CONTENTS

Introduction.......................................................................................................................................................... 5

Chapter I. Some preliminary lemmas............................................................................................................ 8

Chapter II. Weighted $H^p$ spaces of analytic functions.......................................................................... 13
 1. Behaviour at the boundary....................................................................................................................... 13
 2. Maximal function characterization........................................................................................................... 15
 3. Atomic decomposition.............................................................................................................................. 20
 4. Dual spaces............................................................................................................................................... 27

Chapter III. $H^p$ spaces associated with the space of homogeneous type (R, w(x)dx).................... 31
 1. The space $\mathfrak{H}^1(w(x)dx)$...................................................................................................... 31
 2. The spaces $\mathfrak{H}^p(w(x)dx)$ for p < 1.................................................................................... 33

Chapter IV. Applications and examples.......................................................................................................... 40
 1. A weighted Hilbert transform.................................................................................................................... 40
 2. Equivalence between the space of radial functions in $H^1(R^n)$ and the space
 of even functions in $\mathfrak{H}^1(|r|^{n-1}dr)$..................................................................................... 40
 3. Integral operators in the line obtained by restricting to radial functions some systems
 of Riesz transforms in higher dimensions................................................................................................. 44
 4. The kernel $z^{-2}$...................................................................................................................................... 54

References............................................................................................................................................................. 58