# Estimates for the Poisson kernels and a Fatou type theorem applications to analysis on Siegel domains

Banach Center Publications (1995)

- Volume: 34, Issue: 1, page 65-77
- ISSN: 0137-6934

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topHulanicki, Andrzej. "Estimates for the Poisson kernels and a Fatou type theorem applications to analysis on Siegel domains." Banach Center Publications 34.1 (1995): 65-77. <http://eudml.org/doc/251298>.

@article{Hulanicki1995,

abstract = {This is a short description of some results obtained by Ewa Damek, Andrzej Hulanicki, Richard Penney and Jacek Zienkiewicz. They belong to harmonic analysis on a class of solvable Lie groups called NA. We apply our results to analysis on classical Siegel domains.},

author = {Hulanicki, Andrzej},

journal = {Banach Center Publications},

keywords = {Siegel domain; Poisson-Szegö integral; Poisson-Szegö kernel; boundary behaviour; approach regions; solvable Lie groups of type NA; second order degenerate elliptic operators; Poisson kernels; harmonic functions; Fatou type theorems; Furstenberg boundaries},

language = {eng},

number = {1},

pages = {65-77},

title = {Estimates for the Poisson kernels and a Fatou type theorem applications to analysis on Siegel domains},

url = {http://eudml.org/doc/251298},

volume = {34},

year = {1995},

}

TY - JOUR

AU - Hulanicki, Andrzej

TI - Estimates for the Poisson kernels and a Fatou type theorem applications to analysis on Siegel domains

JO - Banach Center Publications

PY - 1995

VL - 34

IS - 1

SP - 65

EP - 77

AB - This is a short description of some results obtained by Ewa Damek, Andrzej Hulanicki, Richard Penney and Jacek Zienkiewicz. They belong to harmonic analysis on a class of solvable Lie groups called NA. We apply our results to analysis on classical Siegel domains.

LA - eng

KW - Siegel domain; Poisson-Szegö integral; Poisson-Szegö kernel; boundary behaviour; approach regions; solvable Lie groups of type NA; second order degenerate elliptic operators; Poisson kernels; harmonic functions; Fatou type theorems; Furstenberg boundaries

UR - http://eudml.org/doc/251298

ER -

## References

top- [A] A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2) 125 (1987), 495-536. Zbl0652.31008
- [Ch1] M. Christ, Hilbert transforms along curves I. Nilpotent groups, Ann. of Math. (2) 122 (1985), 575-596. Zbl0593.43011
- [Ch2] M. Christ, The strong maximal function on a nilpotent group, Trans. Amer. Math. Soc. 331(1) (1992), 1-13. Zbl0765.43002
- [CW] R. R. Coifman and G. Weiss, Operators associated with representations of amenable groups, singular integrals induced by ergodic flows, the rotations method and multipliers, Studia Math. 47 (1973), 285-303. Zbl0297.43010
- [D1] E. Damek, Left-invariant degenerate elliptic operators on semidirect extensions of homogeneous groups, Studia Math. 89 (1988), 169-196. Zbl0675.22005
- [D2] E. Damek, Pointwise estimates on the Poisson kernel on NA groups by the Ancona method, to appear. Zbl0876.22008
- [DH1] E. Damek and A. Hulanicki, Boundaries for left-invariant subelliptic operators on semi-direct products of nilpotent and Abelian groups, J. Reine Angew. Math. 411 (1990), 1-38. Zbl0699.22012
- [DH2] E. Damek and A. Hulanicki, Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group, Studia Math. 101 (1991), 33-68. Zbl0811.43001
- [DH3] E. Damek and A. Hulanicki, Boundaries and the Fatou theorem for subelliptic second order operators on solvable Lie groups, Colloq. Math. 68 (1995), 121-140. Zbl0839.22008
- [DHP1] E. Damek, A. Hulanicki, R. Penney, Admissible convergence for the Poisson-Szegö integrals, J. Geom. Anal. (to appear) Zbl0813.32030
- [DHP2] E. Damek, A. Hulanicki, R. Penney, Hua operators on bounded homogeneous domains in ${C}^{n}$, preprint. Zbl0887.43005
- [DR1] E. Damek and F. Ricci, A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. 27 (1992), 139-142. Zbl0755.53032
- [DR2] E. Damek and F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal. 2 (1992), 213-248. Zbl0788.43008
- [H] L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in Classical Domains, Vol. 6, Translations of Math. Monographs, Amer. Math. Soc., Providence, 1963.
- [JK] K. Johnson, A. Korányi, The Hua operators on bounded symmetric domains of tube type, Ann. of Math. (2) 111 (1980), 589-608. Zbl0468.32007
- [K1] A. Korányi, The Poisson integral for generalized halfplanes and bounded symmetric domains, Ann. of Math. (2) 82 (1965), 332-350.
- [K2] A. Korányi, Boundary behavior of Poisson integrals on symmetric spaces, Trans. Amer. Math. Soc. 140 (1969), 393-409. Zbl0179.15103
- [K3] A. Korányi, Harmonic functions on symmetric spaces, in: Symmetric Spaces, Basel-New York 1972.
- [KM] A. Korányi and P. Malliavin, Poisson formula and compound diffusion associated to overdetermined elliptic system on the Siegel half-plane of rank two, Acta Math. 134 (1975), 185-209. Zbl0318.60066
- [KS1] A. Korányi, E. M. Stein, Fatou's theorem for generalized half-planes, Ann. Scuola Norm. Sup. Pisa 22 (1968), 107-112.
- [KS2] A. Korányi, E. M. Stein, ${H}^{2}$-spaces of generalized half-planes, Studia Math. 44 (1972), 379-388.
- [NS] A. Nagel and E. M. Stein, On certain maximal functions and approach regions, Adv. Math. 54 (1984), 83-106. Zbl0546.42017
- [P] I. I. Pjatecki-Shapiro, Geometry and classification of homogeneous bounded domains in ${C}^{n}$, Uspekhi Mat. Nauk 2 (1965), 3-51; Russian Math. Surv. 20 (1966), 1-48.
- [R] F. Ricci, Singular integrals on ${R}^{n}$, Tempus lectures held at the Institute of Mathematics of Wrocław University, 1991.
- [Sj] P. Sjögren, Admissible convergence of Poisson integrals in symmetric spaces, Ann. of Math. (2) 124 (1986), 313-335. Zbl0646.31008
- [S] J. Sołowiej, The Fatou theorem for NA groups - a negative result, Colloq. Math. 67 (1994), 131-145. Zbl0839.22009
- [St] E. M. Stein, Boundary behavior of harmonic functions on symmetric spaces: Maximal estimates for Poisson integrals, Invent. Math. 74 (1983), 63-83.
- [V] E. B. Vinberg, The theory of convex homogeneous cones, English translation, Trans. Moscow Math. Soc. 12 (1963), 340-403. Zbl0138.43301

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