Absolutely continuous spectrum for one-dimensional Schr\"odinger operators with slowly decaying potentials: some optimal results

The absolutely continuous spectrum of one-dimensional Schr\"odinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic Schr\"odinger operators is preserved under all perturbations $V(x)$ satisfying $|V(x)|\leq C(1+x)^{-\alpha}$, $\alpha >\frac{1}{2}.$ This result is optimal in the power scale. More general classes of perturbing potentials which are not necessarily power decaying are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schr\"odinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on new maximal function and norm estimates and almost everywhere convergence results for certain multilinear integral operators

Weighted lattice polynomials of independent random variables

We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems.Comment: 14 page

Multiple Vector Valued Inequalities via the Helicoidal Method

We develop a new method of proving vector-valued estimates in harmonic analysis, which we like to call "the helicoidal method". As a consequence of it, we are able to give affirmative answers to some questions that have been circulating for some time. In particular, we show that the tensor product $BHT \otimes \Pi$ between the bilinear Hilbert transform $BHT$ and a paraproduct $\Pi$ satisfies the same $L^p$ estimates as the $BHT$ itself, solving completely a problem introduced in a paper of Muscalu, Pipher, Tao and Thiele. Then, we prove that for "locally $L^2$ exponents" the corresponding vector valued $\overrightarrow{BHT}$ satisfies (again) the same $L^p$ estimates as the $BHT$ itself. Before the present work there was not even a single example of such exponents. Finally, we prove a bi-parameter Leibniz rule in mixed norm $L^p$ spaces, answering a question of Kenig in nonlinear dispersive PDE.Comment: 56 pages, 7 figure

On a Biparameter Maximal Multilinear Operator

It is well-known that estimates for maximal operators and questions of pointwise convergence are strongly connected. In recent years, convergence properties of so-called `non-conventional ergodic averages' have been studied by a number of authors, including Assani, Austin, Host, Kra, Tao, and so on. In particular, much is known regarding convergence in $L^2$ of these averages, but little is known about pointwise convergence. In this spirit, we consider the pointwise convergence of a particular ergodic average and study the corresponding maximal trilinear operator (over $\mathbb{R}$, thanks to a transference principle). Lacey and Demeter, Tao, and Thiele have studied maximal multilinear operators previously; however, the maximal operator we develop has a novel bi-parameter structure which has not been previously encountered and cannot be estimated using their techniques. We will carve this bi-parameter maximal multilinear operator using a certain Taylor series and produce non-trivial H\"{o}lder-type estimates for one of the two "main" terms by treating it as a singular integrals whose symbol's singular set is similar to that of the Biest operator studied by Muscalu, Tao, and Thiele.Comment: 32 pages, 1 figur