On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1

Rönning, Jan-Olav

Studia Mathematica, Tome 122 (1997), p. 219-229 / Harvested from The Polish Digital Mathematics Library

Résumé

Let P(z,β) be the Poisson kernel in the unit disk , and let $P_{λ} f (z) = ʃ_{\partial} P {(z, φ)}^{1 / 2 + λ} f (φ) d φ$ be the λ -Poisson integral of f, where $f \in L^{p} (\partial)$ . We let $P_{λ} f$ be the normalization $P_{λ} f / P_{λ} 1$ . If λ >0, we know that the best (regular) regions where $P_{λ} f$ converges to f for a.a. points on ∂ are of nontangential type. If λ =0 the situation is different. In a previous paper, we proved a result concerning the convergence of $P_{0} f$ toward f in an $L^{p}$ weakly tangential region, if $f \in L^{p} (\partial)$ and p > 1. In the present paper we will extend the result to symmetric spaces X of rank 1. Let f be an $L^{p}$ function on the maximal distinguished boundary K/M of X. Then $P_{0} f (x)$ will converge to f(kM) as x tends to kM in an $L^{p}$ weakly tangential region, for a.a. kM ∈ K/M.

Publié le : 1997-01-01

Zbl 0917.42025

EUDML-ID : urn:eudml:doc:216434

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     author = {Jan-Olav R\"onning},
     title = {On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1},
     journal = {Studia Mathematica},
     volume = {122},
     year = {1997},
     pages = {219-229},
     zbl = {0917.42025},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv125i3p219bwm}
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Rönning, Jan-Olav. On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1. Studia Mathematica, Tome 122 (1997) pp. 219-229. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv125i3p219bwm/

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