数理解析研究所講究録
1197 巻 2001 年 108-121

108

Maximal$L_{p^{-}}\mathrm{R}\mathrm{c}^{\mathrm{Y}}\mathrm{g}\mathrm{u}1^{\mathrm{f}}\zeta\iota \mathrm{r}\mathrm{i}\mathrm{t},\mathrm{y}$

Ph.

a,nd

-sectorial$\mathrm{E}_{\text{ノ}}$$O\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}_{)}\mathrm{o}\mathrm{r}\mathrm{s}$$\mathrm{C}\mathrm{l}\mathrm{e}^{\text{ノ}}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}$Department of Applied$\mathrm{b}\mathrm{I}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}$TU Delft

Analysis(デルフト工科大)

Mekelweg 4$\mathrm{N}\mathrm{T}_{I^{-}}2628$

CD Delft, The Netherlands,-mail: Ph.P.J.E.

$\mathrm{E}$$\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{c}a_{-\mathrm{i}\{\mathrm{t}}\backslash \mathrm{S}.\mathrm{u}\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{f}\{.\mathrm{n}1$

February 8, 2001

Acknowledgment. This note$\mathrm{i}_{\iota}\mathrm{s}$based on my lecture presented in the$\mathrm{s}_{\sim}\mathrm{v}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{f}\dot{\mathrm{f}\mathrm{l}}\mathrm{u}\mathrm{m}$”Nonlinear evolution equations and applications” at RTMS, October 2000. I would liketo thank the organizing commirtee, especially Prof. A. Yagi for his invitation and kind

hospitality.

109-boundedness and operator-valued
multiplier theorems

1$\mathrm{R}$Definition 1.1 Let (X,$bouj\iota ded$linear$||\cdot||$) be a real or compl

in $X$ is said to be$ope\mathit{7}uto/\mathrm{L}\mathrm{s}$$R$

Banatih.space. A collection$e_{J}x$$\mathcal{T}$of-bounded (Rondomized bounded) if there

exists a constant $M\geq 0$ such that$\sum_{\underline{c}\in\{-11\}^{N}},||\sum_{k=1}\mathcal{E}_{k}Tkx_{k}|’|^{2}\wedge’\leq \mathit{1}lf^{2}\overline{\mathrm{c}}\in \mathrm{i}\sum_{-}1,13^{N}||\sum_{k=1}\mathcal{E}_{k}X_{k}||^{2}N$

hoi& $for$ ait$\{T_{k}\}_{k}^{\bigwedge_{=1}}r\subset \mathcal{T}_{f}\mathrm{o}ll,$that (1.1) holdthe colle ction

The$\mathrm{f}\mathrm{i}\mathrm{r}\mathrm{S}\mathrm{f}_{\lrcorner}$$\mathrm{S}^{J},i.gc\mathrm{o}$$\mathcal{T}$islled an.

$\mathit{7}l,ot$

$R$

$R$

$\{x_{k},\}^{\mathit{1}\backslash ’}k=1\subset X$-bound for-bounded

explicit definition of$\mathrm{R}$. set$?\mathfrak{l}\mathrm{J}^{\supset}‘$$\mathcal{T}$and all $N=1,2,$

attd thec.otlstant $M\geq 0$ such

one is denoted by$st_{}a\mathit{1}\mathit{1}\epsilon\cdot \mathrm{L}\mathrm{q}t$$R_{2}(\mathcal{T})=\infty\rangle)$$\ldots.\Lambda$(i.1)$\mathcal{R}_{2}(\mathcal{T})$

. (If

.

-boundedness can be found in [2], although this notion

was already used by J. Bourgain in [3]. For a systematic treatment of this notion see [4],[15].

Remarks
1. The notion of [$\mathfrak{i}-\})\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{C}\mathrm{d}\mathrm{n}\mathrm{e}\mathrm{s}^{\backslash }\mathrm{b}$can be trivially extended to a family of operators

acting from a Banach space $X$ into a Banach space
2. By taking $N=1$ in (1.1) one finds that a$\mathrm{R}$$Y$.

-bounded collection of operators is

uniformly bounded.

3. A finite collection of$\mathrm{I}_{\mathrm{J}\mathrm{O}\mathrm{U}\mathrm{n}(}\iota_{\theta(}1$

operators is R-bounded.

110

4.

$\mathrm{B}\mathrm{v}\mathrm{u}\mathrm{H}\mathrm{i}\mathrm{n}_{rightarrow}\sigma$

the generalized parallelogram la.

$\tau \mathrm{v}$$\sum_{k=1}^{N}||_{j\gamma_{Jk}}||2=\frac{1}{2^{N}}\sum\epsilon\in i-\rceil,1\dagger^{\mathrm{N}}||\sum_{=k1}^{N}\overline{\mathrm{c}}_{k}x_{k}||2$

(1.2)

:

one shows that in a Hilbert Hpace, the notion of -boundedness and uniform bound$\mathrm{B}$

cd ness are$\mathrm{e}\mathrm{q}\mathrm{l}\mathrm{l}$ivalent.

5. The RHS of (1.2) can be rewritten as
sequence of independent identically ($|| \sum_{k=0}^{\mathrm{A}’}\epsilon_{k}\prime Xk||^{2}L‘ \mathit{1}(5^{\}}arrow;X)$

where$\{\epsilon_{k}\}_{k-1}^{\infty}$, denotes a$1\mathrm{i}\mathrm{t}\cdot;\mathrm{f}_{\mathrm{J}}\mathrm{r}\mathrm{i}\uparrow \mathrm{J}\mathrm{u}\mathrm{t}\mathrm{f}^{1}\mathrm{d}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{p}\}_{j}\mathrm{r}\mathrm{i}\mathrm{c}\{-1,1\}-\mathrm{v}\mathrm{a}]1\mathrm{l}\mathrm{e}\mathrm{d}$variablcs defined on some probability spacecan be’$(_{-}.\Omega, F, P)$

random. Accordingly condition (1.1)

as$\mathrm{I}^{\cdot}\mathrm{e}\mathfrak{n}\mathrm{T}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{n}$(1. 3)$\cdot$$|| \sum\epsilon_{k}T_{k^{X}}k||_{L_{2(;X}}\Omega)N\leq \mathit{1}1f||\sum_{k=1}^{r}\epsilon kX_{k}|’|\wedge L_{\mathit{1}(}\int).;-\backslash )$$n-1$

In view of Kahane’s inequality we can replace $L_{2}(\Omega;x)$ by
adjusting the constant$\mathit{1}^{\rceil}/f$$L_{p}(_{-}\Omega;^{x)},$$1\leq p<\infty\grave{\text{ノ}}$,appropriately.

It appears that the notion of -boundedness is useful in the context of multiplier theorems$\mathrm{R}$associated$\mathrm{w}\mathrm{i}\mathrm{t}_{\mathrm{J}}\mathrm{h}$Definition 1.2$Schaud,e/\cdot decot\iota$

unconditional Schauder decompositions.
$\Lambda s\mathrm{e}qu$positio,

eitce$l\iota$ofof bounded linear projections in$D=\{D_{k}\}_{k^{-}1}\mathrm{o}\mathrm{o}$$\mathrm{t}vhe,nelJerk\neq l$$x= \sum_{=k1}^{\infty}D_{k}x$gent$:Ot\prime t,poSit’io\mathcal{T}\iota$$fo\gamma$all$\prime x\in X$.

is call$\mathrm{e}d$a$X,\cdot’ if$$D_{k}D_{l}=0$

The de ($X$

is eaiied$unco?\iota dit\dot{?,}onal$

for all, $x\in X$

,

(1.4)

(1.5)

if the series in (1.5) is unconditionally conver-

111

We recall that if $D$ is an unconditional Schauder decomposition of $X$ , then there exists a$\mathrm{c}’ \mathrm{o}\mathrm{n}\mathrm{t}‘,\mathrm{t}\backslash \mathrm{a}\mathrm{n}\mathrm{t}_{}c_{D}>0$

such that(1.6)$\mathrm{c}j-1|D|\sum_{=k1}^{\mathit{7}\mathrm{t}}Dkx||\leq||\sum_{k=1}\mathit{6}_{kk}DX||_{T_{2}(\mathrm{x})}n\Omega;\leq C_{D},||\sum_{1k=}^{n}r)_{k^{Ji}}||$

holds for all $x\in X$ and all $n\geq 1$ .
Letof

$D=\{D_{k}\}_{k=}^{\infty}1$$\mathrm{I}_{\mathrm{J}\mathrm{O}\mathrm{U}}\mathrm{n}(1\mathrm{e}\mathrm{d}$$\mathrm{e}\mathrm{a}\mathfrak{l}\mathfrak{c}.\mathrm{h}k\geq 1$

be a Schauder decomposition of $X$ and let

linear operat$o\mathrm{r}\mathrm{s}$such that$L_{k}$

leaves$X_{D}’$,$D_{k}$, invariant, for

We observe that

$X_{D}’$$D_{k}$

to$X_{D}$

by$D_{k}$$X_{D}$. The operator$T_{L}$

maps, i.e.

(1.8)$X_{D}$

is invariant under$D_{k},$$k\geq 1$. We stilland define(finite sum)

$T_{L’}x:= \sum_{k=\perp}^{\infty}L_{k}D_{k}x$$\prime x\in X_{D}’$$\{\mathcal{R}(D_{k})\}_{k=1}\propto$$\bigcup_{-,k1}\infty R(\sum_{\mathrm{t}=1}^{k}Dt)$

is dense in $X$ and that

denote the restriction of(1.7)$k\geq 1$

be the lincar subspace of $X$ generated by the subspaces$X_{D}:=$

for all, the range of

be a sequence, i.e.

$L_{k}D_{k}=D_{k}L_{k}D_{k}$Let$\hslash(D_{k})$$L=\{L_{k}\}_{k=1}^{\infty}$

into itself and commutes with$\tau_{LkkL}D=DT$,$k\geq 1$(1.9)$D_{k}$:(1.10)

In case the decomposition $D$ is unconditianal. we obtain from (1.6), (1.9) and (1.10):$\mathrm{c}j-D1||T_{L}x||\leq||\sum_{k=1}^{\infty}\overline{\mathrm{b}}_{k}L_{k}Dkx||_{L_{2(:X)}}\Omega\leq c_{D}||T_{L}.T||$(1.11)

112

for all$\prime x\in X_{D}’$. Since$X_{D}$

is dense in $X$ flJld $X$ is c.oIIlplete, the operator. bounded linear operator on $X$ ifr$\mathrm{a}$$T_{L}$

is bounded on$X_{L}$$T_{L}$

extends to. In view of (1.11) we have thefollowi ng ch aracterization.
Theorem 1.1 ([4], [15]). Let$D=\{D_{k}\}_{k=}^{\infty}1^{j}$be an unconditional Schauder deof the Banach.space$L=\{L_{k}\}_{k1i}^{\infty}=$

be a se.quence of bounded linear operators of$Xsati_{S}fi,ng(\mathit{1}.7)$

Let

on

$X_{D}’$and let$X$.

be the $de,nse$ linear subspace of $Xdefi,\cdot?ted$ by (1.8) aitd let$\lambda_{L}’defi_{}J\iota ed$$(j\mathit{0}r\prime po.9ition$$T_{L}$

be the linear opera. $tor$by (1.9).

Then the operator$T_{L}$

is bounded iff there exists a$\mathfrak{c}^{l}.\mathit{0}\prime stantM>0su\zeta^{\wedge}.h$th.at(1.12)$|| \sum_{k=1}^{n}\epsilon_{k}rJkxk,||_{\Gamma,(}2\Omega;^{x})\leq M||\sum_{1k=}\epsilon_{k}\prime X_{k}||nL_{\mathit{1}()}\mathrm{t}^{)}arrow;\sim\backslash$;$hoid_{\iota^{\mathrm{t}}}$for$aii\prime x_{k}\in R(D_{k}),$If this coisdition$idful_{r}f\overline{i}\iota\iota ed$

and all $n\geq 1$ .

$k\geq 1$

then the norm of$T_{\Gamma_{i}}$.sati.sfies(1.13)$||T_{L}||\leq c_{D}^{2}M$$I_{J}^{:}\iota pa\gamma t\prime icu\mathit{1}a’$

if the collection

$L$

is

$R$

-boem $ded$ with eon.stant $R_{2}(L),$$tllej\iota(\mathit{1}.\mathit{1}\mathit{2})$hoids with

$M=R_{2}(L)$ .

$P_{JO}of.\cdot$

If (1.12) holds and$x= \sum_{k=1}^{J\iota}D_{k}x$for some $n\geq$ 1 we have$||T_{L’}x||$$\leq$. where$\mathrm{t}jv||\sum^{n}\epsilon_{k}LkDk\prime k=1x||L_{\mathit{1}}(\mathrm{t}\underline{)};-\backslash )\leq(_{D}’,\mathit{1}\mathrm{t}f||\sum_{k=1}^{n}\mathrm{G}cDkk^{\mathcal{I}||)}L_{2}(\mathrm{f}\iota;X\leq c_{D}^{2}M||\sum_{-k1}^{\mathcal{R}}Dkx||=c_{D}^{2}M||x||’$

we have$\mathrm{u}t\cdot,\mathrm{f}\mathrm{f}\mathrm{l}(1.11)$

and (1.6).

113$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}1_{\mathrm{V}}$if$T_{L}$

is bounded, we have for all $x_{k}\in R(D_{k}),$ $k=1,$$\cdot n$$\ldots,$

and all $n\geq 1$ ,$|| \sum_{1k=}^{}8kLkXk||_{L_{9(;X}}\Omega)n=$$|| \sum_{k=1}\mathcal{E}_{k}LkD_{k}x|n|L_{2(X}\Omega;)\leq C_{\Gamma)}||T_{r}.x|j|$$\leq(jD||\tau_{L}||||\prime x||=\mathfrak{c}i_{D}||\tau_{L}||||\sum_{k=1}^{n}Dk^{X}||$$\mathrm{c}j|2|DT_{L}||||\sum_{k=1}^{n}\mathcal{E}_{k}D_{k}x||_{L\cdot(}\mathit{1}iarrow;_{\mathrm{r}}.)=?1$$\mathrm{r}_{D}^{2}||T_{L}||||\sum_{k=1}^{n}||L\mathit{1}(\int_{arrow}|\backslash ;arrow)$

where

2

$\prime x=\sum_{k=1}^{n}$

Dkxk and with the use of (1.6), (1.11).

Operator-valued Marcinkiewicz and Mikhlin
multiplier theorems

Let (X,$||\cdot||$) be a complex Banach space and let, $L^{p}(0,1, x),$ $1\leq p<\infty$ denote the usualcomplex Banach space equipped with the norm$||?J,||p:=(.[^{1}||u(t)||pdt)1/\rho 0$

Let $N=0,1,2,$$\ldots$

and let,$F_{\mathrm{A}’}n$$:=$$\epsilon_{k}^{},(t)$$:=$$e^{2\pi kit}$$:=$$J^{1}\cdot e^{-2\pi}u(kitt)0dt$$\iota\iota(\wedge k)$$(\epsilon_{k}^{\mathrm{J}},\otimes\{\iota(\wedge k))(t)$$:=$$\sum_{k=-\backslash \mathcal{T}}^{}\mathrm{J}\backslash \Gamma,e_{k}\otimes\hat{u}(k)$$\epsilon_{k}^{},$

;

$t,$

(f)\^u $(k)j$where$\in[0,1\rceil,$,$u\in T_{\lrcorner}^{p}(\mathrm{o}, 1;X)$,$k\in \mathbb{Z}$$k\in \mathbb{Z}$$t\in[0.1]$

,

114

$\mathrm{A}\iota‘ \mathrm{i}$is well-known, the Banach-valued version of Fe.

$|\mathrm{c}\mathrm{r}’ \mathrm{s}$of Cesaro means of the sequence$L^{p}(\mathrm{o}$

. ;
$1$

and every$1\leq p<\infty\grave{\text{ノ}}$$\{F_{N}u\}_{N=}\infty 0$

theorem holds, that is, the sequence

converges to ?4 in$L^{\mathrm{p}}(0,1;X)$for every $u\in$.

It follows in particular that the vector space of trigonometric’$\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\eta \mathrm{o}\mathrm{m}\mathrm{i}\mathrm{a}1_{\llcorner}\backslash$$T(X):=\mathrm{s}_{k\in}\mathrm{p}\mathrm{a}.\mathrm{n}\{\epsilon_{k}\mathrm{z},\}$

is dense in$u$in$L^{\mathrm{p}}(0,1$$L^{p}(\mathrm{o}, 1;X),$;$p\geq 1$. It is also known that the sequence$\{F_{\mathrm{A}^{\Gamma}}u\}_{N=}^{\infty}0$

converges to

iff the Riesz projection $P$ defined on $T(X)$ by

for every $u\in L^{p}(0,1$ ;$Pu:= \sum\epsilon,\otimes k\mathrm{t}\iota(\lambda k\geq 0\wedge i)$is$1_{)\mathrm{O}\mathrm{U}\mathrm{n}}\mathrm{t}1\epsilon 1\mathrm{t}1$

in the $L^{p}(0,1;X)$ norm.

This is the case when $1<p<$ oc and $X=\mathbb{C}[11]$ or more gener.ally iff $X$ has the UMD
property.

TInder these conditions the sequence of bounded projections
by $E_{0}:=F_{0}$ and $E_{k}:=F_{k}-F_{k-1},$$L^{\mathrm{p}}$(, this$0,1_{\mathrm{i}}X1\cdot \mathrm{t}\mathrm{t}^{\mathrm{v}_{\mathrm{h}\mathrm{p}nX}}=\mathbb{C}$$h_{i}>\rceil$)$\mathrm{d}\mathrm{e}.\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{P}^{1}$$\{E_{k}\}_{k=}^{\infty}0$in$L^{\mathrm{p}}(0,1$$k=1,2,$extended to the case $X$ is UMD$\ldots,$$\mathrm{I}_{\mathrm{J}}\mathrm{y}$definedis a Schauder decomposition of the space$\mathrm{S}\mathrm{i}\mathrm{f}_{l}\mathrm{i}o\mathrm{n}\mathrm{i}_{\iota}\mathrm{s}$unconditional iff $p=2$ . In remarkable

papers Paley and Littlewood $[10]j[11]$ showed t,hat the dyadic blocking of$|_{\mathrm{J}_{\mathrm{L}}}\mathrm{Y}^{\mathrm{P}}D_{k}:=F_{2^{k}}-F_{2^{k-1}},$;$D_{0}:=F_{1},$$\mathrm{i}_{l}\mathrm{s}$$\{E_{k}\}_{k0}^{\infty}=$definedunconditional. This property has been

Bourgain [3]. A detailed proof of this fact can be foundin Venni [12]. A careful analysis of this proof$\mathrm{S}\mathrm{l}\mathfrak{l}\mathrm{o}\mathrm{v}\mathrm{V},\mathrm{s}$that the notion of R-boundedness(which is not mentioned explicitly) plays an irnportant role. This$\mathrm{h}\mathrm{a}_{\iota}\mathrm{s}$been the starting

115

point of [4].

In view of Theorem 1.1 a sequence$p<\infty$$\{L_{k}\}_{k-0}^{\infty}$of bounded linear operators insatisfying (1.7) and (1.12) induces a bounded linear operator

In particular condition (1.7) is satisfied when the operatorsof the$\mathrm{f}_{0\mathrm{r}}\mathrm{m}$$:=$$\tau_{k}\lrcorner?l$$\sum_{l=-1}^{1}\epsilon,l\otimes \mathit{1}\mathrm{t}f_{l}$$\{\mathit{1}lf_{l}\}_{larrow\tau}\in$\^u

are diagonal operat.

$()\mathrm{r}\mathrm{s}$$(\iota)$$k=1,2,$$:=$.

$2^{k-1}< \sum_{k ,|l|\leq 2}\epsilon j\iota\otimes M_{\iota}\hat{u}(\iota)$

is a family of bounded operators in$L^{p}(\mathrm{o}, 1;X)$$\ldots$. In a recent work Arendt

and Bu [1] found an interesting sufficient condition on the sequence$\{L_{k}\}_{k=0}^{\infty}$$T_{L^{\mathrm{S}\mathrm{a}}}\mathrm{t}\mathrm{i}\mathrm{S}\mathrm{f}_{\}}\gamma \mathrm{i}\mathrm{n}\mathrm{g}(1.13)$:$L_{o}u$

where$\{T_{r}k\}_{k-}\infty 0$$1<$$L^{\mathrm{p}}((|, 1;x),$to be$\{\mathit{1}\mathrm{t}f_{l}\}_{\iota}\in\prime \mathrm{z}$for the family-bounded, namely$\mathrm{R}$$\Pi_{A\mathit{1}},:=R$($\{\mathrm{i}\nu[_{1\prime}\}_{\tau\in \mathbb{Z}}$and$\{l(M_{l1}+-\mathit{1}\iota f_{\iota});\iota\in \mathbb{Z}\}$

)

(2.1)

$<\infty)$

This leads to the following

Theorem 2.1

Le.t$n\in L^{\mathrm{p}}(\mathrm{o},$Let$\{\text{\^{u}}(k)\}k\in\wedge\cdot\zeta$

of{$bou\mathit{1}\iota dedii$$\mathit{1}lf_{k^{1\iota()\}_{k\in \mathrm{z}}}}\wedge k$and$tl‘ eI^{\cdot}e$$(\Lambda/\mathrm{p}\mathfrak{l}tdt- Bu)$$1;^{x)},$}$1<p<\infty_{i}^{\backslash }XUMD$

be the$\iota$ear$s\mathrm{e}que\mathit{1}\iota \mathrm{r}^{\}}.e$$ope;\mathrm{u}to\prime si_{l}\iota$of$Fourier-\prime j\mathit{0}\mathrm{f}^{\mathrm{J}},fficien\dagger S$

X. Then if theis the equence of Fourier$s$space.

of and let$u$$\{M_{l}\}_{l\in}\mathbb{Z}$$cond,iti_{\mathit{0}}n(\mathit{2}.\mathit{1})h_{c}old\mathrm{J}s$$coeffi_{C}\tilde{l}\epsilon^{},\gamma m$

be a, then the$s\mathrm{e}qu$ence$s\mathrm{e}que\gamma tC\mathrm{e}$of a (unique) function $v\in L^{p}(0,1;x)$

exists a $co\mathit{1}lSta;tC>0$ depcnding only on $p\in(1, \propto)$ and $X$ such that$||v||_{L\beta(}0,1;\backslash ^{r})\leq c,R,.\iota t||u||_{T_{\text{ノ}}}\mathrm{p}(0,1;^{\mathrm{x}})$

(2.2)

116

RemarkTheorem 2.1 is a simplified version of a more involved Marcinkiewicz type theorem due. to\v{S}trkalj and Weis [13], which is a$\mathrm{d}\mathrm{i}\mathrm{s}(\iota \mathrm{e}\mathrm{t}\mathrm{e}$version of the following-valued Mikhlin$\mathrm{o}\mathrm{p}e,\mathrm{r}\mathrm{a}\mathrm{t}0\mathrm{r}$type theorem due to Weis [14].

Theorem 2.2 (Weis ’99)
Let $1<p<\infty$ and $X$ be a $UhID$ space.
Let$M\in C^{1}(\mathbb{R}\backslash \{0\};\mathcal{L}(X))$

the colle ctions. Then $M$ is th.

$\{_{\wedge}h,f(\rho);\beta\in \mathbb{R}\backslash \{0\}\}$

and$\mathrm{e}sy$} $r\iota bol$of a bounded opeiutor in$L^{p}(\mathbb{R}, X)$-bounded in$\{pM’(p);p\in \mathbb{R}\backslash \{0\}\}a7\mathrm{P}R$$\mathcal{L}(X)$if.

Comments
This remarkable theorem iswhere $X$ is$p<\infty$$\underline{\mathrm{n}\mathrm{o}\mathrm{t}}$$\mathrm{t}_{}\mathrm{h}\mathrm{e}$first, operator-valued multiplier theorem in$\mathrm{I}$$\mathrm{e}$

3

$1<$

isomorphic t,o a Hilbert space. Another proof can be found in[5]. It should ) mentioned that theL.

$L^{\mathrm{p}}(\mathbb{R}, X),$$\mathrm{c}o\mathrm{n}\mathrm{t}\mathrm{e}p\eta \mathrm{t}$of [4] has been made available to Professor; in December ’98.

$\backslash \cdot 1^{\vee}\mathrm{e}\mathrm{i}_{t}\cdot$

Converse theorems,-sectoriality and$\mathrm{R}$$L_{p}$-Maximal

Regularity
In thiswe$t\cdot;\mathrm{e}\mathrm{c}:\mathrm{t}\mathrm{i}0\mathrm{n}$]$\mathrm{f};\mathrm{h}.\mathrm{a}1$

present some results showing t,hat the; a necessa,ry ($,\backslash \mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{p}^{]}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{h}’ \mathrm{i}t\cdot$of]$\mathrm{R}- \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{a}$.

$\mathrm{i}\mathrm{t}_{\sim}\mathrm{v}$$;\cdot \mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}_{0\mathrm{n}\mathrm{t}}:\mathrm{n}\mathrm{d}$$\mathrm{B}_{\ulcorner}\mathrm{b}\mathrm{o}\iota 1\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}$of thethat this notion naturally leads to the notion

117

Theorem 3.1
Let$(X, ||\cdot||)$$If\wedge’ \mathrm{t}\prime f$$(\mathrm{L}7’e.is f\mathit{9}\mathit{9})I^{7}4]$

be a$\zeta’.O$} $\gamma|,pie.x$

Banach space (not neccs.sari $lyUMD$)

is the symbol of a bounded operator in$\{M(\rho):\rho\in \mathbb{R}\backslash \{\mathrm{o}\}\}$$\Gamma\nearrow(\mathbb{R};^{x})$,$an_{\mathrm{f}}l$let$M\in c_{\text{ノ}^{}1}(\mathbb{R}\backslash \{0\};c(X))$, 1 $<p<$ oo then th, famiiy$e$is R-bouitded.

Remarks
1. ilnotllPr proof of this result can$\mathrm{I}\mathrm{J}\mathrm{e}$found in [5].

2. An analogue of this result in the discrete cnse
Bu [1], namely if {$\mathrm{R}- \mathrm{I}_{\mathrm{J}\mathrm{O}}\mathrm{u}\mathrm{n}(\iota \mathrm{p}\mathrm{d}$in$\lambda/f_{\iota\}_{\iota\in \mathbb{Z}}}$$\mathcal{L}(X)$; been established by Arendt and$\mathrm{h}\mathrm{a}\{\cdot$is a”multiplid: in $L^{p}(0,1;X)$ then the family$(X_{i}||\cdot||)$$the\gamma e$

exists a$L_{p}$-maximal regularity theorem.

be a complex Ranach spacea sectoriai $ope/ato/\cdot$ with spectral

If$\mathit{0},ngle\omega_{A}<\pi/2$$‘\wedge.or\iota \mathrm{L}9tant\perp ff>0.\mathrm{s}uch$$a\mathit{1}\iota d$

let $A:D(J1)\subset Xarrow X$ be, and let $1\leq p<\infty$ .

that

(3.1)

$||u’||L^{p}\mathrm{f}^{\rfloor \mathrm{L}_{:}X)}.+||Au||L\nu_{()}\mathrm{J}\mathrm{R};^{x}\leq M||u’+Au||_{Lp(X}\mathbb{R};)$for everyis.

An application of Theorcm 3.1 leads to the following
Theorem 3.2 Let$\{\mathit{1}lf_{l}\}\iota\in \mathbb{Z}$$u\in \mathrm{t}^{i}\mathrm{f}^{r1,\mathrm{p}}(\mathbb{R};X)\backslash \cap L^{\mathrm{p}}(\mathbb{R}, D(\lrcorner 4)),$$t,h,en$the $fami,ly$(3.2)$\{\rho(i\rho+A)-1; \rho\in \mathbb{R}\backslash \{\mathrm{o}\}\}$

is R-bounded.

Co’rverseiy if $1<p<\infty\rangle$ $a?ldx$ is [,$r_{\mathrm{A}}\eta,ID$

to hoid.

, then $con($],$itiJon(\sigma d.\mathit{1})$is also$s\uparrow l,ffiCient$for (3.2)

118

Comments

1. The characterization of$L_{p}$-maximal$\mathrm{r}e,\mathrm{g}_{111}\mathrm{a}\Gamma \mathrm{i}\mathrm{t}_{\mathrm{V}}$for abstract differential equations in

a UMD space has been obtained independently$\}_{)}\mathrm{y}$Kalton and NVeis (see [14]).

2. The characterization in the case $X=L^{q}(-\Omega; \mu),$ $1\leq q<\infty$ has been presented in aseminar in Delft,$\mathrm{D}\mathrm{e}\mathrm{C}^{\cdot}\mathrm{e}\mathrm{m}\mathrm{I}_{\mathrm{J}\mathrm{e}\mathrm{r}}’ 98$

by Professor L. Weis in$\mathrm{t}_{l}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{S}$of a$\mathrm{s}\mathrm{q}\iota 1\mathrm{a}\mathrm{r}\mathrm{c}-\mathrm{f}_{1}1\eta \mathrm{C}\mathrm{t}\mathrm{i}_{0}\mathfrak{n}$estimate:

Theire exists a$\mathrm{c}\mathrm{o}\eta \mathrm{S}\mathrm{t},\mathrm{a}\mathrm{n}\mathrm{t}_{l}C>0$

such that for all $f\in L^{q}(L^{2})$$||(J^{\cdot}|tR(?.\prime t,, A)f(t)\mathbb{R}|2df)^{1}/2||_{\backslash }\vee\leq C_{\text{ノ}}||(J^{\cdot}|f(t)|2d\mathrm{f})1/2||_{arrow 1}\mathrm{K}$This notion is in this special case equivalent to condition (3.2).

3. It is natural to call a sectorial operator $A,$
riality, the notion of$\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}o\mathrm{r}\gamma \mathrm{r}\mathrm{l}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\theta \mathrm{d}\mathrm{n}‘ 1\mathrm{A}\mathrm{S}_{\iota \mathrm{S}}$[14] for definitions and examples ofsectorial operators which$\mathrm{R}$-sectorial if in the definition of secto-

is replaced-boundedness. See [5],$\mathrm{I}_{\mathrm{J}}\mathrm{y}\mathrm{R}$-sectorial operators and [8] for examples of$\mathrm{R}$-sectorial in $T^{q},(([0,1]), 1<q<\infty),$ $q\neq 2$ .

$\dot{\mathrm{r}}\iota \mathrm{r}\mathrm{e}\underline{\mathrm{n}\mathrm{o}\mathrm{t},}\mathrm{R}$4. As is shown in [1] Theorem 2.1 and its partial convprlse are strong enough to char-acterize$L_{\mathrm{p}}$-maximal regularity in UMD spaces.

Concluding remarks
It appears that the notion of$\mathrm{R}$-boundedness is an appropriate notion for operator-

valued multiplier theorems. Applications of this notion has been made by B. de Pagter,

119

F. Sukochev and H. Witvliet,the$\mathrm{R}-\mathrm{b}\mathrm{o}\iota\ln[]\theta \mathrm{t}\mathrm{l}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{S}^{}$$|_{\mathrm{J}\mathrm{o}\mathrm{u}\mathrm{n}}\mathrm{d}(\backslash \mathrm{d}\mathrm{n}\mathrm{f}\wedge\aleph \mathrm{s}$not$\mathrm{R}$of$\mathrm{t}_{\mathrm{J}}\mathrm{h}\mathrm{e}$$\mathrm{f}_{\mathfrak{l}}\mathrm{o}$Schur type multipliers [6]. We recall that in Theorem 1.1,of the family
operator$T_{L}$$\{T_{k},\}^{\infty}k=1$

is in general only :1,. Examples where the family-bounded and the operator$T_{L}$$\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{i}\iota\cdot,\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}$$\{L_{k}\}_{k}\mathrm{K}=0$

condition for thein Theorem 1.1 isis bounded can be found in [15].

References[1] W. Arendt and Shangquan Bu. The operator-valued Marcinkiewicz multiplier the o-rem and maximal regularity. Technical Report 5, Ulmer Seminare, 2000.
[2] E. Berkson and T.A. Gillespie. Spectral decompositions and harmonic analysis on

UMMD spaces. Studia Math., $112(1):13-49$ , 1994.
[3] J. Bourgain. Vector-valued singular integrals and the

Chao aml XV.A.

, editors, Probabj,lity$\mathrm{t}\prime 1^{7}\mathrm{o}\mathrm{V}\mathrm{C}\sim \mathrm{V}\prime \mathrm{z}_{\mathrm{L}}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$$H^{1}$$Ti\iota eoY\eta \mathrm{J}$-BMO duality. In J.-A.

and Haivn, 0}$\iota i_{C}$Analysis,pages 1-19. Marcel Dekker. Inc. New York and Basel, 1986.
[4] Ph. Cl\’ement, B. de Pagtcr: F.A. StlkclcheV7 and H. Witvliet. Schauder decomposition

and multiplier theorems. Studia$M_{Cl}th.,$$138(2):135-163$ , 2000.

[5] Ph. Cl\’ement and J. Pr\"uss. An operator-valued transference principle and maxi-

mal regularity on vector-valued$L_{p^{-l^{\backslash }}}\nwarrow \mathrm{p}\mathrm{a}\mathrm{C}^{\mathrm{l}}‘ j\mathrm{s}$. In Lumer and Weis, ffiitors, Evolution

Equatioits azul Tilei’ Applications in Physics 0,nd Life Sciences,

Dekker, 2000.

$\mathrm{p}\mathrm{a}\mathrm{g}\mathrm{t}_{\Lambda}1\mathrm{S}$67-88. Marcel

120[6] B. de Pagter, F. Sukochcv, and H. Witvliet. Double operator integrals. preprint,

2000.
[7] B. de Pagter and H.
$\mathrm{d}^{i}\mathrm{E}\mathrm{t}\acute{\mathrm{e},}$$(.711\eta \mathrm{e}\mathrm{c} 1998)$$\mathrm{W}\mathrm{i}\mathrm{f}_{\mathrm{J}\mathrm{V}}$16,]$\mathrm{i}\mathrm{e}\mathrm{t}$.

$\mathrm{U}\mathrm{n}\mathfrak{c}\cdot \mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}l\mathrm{i}\mathrm{o}\eta$$\mathrm{P}\mathrm{u}[_{\mathrm{J}}]\mathrm{i}\mathrm{C}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}$

al decompositions and UMD spaces.

,$\mathrm{E}\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{r}\backslash$

Mathe’matiques de T’IJFR Sciences et Techniquoe

de Besanqon, 1998.
[8]$\mathrm{N}.\mathrm{J}$. Kalton and G. Lancien. A solution to the problem of the$L^{p}$-maximal$\mathrm{r}e$gularity.

Math. Z. (to appear).

[9]$\mathrm{J}.\mathrm{E}$. Littlewood and. Paley. Theorems on Fourier series and power series, (I).

$\mathrm{R}.\mathrm{F}_{\lrcorner}$

Jour: London Math. Soc., 6:230-233, 1931.
[10]$\mathrm{J}.\mathrm{E}$. Littlewood and$\mathrm{R}.\mathrm{F}_{J}$. Paley. Theorems on Fourier series and power series, (II).

Proc. London Math. Soc.,[11] M. Riesz. Sur les$42:_{\mathrm{e}}52^{-}8\mathrm{g}$, 1936.

. Math.

$\iota_{(\mathrm{n}\mathrm{C}\mathrm{t}}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{n}.|\mathrm{u}\mathrm{g}\mathrm{u}\acute{e}\mathrm{e}\mathrm{S}$[12] A. Venni. Banach spaces with the Hilbert$ZeitSCh\gamma\cdot.$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}_{\iota}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}$, 27:218-244, 1927.

property. Departimento di

Mathematica, Universit\‘a di Bologna.
[13] Z.

$\check{\mathrm{S}}\mathrm{t}\mathrm{r}\mathrm{k}\mathrm{a}\mathrm{J}\mathrm{j}$and L. lVeis. On operator-valued Fourier multiplier theorems. preprint,

2000.
[14] L. Weis. A new approach to maximal

Evolution Equatioits$a\cdot nd$$L_{p}$-regularity. In$\mathrm{T}_{\mathrm{J}}\mathrm{u}\mathrm{m}e\mathrm{r}$and Weis, editors,

Their Application.9 in Physt $cs$ and Life Sciences, pages 195-

214. Marcd Dekker, 2000.

121[15] H. Witvliet. U,tconditional $Scho,uder$ Decompositio $ns$ andthesis, TU$\mathrm{D}\mathrm{e}\mathrm{l}\mathrm{f}\mathrm{t}_{7}$

2000.

.

$kIuitip\iota ieJ^{\cdot}\tau i\mathrm{A}eoT^{\cdot}elr|‘ s$$\mathrm{P}\mathrm{h}\mathrm{D}$