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Weighted $H^p$ spaces

100%

García-Cuerva J.

Instytut Matematyczny Polskiej Akademii Nauk

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

Boundedness properties of fractional integral operators associated to non-doubling measures

61%

García-Cuerva J., Gatto A.

Studia Mathematica

2004

tom 162

nr 3

245-261

The main purpose of this paper is to investigate the behavior of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non-doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on Euclidean space and their known extensions for doubling measures. We start by analyzing the images of the Lebesgue spaces associated to the measure. The Lipschitz spaces, defined in terms of the metric, also play a basic role. For a Euclidean space equipped with one of these measures, we also consider the so-called regular BMO space introduced by X. Tolsa. We show that it contains the image of a Lebesgue space in the appropriate limit case and also that the image of the regular BMO space is contained in a suitable Lipschitz space.

Ograniczanie wyników

1 Studia Mathematica

2 García-Cuerva J.

1 Gatto A.

1 2004

1 1979

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Weighted $H^p$ spaces

Boundedness properties of fractional integral operators associated to non-doubling measures