be removed, answering a question posed in [13] about the universality of the limiting spectral distribution.

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We study the asymptotic behavior of oscillatory Riemann-Hilbert problems arising in the AKNS hierarchy of integrable nonlinear PDE's. Our method is based on the Deift-Zhou nonlinear steepest descent method in which the given Riemann-Hilbert problem localizes to small neighborhoods of stationary phase points. In their original work, Deift and Zhou only considered analytic phase functions. Subsequently Varzugin extended the Deift-Zhou method to a certain restricted class of non-analytic phase functions. In this paper, we extend Varzugin's method to a substantially more general class of non-analytic phase functions. In our work real variable methods play a key role.