Multiparameter Singular Integrals: Product and Flag kernels.

This thesis is concerned with the study of multi-parameter singular integrals on the Euclidean space. The Schwartz Kernel Theorem states that translation invariant continuous linear operators with minimal smoothness conditions are convolution operators. Singular integral operator theory is concerned with the study of the singular kernels associated with such operators. A well developed theory exists for the class of Calderón-Zygmund operators and the associated kernels. This kind of kernels can be seen as a natural generalization of the Hilbert kernel on $\R$, of the Riesz kernels in $\R^n$, and, more generally, of kernels of homogeneous degree $-n$ in $\R^n$. Calderón-Zygmund theory is a one-parameter homogeneous theory since the kernels of interest are well-behaved with respect to a family of homogeneous dilation with one parameter. Calderón-Zygmund kernels arise from many problems in linear PDEs and complex analysis. $L^p$ boundedness for $p\in (1,+\infty)$ and stability under composition are well known results for such kernels. Product-type kernels arise naturally in analysis in several complex variables and PDEs. As a matter of fact joint spectral functional calculus for more than one differential operator naturally produce to product structures. Product spaces occur naturally in the heat equation or in the Shrödinger equation. Abstractly, product kernels are the result of the extension of Calderón-Zygmund theory to product spaces. The tensor product of two or more Calderón-Zygmund kernels gives a singular kernel defined on the product space. The new kernel has a singularity not only in the origin but also along all coordinate sub-spaces. From the point of view of the associated operators, the tensor product corresponds to the composition of the original operators acting independently on the coordinates of the product space. Product kernel theory aims to extend the space of tensor products of Calderón-Zygmund kernels to a suitably defined completion. This is done mainly by using multi-parameter dilation techniques, with one parameter for each factor of the product space. An other idea that is pursued is that product theory can be inspired by vector valued functional analysis and integration. While avoiding a too abstract approach to such functional analysis in this thesis, some ideas are shown to be very useful. This thesis illustrates the adaptation of some important results inspired by Calderón-Zyg-mund theory to product kernels. These include decomposing a kernel into a multi-parameter dyadic series of homogeneous dilates of smooth functions concentrated on essentially disjoint scales and, conversely, finding conditions when such dyadic sums converge to product kernels. Furthermore, since tensor products of bounded operators on $L^p$ remain bounded on $L^p$ one can suppose that this remains true for general product kernels. However, the proof usually used for Calderón-Zygmund operators does not seem to be generalized to product kernels since weak $L^1-L^{1}_{w}$ boundedness fails. A finer technique based on product square function estimates and product Littlewood-Payley theory is developed to solve this problem. This idea is based on the quasi-orthogonality of the dyadic decomposition for the kernels. The second part of this thesis deals with a certain sub-class of product kernels given by flag kernels. While product kernels are the most intuitive generalization of Calderón-Zygmund theory to a multi-parameter setting, the singularities are generally too many to work with directly. Flag kernels have singularities concentrated on a flag or filtration of the space, and not along all coordinate subspaces. Kernels with such an ordered structure of singularities appear more often from concrete problems than general product kernels. A multi-parameter theory for flag kernels similar to the one for product kernels is developed. We also show that even though flag kernels form a sub-class of product kernels any product kernel can be written as a sum of flag kernels adapted to different flags. These results were already present in literature. Flag kernels were introduced by Nagel, Ricci, and Stein in “Singular integrals with flag kernels and analysis on quadratic CR mani- folds”, Journal of Functional Analysis, 2001. A large portion of the above paper is dedicated to applications of product-type singular integral operators. Here we develop the results and provide detailed proofs based on the ideas contained in the part of that paper dedicated to the general theory of flag kernels. In this thesis we also establish several new results. While the question of whether changes of variables conserve product and flag kernels will be addressed in a forthcoming paper by Alexander Nagel, Fulvio Ricci, Elias Stein, and Richard Wainger, they deal only with polynomial changes of variable. We show that the classes of Calderón-Zygmund, product and flag kernels with compact support are stable with respect to generic smooth changes of variable that have the geometric property of fixing the singular subspaces. These results and the techniques we use can be the first step to studying product-type singular integral operators on manifolds. Furthermore an attempt is made to develop a basic functional calculus for product singular integrals with respect to derivation, multiplication and convolution. This is done by introducing kernels of generic pseudo-differential order. We establish some useful facts but show that some properties may fail except for a restricted range of pseudo-differential orders. Finally we show how this functional calculus can be used to establish an approximation result for kernels composed with changes of variable

On the Distributional Hessian of the Distance Function

We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. In doing this we also discuss some geometrical properties of the cutlocus of a point and we compare some different weak notions of Hessian and Laplacian

Time-Frequency Analysis of the Variational Carleson Operator using outer-measure <em>L^p</em> spaces

In this thesis we are concerned with developing a systematic framework for dealing with the Carleson and the Variational Carleson Operators. We extend the framework of outer measure Lp spaces of Do and Thiele to an iterated version. The first chapter contains an intuitive introduction and a simplified dyadic model for the results of the rest of the thesis. We formulate embedding maps into time-frequency space and we obtain bounds for these embedding maps in terms of iterated outer Lp spaces, the main novelty of this thesis. Finally we show how the above embedding bounds can be used to easily obtain sharp weighted estimates for the Carleson operator and its variational counterpart

Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators

In this paper, we study the local well-posedness of the cubic Schr\" odinger equation: $$(i\partial_t - P) u = \pm |u|^2 u\quad \textrm{ on }\ I\times \R^d ,$$ with randomized initial data, and $P$ being an operator of degree $s \geq 2$. Using careful estimates in anisotropic spaces, we improve and extend known results for the standard Schr\"odinger equation (that is, $P$ being Laplacian) to any dimension under natural assumptions on $P$, whose Fourier symbol might be sign changing. Quite interestingly, we also exhibit the existence of a new regime depending on $s$ and $d$, which was not present for the Laplacian