Mathematics

We prove L p-bounds for the bilinear Hilbert transform acting on functions valued in intermediate UMD spaces. Such bounds were previously unknown for UMD spaces that are not Banach lattices. Our proof relies on bounds on embeddings from Bochner spaces L p (R; X) into outer Lebesgue spaces on the time-frequency-scale space R 3 + .

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 co...

We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. At the same time we discuss some geometrical properties of the cut locus of a point, and compare some different weak notions of the Hessian and Laplacian.

We prove modulation invariant embedding bounds from Bochner spaces$$L^p(\mathbb {W};X)$$Lp(W;X)on the Walsh group to outer-$$L^p$$Lpspaces on the Walsh extended phase plane. The Banach spaceXis assumed to be UMD and sufficiently close to a Hilbert space in an interpolative sense. Our embedding bounds imply$$L^p$$Lpbounds and sparse domination for the Banach-valued tritile operator, a discrete model of the Banach-valued bilinear Hilbert transform.

Due to its nonlocal nature, the r-variation norm Carleson operator C r does not yield to the sparse domination techniques of Lerner [15, 17], Di Plinio and Lerner [6], Lacey [14]. We overcome this difficulty and prove that the dual form to C r can be dominated by a positive sparse form involving L p averages. Our result strengthens the L p-estimates by Oberlin et. al. [18]. As a corollary, we obtain quantitative weighted norm inequalities improving on [8] by Do and Lacey. Our proof relies on the localized outer L p-embeddings of Di Plinio and Ou [7] and Uraltsev [19].

In this paper, we study the local well-posedness of the cubic Schrödinger equation: (i∂t − L)u = ±|u| 2 u on I × R d , with randomized initial data, and L being an operator of degree σ ≥ 2. Using estimates in directional spaces, we improve and extend known results for the standard Schrödinger equation (i.e. L = ∆) to any dimension and obtain results under natural assumptions for general L .

We prove L p-boundedness of variational Carleson operators for functions valued in intermediate UMD spaces. This provides quantitative information on the rate of convergence of partial Fourier integrals of vectorvalued functions. Our proof relies on bounds on wave packet embeddings into outer Lebesgue spaces on the time-frequency-scale space R 3 + , which are the focus of this paper. Contents 1. Introduction Outline of the paper Acknowledgements Notation 2. Preliminaries 2.1. Notions and lemmas in Banach-valued analysis 2.2. Outer Lebesgue spaces 2.3. The time-frequency-scale space 2.4. (Truncated) wave packets and V c 3. Reduction of the main theorem 3.1. Reduction to wave packet embedding bounds 3.2. Reduction to domination by a scalar embedding 4. Scalar domination of the truncated wave packet embedding 4.1. The Lebesgue term 4.2. Splitting the randomised term 4.3. Controlling H * 4.4. Bounding the variation term 4.5. Bounding the first error term Err (1). 4.6. Bounding the second error term Err (2) t(•) 4.7. Bounding the third error term Err (3) 5. A discussion of Banach function spaces References