Strong q -variation inequalities for analytic semigroups

Christian Le Merdy[1]; Quanhua Xu[2]

  • [1] Laboratoire de Mathématiques Université de Franche-Comté 25030 Besançon Cedex France
  • [2] School of Mathematics and Statistics Wuhan University Wuhan 430072 Hubei China and Laboratoire de Mathématiques Université de Franche-Comté 25030 Besançon Cedex France

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 6, page 2069-2097
  • ISSN: 0373-0956

Abstract

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Let T : L p ( Ω ) → L p ( Ω ) be a positive contraction, with 1 < p < ∞ . Assume that T is analytic, that is, there exists a constant K ≥ 0 such that ∥ T n - T n - 1 ∥ ≤ K / n for any integer n ≥ 1 . Let 2 < q < ∞ and let v q be the space of all complex sequences with a finite strong q -variation. We show that for any x ∈ L p ( Ω ) , the sequence [ T n ( x ) ] ( λ ) n ≥ 0 belongs to v q for almost every λ ∈ Ω , with an estimate ∥ ( T n ( x ) ) n ≥ 0 ∥ L p ( v q ) ≤ C ∥ x ∥ p . If we remove the analyticity assumption, we obtain an estimate ∥ ( M n ( T ) x ) n ≥ 0 ∥ L p ( v q ) ≤ C ∥ x ∥ p , where M n ( T ) = ( n + 1 ) - 1 ∑ k = 0 n T k denotes the ergodic average of T . We also obtain similar results for strongly continuous semigroups ( T t ) t ≥ 0 of positive contractions on L p -spaces.

How to cite

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Le Merdy, Christian, and Xu, Quanhua. "Strong $q$-variation inequalities for analytic semigroups." Annales de l’institut Fourier 62.6 (2012): 2069-2097. <http://eudml.org/doc/251053>.

@article{LeMerdy2012,
abstract = {Let $T\colon L^p(\Omega )\rightarrow L^p(\Omega )$ be a positive contraction, with $1&lt;p&lt;\infty $. Assume that $T$ is analytic, that is, there exists a constant $K\ge 0$ such that $\Vert T^n-T^\{n-1\}\Vert \le K/n$ for any integer $n\ge 1$. Let $2&lt;q&lt;\infty $ and let $v^q$ be the space of all complex sequences with a finite strong $q$-variation. We show that for any $x\in L^p(\Omega )$, the sequence $\bigl ([T^n(x)](\lambda )\bigr )_\{n\ge 0\}$ belongs to $v^q$ for almost every $\lambda \in \Omega $, with an estimate $\Vert (T^n(x))_\{n\ge 0\}\Vert _\{L^p(v^q)\}\le C\Vert x\Vert _p$. If we remove the analyticity assumption, we obtain an estimate $\Vert (M_n(T)x)_\{n\ge 0\}\Vert _\{L^p(v^q)\}\le C\Vert x\Vert _p$, where $M_n(T)=(n+1)^\{-1\}\sum _\{k=0\}^\{n\} T^k\,$ denotes the ergodic average of $T$. We also obtain similar results for strongly continuous semigroups $(T_t)_\{t\ge 0\}$ of positive contractions on $L^p$-spaces.},
affiliation = {Laboratoire de Mathématiques Université de Franche-Comté 25030 Besançon Cedex France; School of Mathematics and Statistics Wuhan University Wuhan 430072 Hubei China and Laboratoire de Mathématiques Université de Franche-Comté 25030 Besançon Cedex France},
author = {Le Merdy, Christian, Xu, Quanhua},
journal = {Annales de l’institut Fourier},
keywords = {Ergodic theory; operators on $L^p$; strong $q$-variation; analytic semigroups; ergodic theory; operators on ; strong -variation},
language = {eng},
number = {6},
pages = {2069-2097},
publisher = {Association des Annales de l’institut Fourier},
title = {Strong $q$-variation inequalities for analytic semigroups},
url = {http://eudml.org/doc/251053},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Le Merdy, Christian
AU - Xu, Quanhua
TI - Strong $q$-variation inequalities for analytic semigroups
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 6
SP - 2069
EP - 2097
AB - Let $T\colon L^p(\Omega )\rightarrow L^p(\Omega )$ be a positive contraction, with $1&lt;p&lt;\infty $. Assume that $T$ is analytic, that is, there exists a constant $K\ge 0$ such that $\Vert T^n-T^{n-1}\Vert \le K/n$ for any integer $n\ge 1$. Let $2&lt;q&lt;\infty $ and let $v^q$ be the space of all complex sequences with a finite strong $q$-variation. We show that for any $x\in L^p(\Omega )$, the sequence $\bigl ([T^n(x)](\lambda )\bigr )_{n\ge 0}$ belongs to $v^q$ for almost every $\lambda \in \Omega $, with an estimate $\Vert (T^n(x))_{n\ge 0}\Vert _{L^p(v^q)}\le C\Vert x\Vert _p$. If we remove the analyticity assumption, we obtain an estimate $\Vert (M_n(T)x)_{n\ge 0}\Vert _{L^p(v^q)}\le C\Vert x\Vert _p$, where $M_n(T)=(n+1)^{-1}\sum _{k=0}^{n} T^k\,$ denotes the ergodic average of $T$. We also obtain similar results for strongly continuous semigroups $(T_t)_{t\ge 0}$ of positive contractions on $L^p$-spaces.
LA - eng
KW - Ergodic theory; operators on $L^p$; strong $q$-variation; analytic semigroups; ergodic theory; operators on ; strong -variation
UR - http://eudml.org/doc/251053
ER -

References

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