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

• Autorzy: Jan-Olav Rönning
• Języki publikacji: EN

Abstrakty

EN

Let P(z,β) be the Poisson kernel in the unit disk 𝕌, and let $P_{λ}f(z) = ʃ_{∂𝕌} P(z,φ)^{1/2+λ} f(φ)dφ$ be the λ -Poisson integral of f, where $f ∈ L^p(∂𝕌)$. 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 ∂𝕌 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 $f ∈ L^p(∂𝕌)$ 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.

Słowa kluczowe

EN

maximal function; square root of the Poisson kernel; convergence region; symmetric space of rank 1

Kategorie tematyczne

• 42B25: Maximal functions, Littlewood-Paley theory
• 43A85: Analysis on homogeneous spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1997
• Tom: 125
• Numer: 3
• Strony: 219-229

Daty

wydano

1997

otrzymano

1996-05-20

poprawiono

1997-04-01

Twórcy

autor

Jan-Olav Rönning

• Institutionen för Naturvetenskap, Högskolan i Skövde, Box 408, 541 28 Skövde, Sweden

Bibliografia

• [He78] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math. 80, Academic Press, 1978.
• [DR92] E. Damek and F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal. 2 (1992), 213-248.
• [Kor85] A. Korányi, Geometric properties of Heisenberg-type groups, Adv. Math. 56 (1985), 28-38.
• [JOR] J.-O. Rönning, Convergence for square roots of Poisson kernels in weakly tangential regions, Math. Scand., to appear.
• [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.
• [Sjö84] P. Sjögren, A Fatou theorem for eigenfunctions of the Laplace-Beltrami operator in a symmetric space, Duke Math. J. 51 (1984), 47-56.
• [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.
• [Sjö88] P. Sjögren, Convergence for the square root of the Poisson kernel, Pacific J. Math. 131 (1988), 361-391.