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

Abstract

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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 ) ≤ t Ω ( 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 ) = s u p t > 0 1 / t | ʃ 0 t h t - s ∗ f ∗ h s ( x ) d s | which, as is proved here, is of weak type (1,1).

How to cite

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Głowacki, 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 -

References

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