# Pointwise estimates for densities of stable semigroups of measures

Paweł Głowacki; Waldemar Hebisch

Studia Mathematica (1993)

- Volume: 104, Issue: 3, page 243-258
- ISSN: 0039-3223

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topGłowacki, Paweł, and Hebisch, Waldemar. "Pointwise estimates for densities of stable semigroups of measures." Studia Mathematica 104.3 (1993): 243-258. <http://eudml.org/doc/215973>.

@article{Głowacki1993,

abstract = {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).},

author = {Głowacki, Paweł, Hebisch, Waldemar},

journal = {Studia Mathematica},

keywords = {semigroup of probability measures; homogeneous group; Haar measure; Lévy measure; Zygmund class; maximal function; weak type },

language = {eng},

number = {3},

pages = {243-258},

title = {Pointwise estimates for densities of stable semigroups of measures},

url = {http://eudml.org/doc/215973},

volume = {104},

year = {1993},

}

TY - JOUR

AU - Głowacki, Paweł

AU - Hebisch, Waldemar

TI - Pointwise estimates for densities of stable semigroups of measures

JO - Studia Mathematica

PY - 1993

VL - 104

IS - 3

SP - 243

EP - 258

AB - 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).

LA - eng

KW - semigroup of probability measures; homogeneous group; Haar measure; Lévy measure; Zygmund class; maximal function; weak type

UR - http://eudml.org/doc/215973

ER -

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