数理解析研究所講究録1102 巻 1999 年 49-63

49

Commutators of Singular Integral Operators
on Morrey Spaces with General Growth Functions
山形大学理学部

水原

昂廣 (Takahiro Mizuhara)

Abstract of the Talk
The talk will be concerned with the boundedness of the commutators of Calderon-

Zygmund singular integral operators on Morrey spaces

satisfying the condition; there exists a constantsuch that for any $(x, r)\in R_{+}^{n+1}$$\Phi(x,r)$(1)$\int_{f}^{\infty}[\Phi(X, t)/ta+1]dt\leq C\Phi(x, r)/r^{a}$$f$, defined on$R^{n}$$C$

with growth fimctions, independent of $(x, r)\in R_{+}^{n+1}$ ,for some $a>0$ .

,In this case, we write $\Phi\in G_{a}$ simply. We denote byof locally integrable functions$IP^{\Phi},(R^{n})$$IP^{\Phi},(R^{n}),$$0<p<\infty$ , the space, for which there exists a constant $C$ ,independent of balls $B=B(x, r)$ , such that(2)$\int_{B(x,\mathrm{r})}|f(y)|pdy\leq cp\Phi(x,r)$

for all balls $B=B(x,r)$ . Let
be the smallest constantspace
becomes a quasi-Banach space with quasi-norm$||f||_{p,\Phi}$$c_{\mathrm{s}\mathrm{a}\mathrm{t}}\mathrm{i}\mathrm{S}6^{r}$$IP^{\Phi},(R^{n})$$1\leq p<\infty$Let, then the space$lP^{\Phi},(R^{n})$ing (2). Then the$||\cdot||_{p,\Phi}$. In

becomes a Banach space with norm$\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}_{\dagger}$$||\cdot||_{p,\Phi}$

if.

be the space of an functions of bounded mean oscillation and let$BMO(R^{\mathrm{R}})$, be the space of all Lipschitz continuous functions of order a. Let $M$
be the Hardy-Littlewood maximal operator. We need two variants of $M$ . For $0<q<\infty$$\Lambda_{\alpha}(R^{n}),$$0<\alpha<n$

let $M_{q}f(x)=\{(M|f|^{q})(X)\}1/q$. The sharp maximal function $f\#(x)$ is defined by$f^{\#}(x)= \sup_{\in xB}|B|^{-1}\int_{B}|f(y)-f_{B}|dy$

Let

$T$

, where.

be a Calderon-Zygmund singular integral operator defined by $Tf=k*f$ withthe kernel$k\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{S}}\mathfrak{h}r$

ing the conditions :,$||\ovalbox{\tt\small REJECT}||_{\infty}\leq C$[ $k(x)|\leq C|x|^{-}n$ for $0\neq x\in R^{n}$ ,$|k(x)-k(x-y)|\leq C|y|/|x|^{n+1}$

Let$f_{B}=|B|^{-1} \int_{B}f(y)dy$$I_{\alpha},$$0<\alpha<n$for $|y|\leq|x|/2$ .

, be the Riesz Potential of order$\alpha$defined by.

$(I_{\alpha}f)(x)=I_{R} \hslash\frac{f(y)}{|x-y|^{n}-\alpha}dy$50
Related to$I_{\alpha}f$, the fractional maximal function

is defined by$M_{\alpha/n}^{*}f(X)$$M_{\alpha/n}^{*}f(_{X})=f_{\alpha,1}^{*}(X)= \sup_{\in xQ}\frac{1}{|Q|^{1-\alpha}/n}\int_{Q}|f(y)|dy$

For a locally integrable function$b$.

and an operator 8, we define the commutator $[b, S]$ ,between the operator and the multiplication operator by;
We have proved the following$S$

by $[b, S]=bS-Sb$.

$b,$$([\mathrm{M}\mathrm{i}\mathrm{z}2])$

Let $0<p<\infty$ . We assume thatexists a constant $C=C(p, \Phi)>0$ , independent of , such that for all
Theorem$\Phi\in G_{n}$$1(\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}2.1.)$$f$. Then there$f\in L^{\mathrm{p},\Phi}(R^{n})\cap$$L_{c}^{\infty}\{R^{n})$(3)$||Mf||_{p,\Phi}\leq C||f\#||p,\Phi$wherebe the set of all essentially bounded functions on$L_{c}^{\infty}(R^{n})$$H^{\iota}$with compact support.

We use the method due to Di Fazio and Ragusa. Our method is based on weighted
maximal inequality due to Garcia-Cuerva and Rubio de Francia.
$\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}^{\backslash }$this and the pointwise estimmate due to Str\’omberg;$1<q,$ $s<\infty$ ,$\{[b, T](f)\}\#(x)\leq C||b||_{*}\{Mq(Tf)(x)+(M_{\epsilon f)()\}}x,$

for almost all $x\in R^{n}$ , we obtain the boundedness of the commutators $[b, T]$ on Morrey

spaces;$([\mathrm{M}\mathrm{i}_{\mathrm{Z}_{2}}])$

Theorem$\Phi\in G_{n}$$2(\mathrm{T}\mathrm{h}\infty \mathrm{r}\mathrm{e}\mathrm{m}2.2.)$. Then the commutator

Let $1<p<\infty$ and $b\in BMO(R^{n})$ . We assume thatis bounded in$[b, T]$$IP^{\Phi},$. More precisely, there exists a

constant $C=C(p, \Phi)>0$, independent of and , such that for all $b\in BMO(R^{n})$ and
$b$

$f$

$f\in L^{p,\Phi}(Rn)\cap L_{c}^{\infty}(R^{n})$(4)$||[b, \tau](f)||p,\Phi\leq C||b||_{*}||f||_{p},\Phi$

Also we can observe the following
Theoremassume thatfrom

$3(\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}2.3.)$$\Phi\in G_{n-p\alpha}$$IP^{\Phi}’(R^{n})$intoindependent of
(5)

$b$

. If

$([\mathrm{M}\mathrm{i}_{\mathrm{Z}_{2}}])$.

;

Iaet $1<p<q<\infty,$ $0<\alpha=n(1/p-1/q)<n$ . We
, then the commutator $[b, T]$ is a bounded operator$b\in\Lambda_{\alpha}(R^{n})$. More precisely, there exists a constant $C=C(p, q, \Phi)>0$,
such that forand$L^{q,\Phi q/}p(R^{n})$and$f_{f}$$dlb\in\Lambda_{\alpha}(R^{n})$$f\in L^{\mathrm{p},\Phi}(R^{n})\cap L_{c}^{\infty}(R^{n})$$||[b, T]f||q,\Phi q/\mathrm{p}\leq C||b||_{\ (}R^{n})||f||\mathrm{p},\Phi$.

51
This follows from the result (due to Naki [N]) of the boundedness of Riesz potential

on Morrey spaces and the pointwise estimate;.

$|([b, \tau]f)(x)|\leq\int_{R^{\hslash}}|b(_{X\}-}b(y\grave{)}||k(x-y)||f(y)|dy\leq C\dagger|b||_{\mathrm{A}}\circ(R^{n})I_{\alpha}(|f|)(x)$Further we obtain the following result
maximal operator$M_{\alpha/n}^{*}$$([\mathrm{M}\mathrm{i}\mathrm{z}_{2}])\mathrm{h}\mathrm{o}\mathrm{m}$the boundedness of the fractionalon Morrey spaces and the pointwise estimate due to$\mathrm{S}\mathrm{t}\mathrm{r}\acute{\acute{\mathrm{o}}}$mberg;$\{[b, I_{\alpha}](f)\}\#(x)\leq C||b||_{*}\{Mu(I\alpha f)(_{X})+(M^{*}|\alpha t/nf|t)^{1}/t(X)\}$

for almost all $x\in R^{n}$ , where $1<u,$ $t<p<n/\alpha$ .

Theorem
We assume that$4(\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}3.1.)$

Iaet $1<p<q<\infty,$ $0<\alpha=n(1/p-1/q)<n$ .

If $b\in BMO(R^{n})_{f}$ then the commutator
a bounded operator fiom
into
. Morethere exist8
a constant $C=C(p, q, \Phi)>0$ , independent of andthat for all $b\in BMO(R^{n})$$\Phi\in G_{n-\infty}$

and $\Phi^{q/p}\in G_{n}$ .
$IP^{\Phi},(R^{n})$$[b, I_{\alpha}]i_{\mathit{8}}$$L^{q,\Phi^{q/\mathrm{p}}}(Rn)$

$b$

and

$preci\mathit{8}ely_{f}$$f_{f}\mathit{8}uch$$f\in L^{\mathrm{p},\Phi}(R^{n})\cap L_{c}^{\infty}(R^{n})$(6)$||[b, I_{\alpha}]f||q,\Phi^{q}/\mathrm{p}\leq C||b||*||f||_{p},\Phi$

Similarly we can show the following

Theorem$5(\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}3.2.)$$([\mathrm{M}\mathrm{i}\mathrm{z}_{2}])$;

Let $1<p<q<\infty,$$n(1/p-1/q)<n,$ $1<p<n/(\alpha+\beta)$ . We assume thatthen the commutator

is a bounded operator from$[b, I_{\beta}]$.

$0<\alpha,$$\Phi\in G_{n-p(\alpha+}\beta$$IP^{\Phi},(R^{n})$intoprecisely, there exists a constanB $C=C(p, q, \Phi)>0$ , independent offor all(7)$b\in\Lambda_{\alpha}(R^{n})$and$\beta,$).

$0<\alpha+\beta=$

If$b\in\Lambda_{\alpha}(R^{n})_{f}$$L^{q,\Phi^{q/\mathrm{p}}}(Rn\rangle$

$b$

and

$f_{f}$. Moresuch that$f\in L^{\mathrm{p},\Phi}(R^{n})\cap L_{c}^{\infty}(R^{n})$$||[b, I_{\beta}]f||q,\Phi^{q}/\mathrm{p}\leq C||b||_{\Lambda}\alpha(R^{n})||f||_{p},\Phi$.

Our results (Theorems 1, 2 and 4) generalize partly the classical results due to Di
Fazio and Ragusa. Also we obtain the new results (Theorems 3 and 5).
$[\mathrm{D}\mathrm{i}\mathrm{F}\mathrm{R}\mathrm{a}\mathrm{g}]$Acknowledgements. We wish to thank Professor Hideo Kozono for his kindness toinvite me as a speaker of this Congress.

52
1. Introduction.

, that is, a positive and nonLet $\Phi=\Phi(x,r)$ , be a growth function on
satisfies
decreasing function with repect to $r>0$ . We say that the growth function
the -condition (or doubling condition) for $r>0$ if there exists constant $D=D(\Phi)\geq 1$ ,$R_{+}^{n+1}=R^{n}\mathrm{x}B_{\vdash}$$\Phi(x,r\rangle$$\Delta_{2}$independent of $(x, r)$ , such that
(1.1)$\Phi(x, 2r)\leq D\Phi(x,r)$,$(x,$ $r\rangle\in R_{+}^{n+1}$,

or equivalently,$\Phi(x, 2r)/D\leq\Phi(x,r)\leq\Phi(x, 2r)$

In this case, we write$\Phi\in\triangle_{2}$,$(x, r)\in R_{+}^{n+1}$.

simply. We consider the following functions in$\Phi(x, r)=\Psi(x)r^{\lambda}\{\log(1+r)\}^{\mu},$$\Psi(x)\in L^{\infty}(R^{n}),$$0\leq\lambda<\infty,$$\Delta_{2}$;$-\infty<\mu<\infty$.

Remark. Nakai [Nak] assumed a slightly weak condition on $\Phi(x,r)$ replacing (1.1) ;there exists a constant $C>0$ such that, for all $(x, r)\in R_{+}^{n+1}$ ,(1.2)$r\leq t\leq 2r\Rightarrow C^{-1}\leq\Phi(x, t)/\Phi(x, r)\leq C$.

However, for simplicity, we describe the results on the asumption of (1.1). Of course ourresults are also valid under the condition (1.2).

be the -dimensional Euclidean space and let $B=B(x, r)$
be the ball centered at $x\in R^{n}$ and with radius $r>0$ . Let $Q=Q(x, r)$ be the cube

Function Spaces. Let$R^{n}$$\mathrm{n}$centered at $x\in R^{n}$ and with sides of length $r>0$ , where the cube will always mean acompact cube with sides parallel to the axes and nonempty interior.

for the Lebesgue measures of ball$B$and cube$Q$$C$), We denote by $IP^{\Phi},=$$R^{n}$, for which there exists, independent of balls $B=B(x, r)_{f}$ such that

(1.3)

$I_{B(x},f)|f(y)|^{\mathrm{p}}dy\leq C^{p}\Phi(X,r)$for all balls $B=B(x, r)$ .
,$\mathit{8}paceIP^{\Phi}$$Let||f||_{\mathrm{p},\Phi}$

become8 a quasi-Banachbe the smallest constant $Csat\dot{w}$fying (1.3). Then the$\mathit{8}pace$with quasi-no7m[TYi]. In particular, if $1\leq p<\infty$ , then the space$||\cdot||_{p,\Phi}$$[\mathrm{M}\mathrm{i}\mathrm{z}_{1}]$the space of locally integrable functions $f$ , defined ona constantand $|Q|$ stand, respectively. Let $0<p<\infty$ .

Definition 1.1 (Morrey spaces). (Confer Mizuhara$IP^{\Phi}’(R^{n})$

$|B|$

,

$If^{\Phi}$$||\cdot||_{\mathrm{p},\Phi}$in the sense of Triebel

becomes a Banach$\mathit{8}pace$. The bdls $B=B(x, r)$ in (1.3) can be replaced by cubes $Q=Q(x, r)$ .

with norm

53, then , is the classical Morrey space denoted by ,simply. The classical Morrey spaces$0<\lambda<n$ , were originally introduced byMorrey [Mor] in 1938 and used by himself and the others in the problems related to thecalculus of variations and the theory of elliptic PDE’s. We refer to Campanato [Cam],
Giaguinta [Gia],
and Peetre.
,
The some properties ofare known ; If $1\leq p<\infty$ , then
and
When

$\Phi(x, r)=r^{\lambda},$$L^{\mathrm{P}^{t^{\lambda}}}$$\lambda\geq 0$$IP^{\lambda}$$IP^{\lambda},,$$\mathrm{K}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{r}-^{\mathrm{j}}\mathrm{o}\mathrm{h}\mathrm{n}-\mathrm{m}_{\check{C}}\mathrm{i}\mathrm{k}[\mathrm{K}\mathrm{u}\mathrm{f}\mathrm{J}o\mathrm{h}\mathrm{m}\mathrm{c}]$$[\mathrm{P}_{2}]$$IP^{\lambda}$$L^{p,0}=L^{\mathrm{p}}(R^{n})$. If$IP^{n},=L^{\infty}(R^{n})i_{\mathit{8}}\circ met\dot{n}Cdly$

then$IP^{\lambda},doe\mathit{8}$$n<\lambda$

not indutlle nonzero, then$IP^{\lambda},=\{0\}$. If$1\leq p<\infty$

and $0<\lambda<n_{f}$. Hence, in the classical Morrey spaces,$conStant_{\mathit{8}}$$L^{p,\lambda}$

for $0<\lambda<n$ is interesting. Also H\"older’s inequality implies the imbedding theorem; if($n-\lambda\rangle/q=(n-\mu)/p,$ $p\leq q$ , then$L^{q,\lambda}\subset L^{p,\mu}$.

Let $BMO(R^{n}\rangle$ be the John-Nirenberg space of all functions of bounded mean$\mathrm{o}\mathrm{s}\mathrm{c}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}_{r^{-}}$

tion (see John-Nirenberg [JoN]), that is, $BMO(R^{n})$ is a Banach space, modulo constants,
with norm$||\cdot||_{*}$

defined by$||b||_{*}= \sup_{B}|B|^{-1}\int_{B}|b(y)-b_{B}|dy$, where$b_{B}=|B|^{-}1I_{B}^{b}(y)dy$

The space $BMO(R^{n})$ is identified with the dual space of the Hardy spacesense of Fefferman-Stein
Let

a on

$H^{p}(R^{n})$$H^{1}(R^{n})$in the.

$0<\alpha<n$ , be the space of all Lipschitz continuousfunctions of order. The spaceis homogeneous in the sense of dilations. The dual space of
can be identified with the Lipschitz space$\alpha=n(1/p-1)$ .

$\Lambda_{\alpha}(ffl),$$R^{n}$$([\mathrm{F}\mathrm{e}\mathrm{S}_{2}])$.

$\Lambda_{\alpha}(R^{n})$$\Lambda_{\alpha}(R^{n}),$Classical operators. Let be a locally integrable function on
Littlewood maximal operator $M$ is defined by$f$$R^{n}$. The Hardy-$Mf(x)= \sup|x\in BB|^{-1}\int_{B}|f(y)|dy$

where the supremum is taken over all balls

ball

$B$

$B_{\mathrm{C}\mathrm{O}}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{i}\dot{\mathrm{m}}\mathrm{n}\mathrm{g}x$and $|B|$ is the volume of the. We introduce two variants of $M$ . Let $0<q<\infty$ and$M_{q}f(x)=\{(M|f|^{q})(X)\}1/q$.

Then H\"older’s inequality shows that $Mf=M_{1}f\leq M_{q}f$ if $1\leq q<\infty$ and $M_{q}f\leq$$M_{1}f=Mf$ if $0<q\leq 1$ . The sharp maximal function $f^{\#}(x)$ is defined by$f^{\#}(x)= \sup_{\in xB}|B|^{-1}\int_{B}|f(y)-f_{B}|dy$, where$f_{B}=|B|^{-1} \int_{B}f(y)dy$

.

54
Let
kemel

$T$

be a Calderon-Zygmund singular integral operator $Tf=k*f$ defined by the$k\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{p}_{\mathrm{i}}\mathrm{n}\mathrm{g}$the conditions;,$|k(x)|\leq C|x|^{-}n$for $0\neq x\in R^{n}$ ,$|k(x)-k(x-y)|\leq C|y|/|x|^{n+1}$

for $|y|\leq|x|/2$ .

$||\hat{k}||_{\infty}\leq C$For$\epsilon>0$, put$T_{\epsilon}f(x)= \int_{|y|>\epsilon}k(y)f(x-y)dy$

Let
order

$\alpha$$I_{\alpha},$$0<\alpha<n$and$T^{*}f(x)= \sup_{\epsilon>0}|T_{\epsilon}f(x)[$.

, be the fractional integral operator (or Riesz potential operator) ofdefined by$(I_{\alpha}f)(_{X})= \int_{R^{n}}\frac{f(y)}{|x-y|^{n-\alpha}}dy$, the fractional maximal function
for a suitable function . Related to, is defined byaswhich appeared in$f$$I_{\alpha}f$$M_{\alpha/n}^{*}f(\mathcal{I})$,$f_{\alpha,1}^{*}(X)$$[\mathrm{M}\mathrm{u}\mathrm{c}\mathrm{W}\mathrm{h}\mathrm{e}]$$M_{\alpha/n}^{*}f(X)=f \alpha*,1(x)=\sup_{Qx\in}\frac{1}{|Q|^{1-\alpha}/n}\int_{Q}|f(y)|dy$

We define the commutator $[b, S]$ between an operator
by a locally integrable function , by $[b, S]=bS-Sb$ .

$S$.

and the multiplication operator$b$

In this note we show the boundednesss of the commutator [or$b\in\Lambda_{\alpha}(R^{n})$, on Morrey spaces$IP^{\Phi},(R^{n})$$b,$ $\eta$, for $b\in BMO(R^{n})$with some growth function$\Phi$. Our results(Theorems 2.1, 2.2, 3.1) generalize partly the recent results due to Di Fazio and Ragusa$0<\lambda<n,$ $1<p<\infty$ . Further
on the classical Morrey spaces
we obtain the new results (Theorems 2.3 and 3.2). The letterswill denote positive
constants, which may have different values in each line.
$IP^{\lambda},(R^{n}),$$[\mathrm{D}\mathrm{i}\mathrm{F}\mathrm{R}\mathrm{a}\mathrm{g}]$$C’ \mathrm{s}$2. Commutators between Calderon-Zygmund singular integral operatorsand multiplication operator by a function $b\in BMO(P)\cup\Lambda_{\alpha}(R^{n})$ .
$G_{a}$-condition. We consider the following condition on growth function $\Phi(x,r)$ ;$\frac{\Phi(x,t)}{t^{a}}\in L^{1}([r, \infty),dt/t)$

for all $r>0$ and any $x\in R^{n}$ , and, in addition, there exists a constant $C$ , independent of$(x, r)\in R_{+}^{n+1}$(2.1), such that$\int_{f}^{\infty}[\Phi(x, t)/t\circ+1]dt\leq C\Phi(x,r)/r^{a}$,$(x,r)\in R_{+}^{n+1}$,

55
for some $a>0$ . In this case, we write$\Phi\in G_{a}$We can observe the following property ofLemma 2.1. (i) If $0<a<a’<n$ , then(\"u) If$\Phi\in\triangle_{2}$simply.

$G_{a},$$a>0$ ;

$G_{a}\subset G_{a}’\subset G_{n}\subset\Delta_{2}$with doubling constant $D,$ $1\leq D<2^{n}$ , then(iii) $Ifa>0$ , then$G_{a}\subset G_{a\gamma}$

for some

there exist constants $\gamma=\gamma(C, a),$$\gamma,$$0<\gamma<1$$0<\gamma<1_{\mathrm{z}}$.

$\Phi\in G_{n}$.

. More precisely, if$\Phi\in G_{a},$$a>0_{f}$and $C’=C^{\text{ノ}}(c_{a,\gamma})-,>0$ such that for any$(x,r)\in R_{+}n+1$(2.2)$\int_{r}^{\infty}[\Phi(X, t)/t^{a}\gamma+1]dt\leq C’\Phi(x,r)/ra\gamma$.

Proof. (i), (ii) These are easy to see.
(iii) Let$\Phi_{a}(x, r)=\int_{f}^{\infty}[\Phi(x, t)/t^{a+}1]dt$.

Then (2.1) implies$\Phi_{a}(X, r)\leq C\Phi(X, r)/r^{a}$.

For $0<r<R$ , we have, integrating by parts and using (2.1),$\int_{f}^{R}[\Phi(X, t)/ta\gamma+1]dt=l^{[(X}R]\Phi,t)/t^{a}+1t^{a}(\gamma-1)dt$$=[- \Phi_{a}(x, t)ta(1-\gamma)]^{R}t-\int_{\tau}^{R}[-\Phi a(x,t)a(1-\gamma)t^{a}(1-\gamma)-1]dt$$=- \Phi_{a}(x, R)\mathrm{f}\mathrm{f}^{(\gamma)}1-’\Phi_{a}(+x, r)ra(1\neg)+a(1-\gamma)\int_{r}^{R}[\Phi_{a}(x,t)t^{a(\gamma}-)-1]1dt$$\leq C\Phi(x, r)/r^{a}\gamma+a(1-\gamma)c\int^{R}f][\Phi(X, t)/t^{a+1}\prime \mathrm{v}dt$.

Hence we obtain,$\int_{f}^{R}[\Phi(x, t)/t^{a\gamma+}1]dt\leq\frac{C}{1-a(1-\gamma)C}\Phi(x, r)/r^{a’}\mathrm{Y}$and we have (2. 2) with$\cdot$$C’= \frac{C}{1-a(1-\gamma)C}>0$

Thus we have (2.2) for some$\gamma$.

such that $1-(1/aC)<\gamma<1$ .

Q.E.D.
First using this Lemma, we show the following;

56
Theorem 2.1. Iaet $0<p<\infty$ . We assume ffiat$C>0_{f}$. Then there enists a constant

independent of , such that
$f$

(2.3)

$||Mf||_{p,\Phi}\leq C||f^{\#}||p,\Phi$, wherefor all$f\in L^{p,\Phi}\cap L_{\mathrm{c}}^{\infty}(R^{n})$onwith compact support.

$R^{n}$$\Phi\in G_{n}$$L_{c}^{\infty}(P)\dot{w}$theof all$\mathit{8}et$

Proof. We use the method due to Di Fazio-Ragusa
version of the maximal inequality due to Fefferman-Stein$es\mathit{8}entidly$$[\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{R}\mathrm{a}\mathrm{g}]$$C$$[\mathrm{F}\mathrm{S}_{2}]$bounded functions. We recall the weighted; there exists a constantsuch that(2.4)

for all$\int_{R^{n}}\{Mf(X\rangle\}p(wX)dX\leq C\int_{R^{n}}\{f^{\#}(X)\}^{p}w(x)dx$$w\in A_{\infty}$

and all $f\in IP_{w}(R^{n})$ , for $0<p<\infty$ (see Garcia-Cuerva-Rubio de Francia, is the Muckenhoupt class of weight functions.
and a ball. We take $w(x)$ as $(M\chi)^{\gamma}\in A_{1)}0<\gamma<1$ , whereLet$\chi=\chi_{B}(x)$ is the characteristic function of the ball $B=B(x_{0},r)$ .
[; p.410] where$\mathrm{G}\mathrm{a}\mathrm{r}\mathrm{R}\mathrm{u}\mathrm{b}$$)$$A_{q},$$f\in L^{\mathrm{p},\Phi}\cap L_{\mathrm{c}}^{\infty}(R^{n})$$1\leq q\leq\infty$$B$

Then we get by (2.4),$\int_{B}\{Mf(_{X})\}^{p}d_{\mathcal{I}}=\int_{R^{n}}\{Mf(x)\}^{\mathrm{P}}xB(_{X)}d_{X}$$\leq\int_{R^{n}}\{Mf(X)\}p\{M\chi_{B}(x)\}’ldx\leq C\int_{R^{n}}\{f\#(x)\}\mathrm{p}\{MxB(x\mathrm{o},r)(x)\}^{\gamma}d_{X}$$=C \int_{B(x_{0,r}})X\{f\#()\}^{\mathrm{P}}\{MxB(x0,r)(x)\}^{\gamma}dx$$+C \sum_{k=1}^{\infty}\int_{B}(x_{0},2kr)-B(x\mathrm{o},2k-1\mathrm{r})’\}^{\gamma}\{f^{\#}(X)\}^{\mathrm{P}}\{M\chi B(x0^{\prime)}(x)dx$$\leq C\{\int_{B(x0r)}\{f\#(x)\}^{p}dx+\sum(2-kn\gamma)\int_{B(2^{k}r}x\mathrm{o},)\#\{f(X)\}^{\mathrm{p}}k\infty=1dX\}$$\leq C||f^{\#}||^{p}p,\Phi\{\Phi(x_{0},r)+\sum_{k=1}^{\infty}(2-k)^{n}\gamma\Phi(x0,2kr)\}$$\leq C||f^{\#}||^{p}p,\Phi\sum^{\infty}\frac{\Phi(_{X_{0)}2r}\mathrm{k})}{2^{kn\prime}\gamma}k=0\sim c||f^{\#}||pn\gamma\int_{r}p,\Phi\frac{\Phi(x_{0},t)}{t^{n\gamma+1}}rd\infty t$

Since, by Lemma 2.1, the last term is bounded by$C||f^{\#}||_{p}p,(\Phi)\Phi x0,$
$r$

,

.

57
we have$||Mf||_{p,\Phi}\leq C||f\#||p,\Phi$Thus we have (2.3) for some $C>0$ , independent of.

$f\in L^{p,\Phi}\cap L_{\mathrm{C}}^{\infty}(R^{n}\rangle$.
Q.E.D.

Our second aim is to show the following;
Theorem 2.2. Let $1<p<\infty,$ $b\in BMO(R^{n})$ and$T$

be a Calderon-Zygmund singular

integral operator. We assume that $\Phi\in G_{n}$ . Then the commutator $[b,T]$ is bounded in. More recisely, there exists constant independent of and $f,$that$IP^{\Phi},$

$p$

$b$

$C_{f}$(2.5)$\mathit{8}uch$$||[b, T](f)||p,\Phi\leq C||b||_{*}||f||p,\Phi$for all $b\in BMO(R^{n})$ and$f\in L^{p,\Phi}\cap L_{c}^{\infty}(R^{n})$.

To prove the theorem we need Theorem 2.1 and the following three lemmas ;Lemma 2.2. Iaet $1<q,$ $s<\infty,$ $b\in BMO(R^{n})$ andsingular integral operator.

$\mathfrak{M}en$there$exi\mathit{8}tS$

constant$C$$T$

be a Calderon-Zygmund

independent of and
$b$

$f$

such that$\{[b, T](f)\}^{\#}(X)\leq C||b||_{*}\{Mq(\tau f)(x)+(Msf)(x)\}$for almost all$x\in R^{n}$and all $f\in L_{c}^{\infty}(P)$Proof. This is the pointwise estimate due to[Jan; pp.268-269.] .

(see [$\mathrm{S}\mathrm{t}\mathrm{r}\acute{\acute{\mathrm{o}}}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}$$\mathrm{T}\mathrm{o}\mathrm{r}$, p.418.] and Janson$)$

Q.E.D.

Lemma 2.3. Let $0<q<p<\infty$ . We assume that
opemtoris a bounded opemtor inand$M_{q}$$\Phi\in G_{n}$. Then the mmimal$L^{\mathrm{p},\Phi}(R^{n})$$||M_{q}f||_{p,\Phi}\leq C||f||_{p,\Phi}$

for some constant$C$

independent of $f\in L^{p,\Phi}(R^{n})$ .

Proof. The proof depends on the weighted maximal inequality due to Fefferman-Stein$[\mathrm{F}\mathrm{e}\mathrm{f}\mathrm{S}\mathrm{t}\mathrm{e}_{1}]$. In the restricted case $1\leq q<p<\infty$ , the corresponding result is proved by

Nakai [Nak; Theorem 1]. It is not difficult to extend the result to the case $0<q<p<\infty$ .

Confer ako Chiarenza-Frasca$[\mathrm{C}\mathrm{h}\mathrm{i}\mathrm{F}\mathrm{r}\mathrm{a}]$and Mizuhara$[\mathrm{M}\mathrm{i}\mathrm{z}_{1}]$.

58
Q.E.D.
Lemma 2.4. Let $1<p<\infty$ . We assume thatsingular integml opemtor is a bounded opemtor in$\Phi\in G_{n}$

$T$

(2.6)

. Then the Calderon-Zygmund$L^{\mathrm{p},\Phi}(ffl)$

and$||Tf||_{p,\Phi}\leq c||f||p,\Phi$for some constant$C$

independent of$f\in L^{\mathrm{p},\Phi}(R^{n})$.

Proof. This is the result due to Nakai [Nak; Theorem 2] in the setting of more general, Chiarenza-Frasca
and Mizuhara
growth functions. Confer also Peetre$[\mathrm{C}\mathrm{h}\mathrm{i}\mathrm{F}\mathrm{r}\mathrm{a}]$$[\mathrm{P}\mathrm{e}\mathrm{e}_{1}]$$[\mathrm{M}\mathrm{i}\mathrm{z}_{1}]$.

We note that we can give a short proof following the method of the author
which depends on the weighted maximal inequality due to Cordoba-Fefferman$[\mathrm{M}\mathrm{i}\mathrm{z}_{1}]$$[\mathrm{c}_{\mathrm{o}\mathrm{r}}\mathrm{F}\mathrm{e}\mathrm{f}]$(see also$[\mathrm{G}\mathrm{a}\mathrm{r}\mathrm{R}\mathrm{u}\mathrm{b}]$) ;there exists constant $C$, depending only on $T,p$ and $0<\gamma<1$ , such that

(2.7)

for all$\int_{R^{n}}|Tf(x)|^{p}\phi(x)dX\leq C\int_{R^{n}}|f(x)|^{\mathrm{P}}(M\phi)\gamma(X)d_{X}$$f$

anf $\phi(x)\geq 0$ . A standard proof using (2.7) implies (2.6).

Q.E.D.
to our
Proof of Theorem 2.2. We apply the method of Di Fazio-Ragusa
case. We suppose that $b\in BMO(R^{n})$ . Then Theorem 2.1 and Lemma 2.2 imply that,$[\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{R}\mathrm{a}\mathrm{g}]$for $1<q,$ $s<p<\infty$ ,$||[b, T](f)||\mathrm{P},\Phi\leq||M\{[b, T](f)\}||_{p,\Phi}$.

$\leq C||\{([b, \tau](f)\}^{\#}||p,\Phi\leq C||b||_{*}\{||M_{q}(\tau f)||_{p,\Phi}+||M_{\epsilon}f||_{p,\Phi}\}$Since, Lemma 2.3 and Lemma 2.4 imply$||M_{q}(Tf)||\mathrm{P}^{\Phi},\leq C||\tau f||_{p,\Phi}\leq C||f||_{p,\Phi}$and$||M_{s}f||_{p,\Phi}\leq C||f||_{p,\Phi}$,

we obtain$||[b,T](f)||p,\Phi\leq c||b||_{*}||f||_{p},\Phi$for $b\in BMO(p)$ and$f\in L^{\mathrm{p},\Phi}\cap L_{c}^{\infty}(R^{n})$. Thus we have (2.5).
Q.E.D.

When$b\in\Lambda_{\alpha}(R^{n}),$$0<\alpha<n$, we obtain the following. :59
Theorem 2.3. Let $1<p<q<\infty,$ $0<\alpha=n(1/p-1/q)<n$ . We assume
that
. If
, then the commutator $[b, T]\dot{w}$ a unded operator $fmm$
intoand$\Phi\in G_{\mathrm{n}-p\alpha}$$IP^{\Phi},(R^{n})$$\cdot$$b\in\Lambda_{\alpha}(R^{n})$$bo$$L^{q,\Phi^{q/}\mathrm{p}}(R^{n})$(2.8)$||[b, \tau]f||q,\Phi q/\mathrm{p}\leq C||b||_{\Lambda(}\alpha R^{n})||f||_{p,\Phi}$

for some constant$C$

independent of$b\in\Lambda_{\alpha}(R^{n})$and$f\in L^{p,\Phi}\cap L_{c}^{\infty}(R^{n})$To prove the theorem we need the following lenuna.

$,\sim$Lemma 2.5. Let $1<p<q<\infty,$ $0<\alpha=n\langle 1/p-1/q$) $<n$ . We assume that
. Then the fmctional integml opemtor is a bounded opemtor from
into
and

$\Phi\in G_{n-\mu\chi}$$IP^{\Phi},(R^{n})$$I_{\alpha}$$L^{q,\Phi^{q/\mathrm{p}}}(R^{n})$

(2.9)

for

$\mathit{8}ome$$||I_{\alpha}f||q,\Phi q/\mathrm{p}\leq c||f||_{p,\Phi}$

constant $C$ independent of$f\in L^{p,\Phi}$.

Proof. This is the result due to Nakai [Nak; Theorem 3].
Q.E.D.
Proof of Theoren 2.3. Let$b\in\Lambda_{\alpha}(R^{n})$. Then$|([b, T]f)(X)| \leq\int_{R^{\hslash}}|b(x)-b(y)||k(X-y)||f(y)|dy$$\leq C||b||\mathrm{A}_{\alpha}\int_{R^{\hslash}}|x-y|^{\alpha}|x-y|^{-}n|f(y)|dy=C||b||_{\Lambda*}I_{\alpha}(|f|)(x)$.

Hence we have, by Lemma 2.5,$||[b,$ $\mathrm{r}f||q,\Phi \mathrm{p}/q\leq c||b||\Lambda\alpha||I\alpha(|f|)||_{q},\Phi^{p/}q\leq C||b||_{\mathrm{A}_{\alpha}}||f|\dagger_{\mathrm{P}^{\Phi}},\cdot$

Thus we have (2.8) for some $C>0$ , independent of$b\in\Lambda_{\alpha}(R^{n})$and$f\in L^{\mathrm{p},\Phi}\cap L_{c}^{\infty}\langle ffl)$.

Q.E.D.

3. Commutators between the fractional integral operator and multiplication operator by a function.
$b\in BMO(ffl\rangle\cup\Lambda_{\alpha}(R^{n})$In this section first we show the following when $b\in BMO(R^{n})$ ;60
Theorem 3.1. Let $1<p<q<\infty,$ $0<\alpha=n(1/p-1/q)<n$ . We assume thatis a bounded. If $b\in BMO(R^{n})$ , then the commutator
and
and
into
operator fiom$\Phi^{q/p}\in G_{n}$$\Phi\in G_{n-\mathrm{p}\alpha}$$[b, I_{\alpha}]$$L^{q,\Phi^{q/}}p(R^{n})$$IP^{\Phi},(R^{n})$(3.1)$||[b, I_{\alpha}]f||_{q},\Phi^{q}/p\leq C||b||_{*}||f||_{p,\Phi}$

for some constant$C$

inde.pendent of $b\in BMO(ffl)$ and$f\in L^{p,\Phi}\cap L_{c}^{\infty}(R^{n})$.

To prove the theorem, we need the following pointwise estimate and the bounbednessof the fractional maximal operator ;Lemma 3.1. Let $0<\alpha<n,$ $1<u,$ $t<n/\alpha$ and $b\in BMO(R^{n})$ . Then there exists
constant independent of and such that
$C$

$f$

$b$

$\{[b, I_{\alpha}](f)\}\#(x)\leq C||b||*\{Mu(If\alpha)(x)+(M_{\alpha t/}^{*}|nf|^{t})1/t(x)\}$for almost all$x\in R^{n}$

and $dlf\in L_{c}^{\infty}(R^{n})$Proof. This is the pointwise estimate due to Str\’o’mberg (see [p.326, Lemma 2.] .
Fazio-Ragusa [$\mathrm{T}\mathrm{o}\mathrm{r}$, p.419.] and Di$)$$\mathrm{D}\mathrm{i}\mathbb{B}\mathrm{a}\mathrm{g},$Q.E.D.

Lemma 3.2. Let 1 $<p<q<\infty,$ $0<\alpha=n(1/p-1/q)<n$ . We assume. Then the fmctiond maximal operator
is a bounded opemtor $fmm$
that
and
into
$\Phi\in G_{n-\mu_{t}}$$M_{\alpha/n}^{*}$$L^{q,\Phi^{q/p}}(Rn)$$IP^{\Phi},(R^{n})$

(3.2)

for

$||M_{\alpha/}^{*}fn||q,\Phi^{q/p}\leq C||f||_{p,\Phi}$$\mathit{8}ome$

constant$C$

independent of$f\in L^{p,\Phi}$.

Proof. Let $B(z, r)$ be any ball centered at$B(z,r)$$z$

and with radius $r>0$ such that $x\in$. Since$I_{\alpha}(|f|)(x):= \int_{R}h\frac{|f(y)|}{|x-y|^{n-\alpha}}dy\geq\int_{B(z},r)\frac{|f(y)|}{|x-y|^{n-\alpha}}dy$

I $f(y)dy$ ,$\geq\frac{C’}{r^{n-\alpha}}\int_{B(z,f)}|f(y)|dy\simeq\frac{C^{\prime/}}{|B|^{1-}\alpha/n}I_{B(}z,\mathrm{r})|$we have the pointwise estimate ;$\langle M_{\alpha/n}^{*}f)(x)\leq I_{\alpha}(|f|\rangle(x)$61
for almost all $x\in R^{n}$ and all$f\in L^{\mathrm{p},\Phi}$. Hence Lemma 2.5 implies the result.
Q.E.D.

Proof of Theorem 3.1. Let $b\in BMO(R^{n})$ . Then Theorem 2.1 and Lemma 3.1
imply that, for $1<u,$ $t<q<p<\infty$ ,$||[b, I_{\alpha}](f)||q,\Phi^{q}/p\leq||M\{[b, I]\alpha(f)\}||_{q},\Phi^{q}/\mathrm{p}\leq||\{([b, I]\alpha(f)\}^{\#}||q,\Phi q/\mathrm{p}$$\leq C||b||_{*}\{||Mu(I\alpha f)||q,\Phi q/p+||(M^{*}|\alpha l/nf|t)^{1}/t|[q,\Phi^{q}/\mathrm{p}\}$Also, under the assumption on$\Phi$.

, Lemma 2.3 and Lemma 2.5 imply$||M_{u}(I\alpha f)||q,\Phi^{q}/\mathrm{p}\leq C||I\alpha f||q,\Phi q/p\leq c||f||p,\Phi$and Lemma 3.2 imply$||(M_{\alpha t/}^{*}|nf|t)^{1}/t||q,\Phi^{q}/\mathrm{p}|=|M^{*}|\alpha t/nf|^{t}||^{1/t}q/t,\Phi q/p$$\leq C||(|f|^{t})||_{p}^{1}/_{t,\Phi}t=/C||f||p,\Phi$.

Hence we obtain$||[b, I_{\alpha}](f)||q,\Phi^{q}/p\leq C||b||_{*}||f||_{p,\Phi}$

for $b\in BMO(R^{n})$ and$f\in L^{p,\Phi}\cap L_{c}^{\infty}(P)$. Thus we have (3.1).
Q.E.D.

We close this section showing the following;

Theorem 3.2. Let $1<p<q<\infty,$. We assume that$n/(\alpha+\beta)$$0<\alpha,\beta,$$\Phi\in G_{n-p(\alpha+\beta}$

a bounded operator from
(3.3)

$L^{p,\Phi}(R^{n})$

into

).

If

$0<\alpha+\beta=n(1/p-1/q)<n,$ $1<p<$$b\in\Lambda_{\alpha}(R^{n})$$L^{q,\Phi^{q/p}}(Rn)$, then the commutator$[b, I_{\beta}]$

is

and

$||[b, I_{\beta}]f||q,\Phi^{q}/p\leq C||b||_{\mathrm{A}}\alpha||f||_{p,\Phi}$

for some constant$C$

independent of

Proof of Theorem 3.2. Let$b\in\Lambda_{\alpha}(R^{n})$$b\in\Lambda_{\alpha}(R^{n})$and$f\in L^{p,\Phi}\cap L_{c}^{\infty}(R^{n})$.

. Then$|([b, I_{\beta}]f)(x)|\leq C||b||\mathrm{A}\alpha I_{\alpha}+\beta(|f|)(x)$.

for almost all $x\in R^{n}$ . Hence we have, by Lemma 2.5,.

$||[b,I_{\beta}|f||q,\Phi q/p\leq C||b1|\Lambda_{\alpha}||I_{\alpha+\beta(1}f|)||_{q},\Phi q/\mathrm{p}\leq c_{||\mathrm{t}|_{\Lambda_{q}}}b\mathrm{I}\mathrm{I}f|\mathrm{I}p,\Phi$Thus we have (3.3) for some $C>0$ , independent of$b\in\Lambda_{a}(ffl)$

and$f\in L^{\mathrm{p},\Phi}\cap L_{c}^{\infty}(R^{n})$.

Q.E.D.

62
References.
[Caml Campanato,S. Proprieta di una famiglia di spazi funzionali, Ann. Scuola Norm.

Sup. Pisa, 18(1964), 137-160.
[ChiFra] Chiarenza,F. and Frasca,M. Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat.(7), 7(1987), 273-279.

ICorFefl Cordoba,A. and Fefferman,C. A weighted norm inequality for singular integrals}
Studia Math. 57(1976), 97-101.
[Difflag] Di Fazio,G. and Ragusa,M.A. Commutators and Morrey spaces, Boll.

$\mathrm{U}.\mathrm{M}.\mathrm{I}.(7)$,5-A(1991), 323-332.
$[\mathrm{F}\mathrm{e}\mathrm{f}\mathrm{s}\mathrm{t}\mathrm{e}_{1}]$Fefferman,C. and Stein,E.M. Some maximal inequalities, Amer. J. Math. 93(1971),

107-115.
$[\mathrm{F}\mathrm{e}\mathrm{f}\mathrm{S}\mathrm{t}\mathrm{e}_{2}]$Fefferman,C. and Stein,E.M.

$H^{p}$

spaces of several variables, Acta Math. 129(1972),

127-193.
[GarRub] Garcia-Cuerva,J. and Rubio de Francia,J.L. Weighted norm inequalities andrelated topics, North-Holland, 1985.
[Gia] Giaquinta,M. Multiple integrals in the calculus of variations and nonlinear elliptic

systems, Annals of Mathematics Studies, No.105, Princeton Univ. Press, Princeton, New
Jersey, 1983.
[Jan] Janson,S. Mean oscillation and commutators of singular integral operators, Ark.

Mat. 16(1978), 263-270. [KufJohKC] Kufner,A., John,O. &Fu\v{c}ik,S. Function spaces,

Academia, Praha, 1977.
$[\mathrm{M}\mathrm{i}\mathrm{Z}_{1}]$Mizuhara,T. Boundedness of some classical operators on generalized Morrey spaces,

Harmonic Analysis (S.Igari, editor), ICM-90 Satellite Proceedings, Springer-Verlag, Tokyo(1991), 183-189.
[$\mathrm{M}\mathrm{i}\mathrm{z}_{2}1$

Mizuhara,T. Commutators of singular integral operators on Morrey spaces withgeneral growth functions, Preprint. (Also Confer: 第 34 回実函数論函数解析学合同シンポジウム講演集録 1995, p6tkp77)[Nak] Nakai,E. Hardy-Littlewood maximal operator, singular integral operators and theRiesz potentials on generalized Morrey spaces, Math. Nachr. 166(1994), 95103.
$[\mathrm{P}\mathrm{e}\mathrm{e}_{1}]$Peetre,J. On convolution operators leaving$L^{\mathrm{p},\lambda}$

spaces invariant, Ann. Mat. Pura

App1.(4), 72(1966), 295-304.
$[\mathrm{P}\mathrm{e}\mathrm{e}_{2}]$Peetre,J. On the theory of$L_{\mathrm{p},\lambda}$

spaces, J. Funct. Anal. 4(1969), 71-87.

[Ste] Stein,E.S. Singular integrals and differentiability properties of functions, Princeton

63
University Press, Princeton, New Jersey, 1970.
[Tor] Torchinsky,A. Real-Variable Methods in Harmonic Analysis, Academic Press, Inc.
1986.
[Tri] ’biebel,H. Theory of Function Spaces, Monographs in Math.Vol.78,, 1983.
$\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{k}\mathrm{h}\acute{\acute{\mathrm{a}}}\mathrm{u}\mathrm{S}\mathrm{e}\mathrm{r}$TAKAHIRO MIZUHARA:
Department of Mathematical Sciences,
Faculty of Science, Yamagata University,
Yamagata 990-8560, Japan.