数理解析研究所講究録 1399 巻 2004 年 144-156

144

Orlicz-Morrey spaces and some integral operators
中井英一 (Eiichi Nakai)

大阪教育大学教育学部

Department of Mathematics
Osaka Kyoiku University
enakai@cc.osaka-kyoiku.ac.jp

1. INTRODUCTION
For the Hardy-Littlewood maximal operator $M$ , Calder\’on-Zygmund operator
and fractional integral operator Ia, $0<\alpha<n,$ it is well known that

$I_{\alpha}$

:

$M$

:

$L^{1}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{1}(\mathbb{R}^{n})$$T$:

L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{1}($$L^{1}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{1}(\mathbb{R}^{n})$

,

$M$

’ ,
$n)$

$L^{1}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{n/(n-\alpha)}(\mathbb{R}^{n})$,:$L^{p}(\mathbb{R}^{n})arrow L^{p}(\mathbb{R}^{n})$$T:L^{\mathrm{p}}(\mathbb{R}^{n})arrow L^{p}(\mathbb{R}^{n})$$I_{\alpha}$:$L^{p}(\mathbb{R}^{n})$$L^{p}(\mathbb{R}^{n})arrow$arrow L^{q}(\mathbb{R}^{n})$L^{q}(\mathbb{R}^{n})$,,$1<p\leq\infty$$T$,$1<p<\infty$ ,$1<p<q<\infty$ ,,$\{$$\{_{-n/p+\alpha=-n/q}1<p<q<\infty$$-n/p+\alpha=-n/q$ ,These boundedness extended to several function spaces which are generalizations of
ZAspaces, for example, Orlicz spaces, Morrey spaces, Lorentz spaces, Herz spaces,
etc.
, Kita [10] (1997) proved a necFor boundedness of $M$ on Orlicz spaces
essary and sufficient condition on 4 and for$L^{p}$$L^{\Phi}(\mathbb{R}^{n})$$\Phi$$M$:$\Psi$$L^{\Phi}(\mathbb{R}^{n})arrow
L^{\Psi}(\mathbb{R}^{n})$$L^{\varphi}(\mathbb{R}^{n})arrowL^{\mathrm{u}}\mathrm{r}(\mathbb{R}^{n})$.

Cianchi [3] (1999) proved necessary and sufficient conditions on (1) and I for$\Phi$, ,$M$ $T$

$M$

, ,
$T$

$I_{\alpha}$$I_{\alpha}$:$L^{\Phi}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Psi}(\mathbb{R}^{n})$:$L^{\Phi}(\mathbb{R}^{n})arrow L^{\Psi}(\mathbb{R}^{n})$$\Psi$,.

For boundedness of on Morrey spaces, see Peetre (Spanne) [18] (1969)
Adams [1], 1975 Chiarenza and Frasca [2] (1987), Nakai (1995). Chiarenza andFrasca showed$L^{p,\lambda}(\mathbb{R}^{n})$$IP^{\lambda}’(\mathbb{R}^{n})$$I_{\alpha}$$M,T:L^{p,\lambda}(\mathbb{R}^{n})arrow L^{p,\lambda}(\mathbb{R}^{n})$$I_{\alpha}$:$L^{p,\lambda}(\mathbb{R}^{n})arrow L^{q,\lambda}(\mathbb{R}^{n})$,.

The author studied boundedness of genaralized fractional integral operators on
Orlicz-Morrey spaces in [17]. Orlicz-Morrey spaces are useful to estimate genaralized fractional integral operators. In this paper we investigate boundedness and145

weak boundedness of the Hardy-Littlewood maximal operator, singular integtral
operators and generalized fractional integral operators. We refine on the results in[17].

2. DEFNITIONS
A function :is said to be almost increasing (almost decreasing) if there exists a constant $C>0$ such that$(0, +00)arrow(0, +\mathrm{o}\mathrm{o})$$\theta$$\theta(r)\leq C\theta(s)$for$(\theta(r)\geq C\theta(s))$$r\leq s.$

A function :is said to satisfy the doubling condition if there
exists a constant $C>0$ such that$\theta$$(0, +\mathrm{o}\mathrm{o})arrow$$(0, +\mathrm{o}\mathrm{o})$for$C^{-1} \leq\frac{\theta(r)}{\theta(s)}\leq c$$\frac{1}{2}\leq\frac{r}{s}\leq 2.$

For functions 0, :, we denotethere exists a constant $C>0$ such that$(0, +00)arrow(0, +\mathrm{o}\mathrm{o})$$\kappa$$C^{-1}/1/(r)$$\leq\kappa(r)\leq C\theta(r)$$\theta(r)\sim\kappa(r)(\mathrm{O}(\mathrm{r})\approx\kappa(r))$

for$(\theta(C^{-1}r)\leq\kappa(r)\leq\theta(Cr))$

if$r>0.$

Let be the set of all convex functions (I) :such that $\Phi(r)>0$
for $r>0$ . If)), thenis increasing, absolutely continuous and bijective.
For a measurable set, we denote the characteristic function of by
and the Lebsgue measure of 0 by
. For a measuralbe set, a measuralbefunction and $t>0,$ let$[0, +00)arrow[0, +\mathrm{o}\mathrm{o})$$\mathcal{Y}$$\Phi\in$$\Phi$$\Omega\subset \mathbb{R}^{n}$$\Omega$$\chi_{\Omega}$$|\Omega|$$\Omega\subset \mathbb{R}^{n}$$f$$m(\Omega, f, t)=|\{x\in\Omega :In the case$\Omega=\mathbb{R}^{n}$|f(x)|>t\}|$.

, we shortly denote it by $m(f, t)$ .

Definition 2.1 (Orlicz space). For$L^{\Phi}(\mathbb{R}^{n})=$$\Phi$$\in \mathcal{Y}$let$\{f\in L_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R}^{n}) :||f||_{L^{\Phi}}<+\mathrm{o}\mathrm{o}\}$,$||f||_{L^{\Phi}}= \inf\{\lambda>0:\int_{\mathrm{R}^{n}}\Phi(\frac{|f(x)|}{\lambda})dx\leq 1\}$

$L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Phi}(\mathbb{R}^{n})=\{f\in L_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R}^{n})$$||f11_{L\mathrm{Z}\epsilon \mathrm{U}}$$=$ $\inf$ $\{\lambda>0$

If $\Phi(r)=r^{p}$ , $1\leq p<\infty$ , thenIf $\Phi(r)\leq\Psi(Cr)$ , thenand:$||f||\mathrm{z}\mathrm{H}\mathrm{e}*\mathrm{k}$$\sup_{\mathrm{t}>0}\Phi(t)m(\frac{f}{\lambda},$$L^{\Phi}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n})$$L^{\Phi}(\mathbb{R}^{n})\supset L^{\Psi}(\mathbb{R}^{n})$$L^{\Phi}(\mathbb{R}^{n})=L^{\Psi}(\mathbb{R}^{n})$:$||f||_{L^{\Phi}}\sim||f||_{L}\mathrm{v}$

.

and

and

$<+\mathrm{o}\mathrm{o}\}$,,$t)\leq 1\}$y

$L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Phi}(\mathbb{R}^{n})=L^{p}$$||f||_{L^{\Phi}}\leq C||f||_{L^{\Psi}}$. If$\mathrm{e}\mathrm{a}\mathrm{k}($’ .
, then$\Phi\approx\Psi$

$n)$

14El

Let $B(a, r)$ be the ball$\{x\in \mathbb{R}^{n} :with center|x-a\mathit{9}|<r\}$Definition 2.2 (Morrey space). For $1\leq p<\infty$ and$L^{p,\lambda}(\mathbb{R}^{n})=$||f||L\mathrm{p},’ <+\mathrm{o}\mathrm{o}\}$

(

$dx$

$\frac{1}{r^{\lambda}}\int_{B}|f(x)|p$:$=\{f\in L_{1\mathrm{o}\mathrm{c}}^{p}(\mathbb{R}^{n})$and$U^{\lambda},(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n})$$L^{p,\lambda}(\mathbb{R}^{n})=L^{\infty}(\mathbb{R}^{n})$

let

,

),

$||f||L\mathrm{p}\mathrm{w}’ ex\mathrm{k}<+\infty\}$$||f||_{L_{\mathrm{w}\epsilon \mathrm{a}\mathrm{k}}^{\mathrm{p}.\lambda}}= \sup_{B=B(a,t)}\sup_{t>0}\frac{t^{p}m(B,f,t)}{r^{\lambda}}$If A $=0,$ thenand of radius $r>0.$$\leq n,$$\{f\in L_{1\mathrm{o}\mathrm{c}}^{p}(\mathbb{R}^{n}) :$||f||_{L^{\mathrm{p},\lambda}}= \sup_{B=B(a,r)}$$L\mathrm{w}\mathrm{e}_{\mathrm{a}\mathrm{k}}(’ n)$X$0\leq$$a$,.

$L_{\mathrm{w}\acute{\mathrm{e}}\mathrm{a}\mathrm{k}}^{p\lambda}(\mathbb{R}^{n})=L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{p}(\mathbb{R}^{n})$. If A $=n,$ then.

Let be the set of all functions :is almost increasing. If
decreasing andcondition.
$\phi$$\mathcal{G}$$\phi(r)r^{n}$Definition 2.3. For $1\leq p<\infty$ and a function$L^{(p,\phi)}(\mathbb{R}^{n})=$such that 6 is almost
then 6 satisfies doubling$(0, +00)arrow(0, +\mathrm{o}\mathrm{o})$$\{f\in L_{1\mathrm{o}\mathrm{c}}^{p}(\mathbb{R}^{n}) :$\phi\in \mathrm{C}\mathrm{i}$$\phi\in \mathcal{G}$,||f||_{L}(\mathrm{p},\phi)<+\mathrm{o}\mathrm{o}\}$$( \frac{1}{|B|\phi(r)}\int_{B}|f(x)|^{p}dx)^{1/p}$$||f||_{L^{(\mathrm{p},\phi)}}$,,$=\mathrm{s}B=\mathrm{u}_{(\mathrm{a},\mathrm{r})}^{\mathrm{P}}$$L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(p,\phi)}(\mathbb{R}^{n})=\{f\in L_{1\mathrm{o}\mathrm{c}}^{p}(\mathbb{R}^{n})$:$||f||_{L}$Pe&2$<+\mathrm{o}\mathrm{o}\}$$||f||_{L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(\mathrm{p},\phi)}}= \sup_{B=B(a,r)}\sup_{t>0}\frac{t^{p}m(B,f,t)}{|B|\phi(r)}$If $\phi(r)=r^{-n+}$ ’, $0\leq$ A $\leq n,$ then$L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{p,\lambda}(\mathbb{R}^{n})$.

$L^{(p,\phi)}(\mathbb{R}^{n})=L^{p,\lambda}(\mathbb{R}^{n})$and$L_{\mathrm{w}\acute{\mathrm{e}}\mathrm{a}\mathrm{k}}^{(p\phi)}(\mathbb{R}^{n})=$.

If $p\leq q,$ then$C\psi(r)$ , then$L^{(p,\phi)}(\mathbb{R}^{n})\supset L^{(q,\phi^{q/\mathrm{p}})}(\mathbb{R}^{n})$$L^{(p,\phi)}(\mathbb{R}^{n})\subset L^{(p,\psi)}(\mathbb{R}^{n})$and)), a function$L^{(\mathrm{p},\phi)}(\mathbb{R}^{n})=L^{(p,\psi)}(\mathbb{R}^{n})$For,$\Phi\in$$||f||_{L^{(\mathrm{p},\phi)}}$$\phi\in(\mathrm{i}$$|\mathrm{V}||$,,\phi ,B $=$ inf$||f||_{\Phi,\phi,B,\mathrm{w}\mathrm{e}}$$\mathrm{k}$

$=$

$\{$

$||f||_{L(\mathrm{p},\phi)}\leq||f||_{L^{(q}}$$||f||_{L(\mathrm{p},\phi)}$$\sim||f||_{L^{(\mathrm{p},\psi)}}$$\mathit{2}$$C^{-1/p}||f||_{L(\mathrm{p},\psi)}$,4q/p).

. If.

and a ball $B=B(a, r)$ , let

A $>0$ :

inf

and

and

$\frac{1}{|B|\phi(r)}\int_{B}\Phi(\frac{|f(x)|}{\lambda})dx\leq 1\}$$\{\lambda>0:\sup_{t>0}\frac{\Phi(t)m(B,f/\lambda,t)}{|B|\phi(r)}\leq 1\}$,

If$\phi(r)\leq$$6\sim\psi$, then

147

We note that$\sup_{t>0}\Phi(t)m(\Omega, f, t)=\sup_{t>0}tm(\Omega, f, \mathrm{I}^{-1}(t))=\sup_{t>0}tm(\Omega, \Phi(17|)$

Definition 2.4 (Orlicz-Morrey space). For$L^{(\Phi,\phi)}(\mathbb{R}^{n})=$)) and a function$\Phi\in$$\{f\in L_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R}^{n}) :$||f||_{L^{(\Phi,\phi)}}$|| 7 ||_{L}(\Phi,\phi)<+\mathrm{o}\mathrm{o}\}$$= \sup_{B}||f||_{\Phi}$$L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(\Phi,\phi)}(\mathbb{R}^{n})=\{f\in L_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R}^{n})$:$\Phi \mathrm{z}$$r^{-n}$$6\sim\psi$$\Psi$, then)$||f||L\mathrm{y},\mathrm{z}$$L^{(\Phi,\phi)}(\mathbb{R}^{n})=L^{(\Psi,\psi)}(\mathbb{R}^{n})$$L^{(\Phi,\phi)}(\mathbb{R}^{n})=L^{\infty}(\mathbb{R}^{n})$$\Phi\in$$\mathcal{Y}$$\Phi\in$$\mathcal{Y}$is said to satisfy the$\in \mathcal{Y}$$\Phi^{+}(r)$)$\Phi^{+}\in$$\Phi^{+}\approx\Phi$,$\nabla_{2}$with-condition, denoted

,

and

a

$\geq 0,$let$\Phi\in \mathcal{Y}$$r(\log(1/r))^{-\epsilon-1}$$\mathcal{Y}+$a,$r(\log(1/r))^{-\epsilon}$$r(\log r)^{\delta+1}$$\in 7_{2}$, if$\in J\mathit{1}^{+}$, let$\mathrm{I}$$r\geq 0.$$\nabla_{2}$-condition, then $\Phi\in y+$withfor small $r>0,$
for large $r>0.$$\mathrm{d}$$\Phi^{+}(r)\approx\{$, if. For (I).
$\epsilon>0$$\Phi\in\Delta_{2}$.

$r\geq 0,$$=r \int_{0}^{f}\frac{\Phi(t)}{t^{2}}dt$$r(\log r)^{\delta}$$\Phi\in$. If $\phi(r)=$. If $\phi(r)\equiv 1,$ then$r\geq 0,$$\mathrm{r}\mathrm{o}^{1}\Phi(t)t^{-2}dt<+\mathrm{o}\mathrm{o}$$\Phi(r)=\{$

Then

.

and $\Phi(r)\leq\Phi^{+}(2r)$ . If 4 satisfies the

Example 2.1. For,$L^{(\Phi,\phi)}(\mathbb{R}^{n})=L^{p,\lambda}(\mathbb{R}^{n})$$\Phi(r)\leq\frac{1}{2k}\Phi(kr)$

Then
and

$\}$

$\Delta_{2}$for some $k>1.$
Let $y+$ be the set of all (It:$L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(\Phi,\phi)}(\mathbb{R}^{n})=L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(p,\phi)}(\mathbb{R}^{n})$$=r^{p}$$\Phi(r)$

, let

.

$L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{(\Phi,\phi)}(\mathbb{R}^{n})=L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Phi}(\mathbb{R}^{n})$$\Phi(2r)\leq C\Phi(r)$for some $C>0.$
A function$<+$ ooand. If
and $\phi(r)=r"-n$ , thenis said to satisfy the -condition, denoted$L^{(\Phi,\phi)}(\mathbb{R}^{n})=L^{\Phi}(\mathbb{R}^{n})$A functionand$L^{(\Phi,\phi)}(\mathbb{R}^{n})=L^{(p,\phi)}(\mathbb{R}^{n})$$\phi\in \mathcal{G}$.

$\phi$$=\mathrm{s}\mathrm{u}^{\mathrm{P}}$Ifand
If $\Phi(r)=r^{p}$ , then
, then

$t)$

, ,B,

$||f||_{\Phi,\phi,B,\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}$$||f||L\mathrm{r}’ \mathrm{b})$,for small $r>0,$
for large $r>0.$148

Example 2.2. For $1<p<\infty$ ,$\epsilon\in \mathbb{R}$and$\delta\in \mathbb{R}$$r^{p}(\log r)^{\delta}$$\Phi\in\nabla_{2}$andFor a function$\Phi^{+}\approx\Phi$$\theta$$\Phi\in \mathcal{Y}$.

: $(0, +00)arrow(0,$$+\mathrm{o}\mathrm{o}$$\theta^{*}(r)=\int_{0}^{f}\frac{\theta(t)}{t}dt$

, let

,

$\theta_{*}(r)=\int_{f}^{+oo}$

If 0 satisfies the doubling condition, thenis almost increasing for somethendereasing for somethen.
, then
If
and
. If
then
: .

$\frac{\theta(t)}{t}dt$$\epsilon>0,$$\theta(r)r^{-\epsilon}$$\theta^{*}(r)\leq C\theta(r)$$\theta(r)\leq C\theta_{*}(r)$. If$\theta(r)r^{\epsilon}$. If

is almost$\theta_{*}(r)\leq C\theta(r)$$\epsilon>0,$$\phi_{*}$.

and$\theta(r)\leq C\theta^{*}(r)$$7_{0}^{1}6(t)dt<+\mathrm{o}\mathrm{o}$$6\in \mathcal{G}$withfor small $r>0,$
for large $r>0.$$r^{p}(\log(1/r))^{-\epsilon}$$\Phi(r)=\{$

Then

, let

$\phi^{*}\in \mathcal{G}$$\phi\in$$\mathcal{G}$and$\int_{1}^{+\infty}\phi(t)dt<+\mathrm{o}\mathrm{o}$$\mathcal{G}$3. MAIN RESULTS
The Hardy-Littlewood maximal function of$f\in L_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R}^{n})$$Mf(x)= \sup_{B\ni x}|\mathrm{g}|\int_{B}|f(y)|dy$

is defined by,where the supremum is taken over all balls $B$ containing .
The following are our main results.
$x$

Theorem 3.1. Let$\phi$$\in$(i and:$M$

If (I)$\in))^{+}$$\Phi\in\nabla_{2}$Next let

$T$

Then

$L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\acute{\mathrm{a}}\mathrm{k}}^{(\Phi\phi)}(\mathbb{R}^{n})$.

:$L^{(\Phi^{+},\phi)}(\mathbb{R}^{n})arrow L(")(\mathbb{R}^{n})$.

:$L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L^{(\Phi,\phi)}(\mathbb{R}^{n})$.

be a singular integral operator with a kernel $K(x, y)$ which satisfies

(3.1)

(3.2)

$)$

, then$M$and, for) .

, then$M$

If$\Phi\in$$f\in C_{\mathrm{c}}$

,

$x$

,

$Tf(x)= \int_{\mathrm{R}^{n}}K(x, y)f(y)dy$,$|K(x, y)| \leq\frac{C}{|x-y|^{n}}$$y\in \mathbb{R}^{n}$, $x\neq y,$o ’ ,$\mathrm{m}\mathrm{p}($$n)$$\mathrm{a}.\mathrm{e}.x\not\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}f$

.

,

148

(; and, ,Theorem 3.2. Let
there exists a constant $A>0$ such that$\Phi\in$$\Delta_{2}$$\phi$$\psi\in$$\mathit{7}_{1}^{+\infty}\Phi^{-1}(\phi(t))/tdt<+\mathrm{o}\mathrm{o}$r $>0.$(i) $((\Phi^{-1}0\phi)_{*}(r))\leq A\psi(r)$ ,(i) If$T$:$L^{\Phi}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Phi}(\mathbb{R}^{n})$, then.

$T:L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\acute{\mathrm{a}}\mathrm{k}}^{(\Phi\psi)}(\mathbb{R}^{n})$(ii) If$T:L^{\Phi}(\mathbb{R}^{n})arrow L^{\Phi}(\mathbb{R}^{n})$, then.

$T:L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L^{(\Phi,\psi)}(\mathbb{R}^{n})$Remark 3.1. Since

satisfies doubling condition, $\phi(r)=$$\Phi(C(\Phi^{-1}\circ\phi)_{*}(r))\leq C’\psi(r)$ for all $r>0.$

For a function$\rho$$\Phi^{-1}\circ\phi$:. Assume that$(0, +00)arrow(0, +\mathrm{o}\mathrm{o})$$1$$(!-1\circ\phi(r))$$\leq$, let$I_{\rho}f(x)=7n$ $f(y) \frac{\rho(|x-y|)}{|x-y|^{n}}dy$.

We consider the following conditions on :$\rho$(3.3)$\int_{0}^{1}\frac{\rho(t)}{t}dt<+\mathrm{o}\mathrm{o}$(3.4)$\frac{1}{A_{1}}\leq\frac{\rho(s)}{\rho(r)}\leq A_{1}$

(3.5)

If

$\frac{\rho(r)}{r^{n}}\leq A_{2}\frac{\rho(s)}{s^{n}}$$\rho(r)=r^{\alpha}$, $0<\alpha<n$ , thenTheorem 3.3. Let,
constant $A>0$ such that$D$$/\in \mathcal{G}$$I_{\rho}$,fo$\mathrm{r}$for,$\frac{1}{2}\leq\frac{s}{r}\leq 2,$$s\leq r.$

is the fractional integral denoted by$If\in \mathcal{Y}$. Assume that$\phi$$I_{\alpha}$.

is bijective. If there exists$\Psi(\frac{\Phi^{-1}\circ\phi(r)\rho^{*}(r)+((\Phi^{-1}0\phi)\rho)_{*}(r)}{A})\leq\phi(r)$,$r>0,$

then:$I_{\rho}$

If$\Phi\in$$\mathcal{Y}+and$$L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\acute{\mathrm{a}}\mathrm{k}}^{(\Psi\phi)}(\mathbb{R}^{n})$.

there exists a constant $A>0$ such that

I $( \frac{(\Phi^{+})^{-1}0\phi(r)\rho^{*}(r)+(((\Phi^{+})^{-1}0\phi)\rho)_{*}(r)}{A})\leq\phi(r)$,then$I_{\rho}$:$L^{(\Phi^{+},\phi)}(\mathbb{R}^{n})arrow L^{(\Psi,\phi)}(\mathbb{R}^{n})$.

$r>0,$$a$150

Moreover, if$\Phi\in \mathit{7}_{2}$, then$I_{\rho}$:$L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L^{(\Psi,\phi)}(\mathbb{R}^{n})$.

Example 3.1. Let $0<\alpha<n$ , $1<p$ $<q$ $<oo$ , $-1/p+$$\rho(r)=r^{\alpha}$$\Phi(r)=r^{p}$,$\Phi\in\nabla_{2}$$=-1/q$ ,,$\Psi(r)=r^{q}$$\phi(r)=r"-n$Then$\mathrm{a}/(n-\lambda)$

,

.

and

$\Phi^{-1}05(7)\mathrm{p}’(r)$” $=r(\lambda-n)/q$$+((\Phi^{-1}0\phi)\rho)_{*}(r)\sim r^{(-n)/p+}$.

We have:$I_{\alpha}$$L^{p,\lambda}(\mathbb{R}^{n})arrow L^{q,\lambda}(\mathbb{R}^{n})$.

This is the result of Adams [1] (1975).
Example 3.2. Let:$l$$(0, +00)arrow(0, +\mathrm{o}\mathrm{o})$satisfy the doubling condition andfor small $r>0,$
for large $r>0.$$(\log 1/r)^{-1}$$\ell(r)=\{$$\log r$

For $\beta>0,$ let
for small $r>0,$
for large $r>0.$$(\log 1/r)^{-\beta-1}$$\rho(r)=\{$$(\log r)^{\beta-1}$

Then$\rho$

satisfies (3.3)-(3.5) and$\rho^{*}(r)=7’$ $\frac{\rho(t)}{t}dt\sim l^{\beta}(r)$.

Let$\Phi(r)=r^{p}(1\leq p<\infty)$

and $r)=r^{\lambda-n}$ ($0\leq$,$\Psi(r)=r^{p}\Psi^{\beta}(r)$A $<n$ ). Then we have
$I_{\rho}$$I_{\rho}$:$L^{1,\lambda}(\mathbb{R}^{n})=L^{(1,\phi)}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\acute{\mathrm{a}}\mathrm{k}}^{(\Psi\phi)}(\mathbb{R}^{n})$,:$L^{p,\lambda}(\mathbb{R}^{n})=L^{(p,\phi)}(\mathbb{R}^{n})arrow L^{(\Psi,\phi)}(\mathbb{R}^{n})$,$1<p<\infty$

Example 3.3. Let$\ell$,

and$\rho$.

be as in Example 3.2. For $p>0,$ let$1/\exp(1/r^{p})$$e_{p}(r)=\{$$\exp(r^{p})$fo small $r>0,$
for large $r>0.$$\mathrm{r}$

151

Let
I(r) $=e_{p}(r)$ ,I(r) $=e_{q}(r)$ ,$-1/p+\beta=-1/q$ $<0,$$,(r)$ $=r_{:}^{-\mu}0<\mu\leq n.$

Then we have:

$I_{\rho}$

Example 3.4. Let$\ell$

and$\rho$$L^{(\Phi,\phi)}(\mathbb{R}^{n})arrow L^{(\Psi,\phi)}(\mathbb{R}^{n})$be as in Example 3.2. For$r(\log(1/r))^{-\epsilon-1}$$\Phi(r)=\{$$r(\log r)^{\delta}$$r(\log(1/r))^{-\epsilon-\beta}$$\Psi(r)=\{$$r(\log r)^{\delta+1+\beta}$$6(r)$$=r^{-\mu}$.
$\epsilon>0$,$\delta$$\geq 0$and$\mathrm{a}$

$>0,$

let

for small $r>0,$
for large $r>0,$

for small $r>0,$
for large $r>0,$, $0<\mu\leq n.$Then$r(\log(1/r))^{-\epsilon}$$\Phi^{+}(r)\approx\{$$r(\log r)^{\delta+1}$

for small $>0,$
for large $r>0.$$r$

Then we have
$I_{\rho}$:$L^{(\Phi^{+},\phi)}(\mathbb{R}^{n})arrow L^{(\Psi,\phi)}(\mathbb{R}^{n})$4.

.

$\mathrm{p}_{\mathrm{R}\mathrm{O}\mathrm{O}\mathrm{F}\mathrm{S}}$Lemma 4.1. For $B=B(a, r)$ ,$\int_{B}f(x)g(x)dx\leq 2|B|\phi(r)||f||\Phi,B,\phi||g1_{\mathrm{i},B}$

Proofs By Holder’s inequality for the Orlicz spaces$L^{\tilde{\Phi}}(B, dx/(|B|\phi(r)))$,$\phi$.

$L^{\Phi}(B, dx/(|B|\phi(r)))$and, we have$\int_{B}f(x)g(x)\frac{dx}{|B|\phi(r)}\leq 2||f||_{L^{\Phi}(B,dx/(|B|\phi(\mathrm{r})))}||g||_{L^{\tilde{\Phi}}(B,dx/(|B|\phi(r)))}$$=2||f||_{\Phi,B_{=}}\phi|\mathrm{K}/||\mathrm{i},B,1^{\cdot}$Lemma 4.2. If a Young function$\mathrm{D}^{-1}(\phi(r))/\phi(r)$.

$\Phi$$\square$is bijective and $B=B(a, r)$ , then$||1||_{\tilde{\Phi},B,\phi}\leq$152

Proof. By
have

t, we
for $s<t$ and $\Phi(t)s/t$ $-\Phi(s)\leq 0$ for$(\phi(r))/\phi(r)$ . Then$(i). Let $0(t)=\phi(r)$ and A $=t/\Phi(t)=$$\tilde{\Phi}(\Phi(t)/t)\leq$$s\geq$$\leq\Phi(t)$$\Phi(t)s/t-\Phi(\mathrm{g})$$\mathrm{I}^{-1}$$\frac{1}{|B|}\int_{B}\overline{\Phi}(\frac{1}{\lambda})dx=\tilde{\Phi}(\frac{1}{\lambda})\leq\Phi(t)=\phi(r)$Lemma 4.3. If$f\in L^{(\Phi,\phi)}(\mathbb{R}^{n})$and$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}f\cap B(a, 2r)=l)$$Mf(x)\leq C\Phi^{-1}(\phi(r))||f||_{L^{(\Phi,\phi)}}$.

$\square$. Thenfor $x\in B(a, r)$ .

Proof. For all balls $B=B(b, s)$ , if $s\leq r/2$ then $\int_{B}|f(x)|dx=0$ , and, if $s>r/2$ ,then$\int_{B}|f(x)|dx\leq 2\phi(s)||f||_{\Phi,\phi,B}||1||_{\overline{\Phi},\phi,B}$$\mathrm{E}$$2\phi(s)||f||_{L^{(\Phi,\phi)}}!$”1 $(\phi(s))/\phi(s)$$\leq 2\Phi^{-1}(\phi(s))||f|6$ $(+,\phi)$$\leq C\Phi^{-1}(\phi(r))||f||_{L^{(\Phi,\phi)}}$

Theorem 4.4 (see [3, 10, 12]). Let$\Phi\in \mathcal{Y}$.

$\square$. Then$\sup_{t>0}\Phi(t)m(f, t)\leq c_{1}74_{n}\Phi(c_{1}|f(x)|)dx$

If$\Phi\in \mathcal{Y}^{+}$.

, then.

$7_{\hslash} \Phi(Mf(x))dx\leq c_{1}\int_{\mathrm{R}^{n}}\mathrm{D}^{+}(c_{1}|f(x)|)dx$

If$\Phi\in$V2, then$\int_{\mathrm{R}^{n}}$I$( \mathrm{f}f(x))dx\leq c_{1}\int_{\mathbb{R}^{n}}\Phi(c1|f(x)|)$

$dx$

.

Proof of Theorem 3.1. For all balls $B=B(a, r)$ , let $f=f1+f_{2}$ , $f_{1}=f\chi 2B$ . For$f_{1}$, applying Theorem 4.4, we have

,

if

,

if I E)$\sup_{\mathrm{t}>0}\Phi(t)m(B, f_{1}/\lambda, t)\leq c_{1}\int_{\mathrm{R}^{n}}\Phi(c_{1}|f_{1}(x)|/\lambda)dx$$4$ $\Phi(Mf_{1}(x)/\lambda)dx\leq c_{1}4_{n}\mathrm{D}^{+}(c_{1}|f_{1}(x)|/\lambda)dx$

We may assume that$r\geq s.$ Let$c_{1}\geq 1.$

be a constant such thatLet
. Then we havein the cases (I)$c_{*}\geq 1$$\lambda=2^{n}c_{*}c_{1}^{2}||f||_{L}(+,\phi)$$c_{1} \int_{\mathrm{R}^{n}}D$$\Phi\in \mathcal{Y}$$(c_{1}|f_{1}(x)|/ \lambda)dx=c_{1}\int_{2B}$,$+$$\phi(r)\leq c_{*}\phi(s)$for$\in \mathcal{Y}$I $(c_{1}|f(x)|/\lambda)dx$$\leq 2^{-n_{\mathrm{C}_{*}}-1}\int_{2B}\Phi(|f(x)|/||f||_{L(\epsilon,\phi)})dx\leq 2^{-n}c_{*}^{-1}|2B|\phi(2r)\leq|B|\phi(r)$

.

153

Let A

$=2^{n}c_{*}c_{1}^{2}||f||_{L^{(^{\mathrm{g}1+},\phi)}}$in the case)$\Phi\in$$)^{+}$. Then we have in the same wey$c_{1}\acute{\mathbb{R}}^{n}1^{+}(c_{1}|f_{1}(x)|/\lambda)dx\leq|B|$For$f_{2}$$(7).

, applying Lemma 4.3, we have, for $x\in B,$$Mf_{2}(x)\leq c_{3}\Phi^{-1}(\phi(r))||f||_{L^{(\Phi,\phi)}}$$Mf_{2}(x)\leq c_{3}\Phi^{-1}(\phi(r))||f||_{L^{(\Phi}}+$in the case

Let A
Then we have$=c_{3}||f||\iota(\Phi,\phi)$$\Phi\in)^{7}$

,$)

,

if

$\Phi\in \mathcal{Y}$,if$4!?\in$, or A $=c_{3}||f||_{L}$ (,)$)^{+}$

in the case $\Phi\in y+$ .

,$0+_{\phi)}$$\sup_{t>0}\Phi(t)m(B, f_{2}/\lambda, t)\leq\int_{B}\Phi$ $(Mf_{2}(x)/\lambda)dx\leq|B|\phi(r)$.

This shows the norm inequations.

0Letand is a operator on
there exists a constant $c>0$ such that$\Phi\in$$T$$\mathcal{Y}$(4.1)$L^{\Phi}(\mathbb{R}^{n})$$\int_{\mathrm{R}^{n}}\mathrm{D}(|Tf(x)|)dx\leq c\int_{\mathbb{R}^{n}}$which satisfies! $(c|f(x)|)dx$ ,$|7$( $cf\mathrm{l}=|cTf|$ . If

$f\in L^{\Phi}(\mathbb{R}^{n})$,then there exists a constant $C>0$ such that

V

(4.2)

$||T$

$||L\Phi$$\leq C||f||_{L^{\epsilon}}$,$f\in L^{\Phi}(\mathbb{R}^{n})$

,

i.e.

$T:L^{\Phi}(\mathbb{R}^{n})arrow L^{\Phi}(\mathbb{R}^{n})$.

If there exists a constant $c>0$ such that
(4.3)$\sup_{t>0}CD$$(t)m(Tf, t) \leq c\int_{\mathbb{R}^{n}}\mathrm{i}$$(c|f(x)|)dx$ ,$f\in L$’(R ),$n$then there exists a constant $C>0$ such that

V

(4.1)

If

$\Phi\in$

:
for

$T$

$||T$

$\Delta_{2}$$||_{L_{\mathrm{w}\cdot*\mathrm{k}}^{\Phi}}\leq C||f||_{L^{q}}$$f\in L^{\Phi}(\mathbb{R}^{n})$

,

i.e.

.

$T:L^{\Phi}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Phi}(\mathbb{R}^{n})$, then (4.1) and (4.3) are equivalent to (4.2) and (4.4), respectively. If, $1<p<\infty$ , then (4.1) holds
and :.
V2 and (4.3) holds for$L^{1}(\mathbb{R}^{n})arrow L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{1}(\mathbb{R}^{n})$$\Phi\in$,$T$$\Delta_{2}\cap$$L^{p}(\mathbb{R}^{n})arrow L^{p}(\mathbb{R}^{n})$$\Phi$

Proof of Theorem S. 2. For$\in$$\Delta_{2}$$f\in L^{(\Phi,\phi)}(\mathbb{R}^{n})$and$x\in \mathbb{R}^{n}$, we choose a ball $B$ suchthat $x\in B,$ and let$Tf(x)=Tf_{1}(x)$ $+$ $7_{\mathrm{R}^{n}}K(x, y)f_{2}(y)dy$ ,$f=f_{1}+f_{2}$, $f_{1}=f\chi_{2B}$ .

We show that $Tf(x)$ is finite a.e.x and independent of the choice of . Since
,
is in
or
. Let $B=B(a,r)$ and $B_{k}=B(a, 2^{k}r)$ ,$B$$f_{1}\mathrm{E}$$L^{\Phi}(\mathbb{R}^{n})$$Tf_{1}$$L^{\Phi}(\mathbb{R}^{n})$$L_{\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}^{\Phi}(\mathbb{R}^{n})$154$k=1,2$ ,$\cdots$. Then$\int_{\mathbb{R}^{n}}|K(x, y)f_{2}(y)|dy=\sum_{k=2}^{\infty}B_{k}\backslash B_{h-1}|K(x, y)f_{2}(y)|dy$$\leq\sum_{k=2}^{\infty}\frac{C}{|B_{k}|}\int_{B_{k}}|f_{2}(y)|$dy.

By Lemmas 4.1 and 4.2 we have$\frac{1}{|B_{k}|}\int_{B_{k}}|f_{2}(y)|dy\leq 2\phi(2^{k}r)||f||_{\Phi,\phi,B_{k}}||1||_{\tilde{\Phi},\phi,B_{k}}$$\leq 2\phi(2^{k}r)||f||_{L}(\mathrm{t},\phi)$$\mathrm{F}^{1}(\phi(2^{k}r))/\phi(2^{k}r)$$=2\Phi^{-1}(\phi(2^{k}r))||f||_{L^{(\Phi.\phi)}}$

.

Hence
(4.5)

$\int_{\mathbb{R}^{n}}|K(x, y)f_{2}(y)|dy\mathrm{S}$ $C \sum_{k=2}^{\infty}\Phi^{-1}(\phi(2^{k}r))||f||_{L^{(\Phi,\phi)}}$$\leq C\int_{f}^{\infty})^{-1}"(\phi(t))t^{-1}dt||f||_{L}(\circ,+1$

If $x\in B\subset B’$ and $f=fi+f_{2}=f_{1}’+f_{2}’$ , $f1=f\chi 2B$ , $f_{1}’=f\chi 2B’$ , then$Tf(x)=Tf_{1}’(x)+ \int_{\mathbb{R}^{n}}K(x, y)f_{2}’(y)dy$$=T(f_{1}+f_{1}’-f_{1})(x)+ \int_{\mathbb{R}^{n}}K(x, y)f_{2}’(y)dy$$=$ $T(f_{1})$ $(x)+T(f_{1}’-f_{1})(x)+\int_{\mathbb{R}^{n}}K(x, y)f_{2}’(y)dy$$=T(f_{1})(x)+$ $/ \mathrm{R}_{n}K(x, y)(f_{1}’(y)-f_{1}(y))dy+\int_{\mathbb{R}^{n}}K(x, y)f_{2}’(y)dy$$=Tf_{1}(x)+ \int_{\mathrm{R}^{n}}K(x, y)f_{2}(y)dy$.

Now we show the boundedness. For every ball $B=B(a, r)$ , let $f=fi+f_{2}$ , $f_{1}=$. For $x\in B$ we write$f\chi_{2B}$$Tf(x)=Tf_{1}(x)+Tf_{2}(x)$ ,$Tf_{2}(x)= \int_{1\mathrm{R}^{\hslash}}K(x, y)f_{2}(y)dy$.

Then, in the case (i), by (4.3) we have$\Phi(t)m(B,$$\frac{Tf_{1}}{C||f||_{L^{(\Phi,\phi)}}}$,$t) \leq\Phi(t)m(\frac{Tf_{1}}{C||f||_{L(\Phi,\phi)}}$,$t) \leq C\int_{\mathrm{R}^{n}}\Phi(\frac{|f_{1}(x)|}{||f||_{L^{(\Phi,\phi)}}})dx$

,

155

and, in the case (ii), by (4.1) we have.

$\int_{B}\Phi(\frac{|Tf_{1}(x)|}{C||f||_{L^{(\Phi,\phi)}}})dx\leq\int_{\mathbb{R}^{n}}\Phi(\frac{|Tf_{1}(x)|}{C||f||_{L^{(\Phi,\phi)}}})dx\leq C1_{n}\Phi(\frac{|f_{1}(x)|}{||f||_{L^{(\Phi.\phi)}}})dx$

By the almost decreasingness of$\phi$

and Remark 3.1 we have$\int_{\mathbb{R}^{n}}\Phi(\frac{|f_{1}(x)|}{||f||_{L^{(\Phi.\phi)}}})dx=\int_{2B}\Phi(\frac{|f(x)|}{||f||_{L^{(\Phi,\phi)}}})dx$$\leq|2B|$ $6(2r)$ $\leq C|B|\phi(r)\leq C|B|\psi(r)$

,

Hence
$||’ \mathrm{V}_{1}||_{\Phi,1,B,\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}}\leq C||f||_{L(\Phi,\phi)}$$||Tf_{1}||_{\Phi}$,$\psi,B$$\leq C||f||_{L}(+,\phi)$,,in the case (i)in the case (ii).

By (4.5) we have.

$\int_{B}\Phi(\frac{|Tf_{2}(x)|}{C||f||_{L^{(\Phi,\phi)}}})dx\leq\int_{B}\Phi(\int_{r}^{\infty}\Phi^{-1}(\phi(t))t^{-1}dtt)dx$$\leq|B|\psi(r)$

,

i.e.
$||$

’V2H ,1,B,weak$\Phi$$\leq||7$ $f_{2}||_{\Phi,1B},\leq C||f||_{L^{(\Phi,\phi)}}$.

$\square$

Proof of Theorem 8. 3. Using the pointwise estimate in the proof of Theorem 2.2 in[17];$\Psi(\frac{|I_{\rho}f(x)|}{C_{1}||f||_{L^{(\Phi,\phi)}}})\leq\Phi(\frac{Mf(x)}{C_{0}||f||_{L^{(\Phi,\phi)}}})$and Theorem 3.1 we have the norm inequalities.

,$\square$

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