On square functions associated to sectorial operators

Christian Le Merdy

Bulletin de la Société Mathématique de France (2004)

  • Volume: 132, Issue: 1, page 137-156
  • ISSN: 0037-9484

Abstract

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We give new results on square functions ∥ x ∥ F = ∥ ∫ 0 ∞ F ( t A ) x 2 d t t 1 / 2 ∥ p associated to a sectorial operator A on L p for 1 < p < ∞ . Under the assumption that A is actually R -sectorial, we prove equivalences of the form K - 1 ∥ x ∥ G ≤ ∥ x ∥ F ≤ K ∥ x ∥ G for suitable functions F , G . We also show that A has a bounded H ∞ functional calculus with respect to ∥ . ∥ F . Then we apply our results to the study of conditions under which we have an estimate ∥ ( ∫ 0 ∞ | C e - t A ( x ) | 2 d t ) 1 / 2 ∥ q ≤ M ∥ x ∥ p , when - A generates a bounded semigroup e - t A on L p and C : D ( A ) → L q is a linear mapping.

How to cite

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Le Merdy, Christian. "On square functions associated to sectorial operators." Bulletin de la Société Mathématique de France 132.1 (2004): 137-156. <http://eudml.org/doc/272511>.

@article{LeMerdy2004,
abstract = {We give new results on square functions\[ \Vert \{x\}\Vert \_F = \Big \Vert \{\Big (\int \_\{0\}^\{\infty \} \bigl \vert F(tA)x\bigr \vert ^\{2\} \frac\{\hspace\{0.55542pt\}\{\rm d\} t\}\{t\}\Big )^\{1/2\}\}\Big \Vert \_\{p\} \]associated to a sectorial operator $A$ on $L^p$ for $1&lt;p&lt;\infty $. Under the assumption that $A$ is actually $R$-sectorial, we prove equivalences of the form $K^\{-1\}\Vert \{x\}\Vert _\{G\} \le \Vert \{x\}\Vert _\{F\}\le K\Vert \{x\}\Vert _\{G\}$ for suitable functions $F, G$. We also show that $A$ has a bounded $H^\{\infty \}$ functional calculus with respect to $\Vert \{\, .\, \}\Vert _\{F\}$. Then we apply our results to the study of conditions under which we have an estimate $\Vert \{(\int _\{0\}^\{\infty \}\vert C\{\rm e\}^\{-tA\} (x)\vert ^2 \{\rm d\}t)^\{1/2\}\Vert \}_\{q\} \le M \Vert \{x\}\Vert _\{p\}$, when $-A$ generates a bounded semigroup $\{\rm e\}^\{-tA\}$ on $L^p$ and $C\colon D(A)\rightarrow L^q$ is a linear mapping.},
author = {Le Merdy, Christian},
journal = {Bulletin de la Société Mathématique de France},
keywords = {sectorial operators; $H^\{\infty \}$ functional calculus; square functions; $R$-boundedness; admissibility},
language = {eng},
number = {1},
pages = {137-156},
publisher = {Société mathématique de France},
title = {On square functions associated to sectorial operators},
url = {http://eudml.org/doc/272511},
volume = {132},
year = {2004},
}

TY - JOUR
AU - Le Merdy, Christian
TI - On square functions associated to sectorial operators
JO - Bulletin de la Société Mathématique de France
PY - 2004
PB - Société mathématique de France
VL - 132
IS - 1
SP - 137
EP - 156
AB - We give new results on square functions\[ \Vert {x}\Vert _F = \Big \Vert {\Big (\int _{0}^{\infty } \bigl \vert F(tA)x\bigr \vert ^{2} \frac{\hspace{0.55542pt}{\rm d} t}{t}\Big )^{1/2}}\Big \Vert _{p} \]associated to a sectorial operator $A$ on $L^p$ for $1&lt;p&lt;\infty $. Under the assumption that $A$ is actually $R$-sectorial, we prove equivalences of the form $K^{-1}\Vert {x}\Vert _{G} \le \Vert {x}\Vert _{F}\le K\Vert {x}\Vert _{G}$ for suitable functions $F, G$. We also show that $A$ has a bounded $H^{\infty }$ functional calculus with respect to $\Vert {\, .\, }\Vert _{F}$. Then we apply our results to the study of conditions under which we have an estimate $\Vert {(\int _{0}^{\infty }\vert C{\rm e}^{-tA} (x)\vert ^2 {\rm d}t)^{1/2}\Vert }_{q} \le M \Vert {x}\Vert _{p}$, when $-A$ generates a bounded semigroup ${\rm e}^{-tA}$ on $L^p$ and $C\colon D(A)\rightarrow L^q$ is a linear mapping.
LA - eng
KW - sectorial operators; $H^{\infty }$ functional calculus; square functions; $R$-boundedness; admissibility
UR - http://eudml.org/doc/272511
ER -

References

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