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

Jan-Olav Rönning

Studia Mathematica (1997)

  • Volume: 125, Issue: 3, page 219-229
  • ISSN: 0039-3223

Abstract

top
Let P(z,β) be the Poisson kernel in the unit disk , and let P λ f ( z ) = ʃ ∂ P ( z , φ ) 1 / 2 + λ f ( φ ) d φ be the λ -Poisson integral of f, where f ∈ L p ( ∂ ) . 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 ∈ L p ( ∂ ) 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.

How to cite

top

Rönning, Jan-Olav. "On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1." Studia Mathematica 125.3 (1997): 219-229. <http://eudml.org/doc/216434>.

@article{Rönning1997,
abstract = {Let P(z,β) be the Poisson kernel in the unit disk , and let $P_\{λ\}f(z) = ʃ_\{∂\} P(z,φ)^\{1/2+λ\} f(φ)dφ$ be the λ -Poisson integral of f, where $f ∈ L^p(∂)$. 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 $P_0f$ toward f in an $L^p$ weakly tangential region, if $f ∈ L^p(∂)$ 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.},
author = {Rönning, Jan-Olav},
journal = {Studia Mathematica},
keywords = {maximal function; square root of the Poisson kernel; convergence region; symmetric space of rank 1; convergence regions; maximal functions; weakly tangential region},
language = {eng},
number = {3},
pages = {219-229},
title = {On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1},
url = {http://eudml.org/doc/216434},
volume = {125},
year = {1997},
}

TY - JOUR
AU - Rönning, Jan-Olav
TI - On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1
JO - Studia Mathematica
PY - 1997
VL - 125
IS - 3
SP - 219
EP - 229
AB - Let P(z,β) be the Poisson kernel in the unit disk , and let $P_{λ}f(z) = ʃ_{∂} P(z,φ)^{1/2+λ} f(φ)dφ$ be the λ -Poisson integral of f, where $f ∈ L^p(∂)$. 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 $P_0f$ toward f in an $L^p$ weakly tangential region, if $f ∈ L^p(∂)$ 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.
LA - eng
KW - maximal function; square root of the Poisson kernel; convergence region; symmetric space of rank 1; convergence regions; maximal functions; weakly tangential region
UR - http://eudml.org/doc/216434
ER -

References

top
  1. [He78] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math. 80, Academic Press, 1978. 
  2. [DR92] E. Damek and F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal. 2 (1992), 213-248. Zbl0788.43008
  3. [Kor85] A. Korányi, Geometric properties of Heisenberg-type groups, Adv. Math. 56 (1985), 28-38. Zbl0589.53053
  4. [JOR] J.-O. Rönning, Convergence for square roots of Poisson kernels in weakly tangential regions, Math. Scand., to appear. 
  5. [Sjö83] P. Sjögren, Fatou theorems and maximal functions for eigenfunctions of the Laplace-Beltrami operator in a bidisk, J. Reine Angew. Math. 345 (1983), 93-110. Zbl0512.43005
  6. [Sjö84] P. Sjögren, A Fatou theorem for eigenfunctions of the Laplace-Beltrami operator in a symmetric space, Duke Math. J. 51 (1984), 47-56. Zbl0543.43007
  7. [Sjö84a] P. Sjögren, Une remarque sur la convergence des fonctions propres du Laplacian à valeur propre critique, in: Lecture Notes in Math. 1096, Springer, 1984, 544-548. 
  8. [Sjö88] P. Sjögren, Convergence for the square root of the Poisson kernel, Pacific J. Math. 131 (1988), 361-391. Zbl0601.31001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.