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

• Autorzy: Andrzej Hulanicki
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1995
• Tom: 34
• Numer: 1
• Strony: 65-77

Daty

wydano

1995

Twórcy

autor

Andrzej Hulanicki

• Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Polad

Bibliografia

• [A] A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2) 125 (1987), 495-536.
• [Ch1] M. Christ, Hilbert transforms along curves I. Nilpotent groups, Ann. of Math. (2) 122 (1985), 575-596.
• [Ch2] M. Christ, The strong maximal function on a nilpotent group, Trans. Amer. Math. Soc. 331(1) (1992), 1-13.
• [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.
• [D1] E. Damek, Left-invariant degenerate elliptic operators on semidirect extensions of homogeneous groups, Studia Math. 89 (1988), 169-196.
• [D2] E. Damek, Pointwise estimates on the Poisson kernel on NA groups by the Ancona method, to appear.
• [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.
• [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.
• [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.
• [DHP1] E. Damek, A. Hulanicki, R. Penney, Admissible convergence for the Poisson-Szegö integrals, J. Geom. Anal. (to appear)
• [DHP2] E. Damek, A. Hulanicki, R. Penney, Hua operators on bounded homogeneous domains in $C^n$, preprint.
• [DR1] E. Damek and F. Ricci, A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. 27 (1992), 139-142.
• [DR2] E. Damek and F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal. 2 (1992), 213-248.
• [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.
• [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.
• [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.
• [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.
• [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.
• [S] J. Sołowiej, The Fatou theorem for NA groups - a negative result, Colloq. Math. 67 (1994), 131-145.
• [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.