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1993 | 104 | 3 | 243-258
Tytuł artykułu

Pointwise estimates for densities of stable semigroups of measures

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Języki publikacji
EN
Abstrakty
EN
Let ${μ_t}$ be a symmetric α-stable semigroup of probability measures on a homogeneous group N, where 0 < α < 2. Assume that $μ_t$ are absolutely continuous with respect to Haar measure and denote by $h_t$ the corresponding densities. We show that the estimate $h_t(x) ≤ tΩ(x/|x|)|x|^{-n-α}$, x≠0, holds true with some integrable function Ω on the unit sphere Σ if and only if the density of the Lévy measure of the semigroup belongs locally to the Zygmund class LlogL(N╲{e}). The problem turns out to be related to the properties of the maximal function $ℳ f(x) = sup_{t>0} 1/t |ʃ_{0}^{t} h_{t-s} ∗ f ∗ h_s(x)ds|$ which, as is proved here, is of weak type (1,1).
Słowa kluczowe
Czasopismo
Rocznik
Tom
104
Numer
3
Strony
243-258
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-03-31
poprawiono
1992-11-23
Twórcy
  • Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
  • Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Bibliografia
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  • [4] M. Duflo, Représentations de semi-groupes de mesures sur un groupe localement compact, Ann. Inst. Fourier (Grenoble) 28 (3) (1978), 225-249.
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  • [8] P. Głowacki, Stable semi-groups of measures as commutative approximate identities on non-graded homogeneous groups, Invent. Math. 83 (1986), 557-582.
  • [9] P. Głowacki, Lipschitz continuity of densities of stable semigroups of measures, Colloq. Math., to appear.
  • [10] P. Głowacki and A. Hulanicki, A semi-group of probability measures with non-smooth differentiable densities on a Lie group, Colloq. Math. 51 (1987), 131-139.
  • [11] M. de Guzmán, Differentiation of Integrals in $ℝ^n$, Springer, Berlin 1975.
  • [12] W. Hebisch and A. Sikora, A smooth subadditive homogeneous norm on a homogeneous group, Studia Math. 96 (1990), 231-236.
  • [13] A. Hulanicki, A class of convolution semi-groups of measures on a Lie group, in: Lecture Notes in Math. 882, Springer, Berlin 1980, 82-101.
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  • [15] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin 1966.
  • [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin 1983.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv104i3p243bwm
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