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1991-1992 | 101 | 1 | 33-68
Tytuł artykułu

Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
On the domain S_a = {(x,e^b): x ∈ N, b ∈ ℝ, b > a} where N is a simply connected nilpotent Lie group, a certain N-left-invariant, second order, degenerate elliptic operator L is considered. N × {e^a} is the Poisson boundary for L-harmonic functions F, i.e. F is the Poisson integral F(xe^b) = ʃ_N f(xy)dμ^b_a(x), for an f in L^∞(N). The main theorem of the paper asserts that the maximal function M^a f(x) = sup{|ʃf(xy)dμ_a^b(y)| : b > a} is of weak type (1,1).
Słowa kluczowe
Czasopismo
Rocznik
Tom
101
Numer
1
Strony
33-68
Opis fizyczny
Daty
wydano
1991
otrzymano
1990-11-09
poprawiono
1991-07-31
Twórcy
autor
  • Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
  • Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Bibliografia
  • [B] J. M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1) (1969), 277-304.
  • [D] E. Damek, Left-invariant degenerate elliptic operators on semidirect extensions of homogeneous groups, Studia Math. 89 (1988), 169-196.
  • [DH] E. Damek and A. Hulanicki, Boundaries for left-invariant subelliptic operators on semidirect products of nilpotent and abelian groups, J. Reine Angew. Math. 411 (1990), 1-38.
  • [FS] G. B. Folland and E. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, 1982.
  • [GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983.
  • [He] W. Hebisch, Almost everywhere summability of eigenfunction expansions associated to elliptic operators, Studia Math. 96 (1990), 263-275.
  • [HS] W. Hebisch and A. Sikora, A smooth subadditive homogeneous norm on a homogeneous group, ibid., 231-236.
  • [H1] A. Hulanicki, Subalgebra of L₁(G) associated with Laplacian on a Lie group, Colloq. Math. 31 (1974), 259-287.
  • [H2] A. Hulanicki, A class of convolution semi-groups of measures on a Lie group, in: Lecture Notes in Math. 828, Springer, 1980, 82-101.
  • [HJ] A. Hulanicki and J. Jenkins, Nilpotent groups and summability of eigenfunction expansions of Schrödinger operators, Studia Math. 80 (1984), 235-244.
  • [K] C. Kenig, oral communication.
  • [St] E. M. Stein, Boundary behavior of harmonic functions on symmetric spaces: maximal estimates for Poisson integrals, Invent. Math. 74 (1983), 63-83.
  • [S] D. Stroock, Lectures on Stochastic Analysis: Diffusion Theory, Cambridge Univ. Press, 1987.
  • [SV] D. Stroock and S. R. Varadhan, Multidimensional Diffusion Processes, Springer, 1979.
  • [T] J. C. Taylor, Skew products, regular conditional probabilities and stochastic differential equations: a remark, preprint.
  • [Z] J. Zienkiewicz, in preparation.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv101i1p33bwm
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