The continuous analysis of entangled multilinear forms and applications

Abstract

The quadrilinear singular integral form<br /><br /> Lambda(F_1,F_2,F_3,F_4) = int_R^4 F_1(x,y) F_2(x,y') F_3(x',y') F_4(x',y) K(x-x',y-y') dxdydx'dy'<br /><br /> was motivated by the work of Kovač on the twisted paraproduct, who established boundedness in L^p spaces of a dyadic model of the quadrilinear form Lambda. Here K is a smooth two-dimensional Calderón-Zygmund kernel. In this thesis we introduce a continuous variant of Kovač's approach and address boundedness of the quadrilinear form Lambda. Moreover, we study further related multilinear singular integral forms acting on two- and higher-dimensional functions, and discuss their applications to certain problems in ergodic theory and additive combinatorics. <br /> The content of this thesis is organized into six chapters. <br /> Chapter 1 is an introductory chapter, stating the main results of Chapters 2-6.<br /> In Chapter 2 we prove the estimate <br /><br /> |Lambda(F_1,F_2,F_3,F_4)| <= C_{p_1,p_2,p_3,p_4} ||F_1||_{L^{p_1}(R^2)} ||F_2||_{L^{p_2}(R^2)} ||F_3||_{L^{p_3}(R^2)} ||F_4||_{L^{p_4}(R^2)}<br /><br /> for the exponents p_1 = p_2 = p_3 = p_4 = 4.<br /> In Chapter 3 we extend the range of exponents to 2 < p_1, p_2, p_3, p_4 <= infty, whenever the exponents satisfy the scaling condition 1/p_1 + 1/p_2 + 1/p_3 + 1/p_4 = 1. <br /> In Chapter 4 we study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm, by counting their norm-jumps and bounding their norm-variation. This is a joint work with Vjekoslav Kovač, Kristina Ana Škreb and Christoph Thiele.<br /> In Chapter 5 we study side-lengths of corners in subsets of positive upper Banach density of the Euclidean space. We show that if p is in (1,2) u (2,infty) and d is large enough, an arbitrary measurable set A in R^d x R^d of positive upper Banach density contains corners (x,y), (x+s,y), (x,y+s) such that the l^p norm of the side s attains all sufficiently large real values. This is a joint work with Vjekoslav Kovač and Luka Rimanić. <br /> As a byproduct of the approach in Chapters 4 and 5 we obtain a bound for a two-dimensional bilinear square function related to a singular integral called the triangular Hilbert transform. Boundedness of the triangular Hilbert transform is a major open problem in harmonic analysis. Chapter 6 is devoted to the simplex Hilbert transform, a higher-dimensional multilinear variant of the triangular Hilbert transform. The content of this chapter is a joint work with Vjekoslav Kovač and Christoph Thiele. We show that the L^p bounds for the truncated simplex Hilbert transform grow with a power less than one of the truncation range in the logarithmic scale. Boundedness of the simplex Hilbert transform remains an open problem.