Weighted inequalities for maximal operators and the Hardy space H1 on LCA groups

Abstract

The purpose of this thesis is two fold: to study weighted norm inequalities for maximal type operators such as Hardy{Littlewood maximal operator associated with a family of general sets in a topological space and Fourier maximal operator in the context of the ring of integers of a local eld, and to extend the classical theory of Hardy space and related topics such as the space BMO and the John{Nirenberg space in the setting of Locally Compact Abelian (LCA) groups having a covering family. In chapter 2 we study norm inequalities for the maximal operator ME associated with a family E of general sets from various points of view. Our rst main result is the mixed Ap 􀀀 A1 weighted estimates for the operator ME. The main ingredient to prove this result is a sharp form of a weak reverse H older inequality for the A1;E weights. As an application of this inequality, we also provide a quantitative version of the open property for Ap;E weights. Our second main result in this setting is the establishment of the endpoint Fe erman{Stein weighted inequalities for the operator ME. Furthermore, vector-valued extensions for maximal inequalities are also obtained in this context. Chapter 3 focuses on the weighted norm inequalities for Fourier series in the context of the ring of integers D of a local eld K and some important applications. We establish weighted estimates for the maximal partial sum operator M of Fourier series on the weighted spaces Lp(D;w), 1 < p < 1, where w is a Muckenhoupt Ap weight. As a consequence of this result, we obtain the uniform boundedness of the Fourier partial sum operators Sn; n 2 N, on Lp(D;w). Both these results include the cases when D is the ring of integers of the p-adic eld Qp and the eld Fq((X)) of formal Laurent series over a nite eld Fq, and in particular, when D is the Walsh{Paley or dyadic group 2!. The aim of this chapter 4 is to extend the classical theory of the Hardy space H1 and its dual space of BMO functions with \bounded mean oscillation" to the setting of LCA groups G having covering families. First, we discuss in details the setting of LCA groups where our work is developed. Next, we introduce the notion of atomic Hardy spaces H1;q(G) with atom parameter 1 < q 1 and the notion of the space BMO(G) in this setting. After presenting some basic properties of these spaces, we then establish the main feature for functions in BMO(G), namely the John{Nirenberg inequality. Moreover, we show that the atomic Hardy spaces H1;q(G) are independent of the choice of the parameter q. Finally, we relate H1;q(G) with BMO(G) via duality in this setting. Finally chapter 5 of this thesis explores the theory of John{Nirenberg spaces JNp in the setting of LCA groups having covering families. The main result of this chapter is the John{Nirenberg inequality for functions in JNp spaces which describes, as it happens in Euclidean setting, that JNp can be embedded into weak Lp spaces.