On the $A$-integrability of singular integral transforms

Shobha Madan

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Displaying similar documents to “On the $A$ -integrability of singular integral transforms”

Harmonic spaces associated with adjoints of linear elliptic operators

Peter Sjögren (1975)

Annales de l'institut Fourier

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Let $L$ be an elliptic linear operator in a domain in $R^{n}$ . We imposse only weak regularity conditions on the coefficients. Then the adjoint $L^{*}$ exists in the sense of distributions, and we start by deducing a regularity theorem for distribution solutions of equations of type $L^{*} u =$ given distribution. We then apply to $L^{*}$ R.M. Hervé’s theory of adjoint harmonic spaces. Some other properties of $L^{*}$ are also studied. The results generalize earlier work of the author.

Boundary behaviour of harmonic functions in a half-space and brownian motion

D. L. Burkholder, Richard F. Gundy (1973)

Annales de l'institut Fourier

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Let $u$ be harmonic in the half-space $R_{+}^{n + 1}$ , $n \geq 2$ . We show that $u$ can have a fine limit at almost every point of the unit cubs in $R^{n} = \partial R_{+}^{n + 1}$ but fail to have a nontangential limit at any point of the cube. The method is probabilistic and utilizes the equivalence between conditional Brownian motion limits and fine limits at the boundary. In $R_{+}^{2}$ it is known that the Hardy classes $H^{p}$ , $0 < p < \infty$ , may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms...

Duality on vector-valued weighted harmonic Bergman spaces

Salvador Pérez-Esteva (1996)

Studia Mathematica

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We study the duals of the spaces $A^{p α} (X)$ of harmonic functions in the unit ball of $ℝ^{n}$ with values in a Banach space X, belonging to the Bochner $L^{p}$ space with weight ${(1 - | x |)}^{α}$ , denoted by $L^{p α} (X)$ . For 0 < α < p-1 we construct continuous projections onto $A^{p α} (X)$ providing a decomposition $L^{p α} (X) = A^{p α} (X) + M^{p α} (X)$ . We discuss the conditions on p, α and X for which $A^{p α} (X) * = A^{q α} (X *)$ and $M^{p α} (X) * = M^{q α} (X *)$ , 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.

An inversion formula and a note on the Riesz kernel

Andrejs Dunkels (1976)

Annales de l'institut Fourier

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For potentials $U_{K}^{T} = K * T$ , where $K$ and $T$ are certain Schwartz distributions, an inversion formula for $T$ is derived. Convolutions and Fourier transforms of distributions in $(D_{L}^{'} p)$ -spaces are used. It is shown that the equilibrium distribution with respect to the Riesz kernel of order $α$ , $0 < α < m$ , of a compact subset $E$ of $R^{m}$ has the following property: its restriction to the interior of $E$ is an absolutely continuous measure with analytic density which is expressed by an explicit formula.

On the boundary limits of harmonic functions with gradient in $L^{p}$

Yoshihiro Mizuta (1984)

Annales de l'institut Fourier

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This paper deals with tangential boundary behaviors of harmonic functions with gradient in Lebesgue classes. Our aim is to extend a recent result of Cruzeiro (C.R.A.S., Paris, 294 (1982), 71–74), concerning tangential boundary limits of harmonic functions with gradient in $L^{n} (R_{+}^{n})$ , $R_{+}^{n}$ denoting the upper half space of the $n$ -dimensional euclidean space $R^{n}$ . Our method used here is different from that of Nagel, Rudin and Shapiro (Ann. of Math., 116 (1982), 331–360); in fact, we use the integral representation...

On separately subharmonic functions (Lelong’s problem)

A. Sadullaev (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

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The main result of the present paper is : every separately-subharmonic function $u (x, y)$ , which is harmonic in $y$ , can be represented locally as a sum two functions, $u = u^{*} + U$ , where $U$ is subharmonic and $u^{*}$ is harmonic in $y$ , subharmonic in $x$ and harmonic in $(x, y)$ outside of some nowhere dense set $S$ .

Mapping properties of fundamental operators in harmonic analysis related to Bessel operators

Jorge J. Betancor, Eleonor Harboure, Adam Nowak, Beatriz Viviani (2010)

Studia Mathematica

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We obtain sharp power-weighted $L^{p}$ , weak type and restricted weak type inequalities for the heat and Poisson integral maximal operators, Riesz transform and a Littlewood-Paley type square function, emerging naturally in the harmonic analysis related to Bessel operators.

Displaying similar documents to “On the A -integrability of singular integral transforms”

Harmonic spaces associated with adjoints of linear elliptic operators

Boundary behaviour of harmonic functions in a half-space and brownian motion

Duality on vector-valued weighted harmonic Bergman spaces

An inversion formula and a note on the Riesz kernel

On the boundary limits of harmonic functions with gradient in L p

On separately subharmonic functions (Lelong’s problem)

Mapping properties of fundamental operators in harmonic analysis related to Bessel operators

Displaying similar documents to “On the $A$ -integrability of singular integral transforms”

On the boundary limits of harmonic functions with gradient in $L^{p}$