数理解析研究所講究録 1277 巻 2002 年 56-66

56

The corona type decomposition of Hardy-Orlicz spaces
東大・数理今井隆大 (IMAI Ryuta)

Graduate School of Mathematical Sciences,
The University of Tokyo
Abstract
complex variables has been solved affirmatively by
probleminseveralThe
type-corona
Amar [1], Andersson [2], Andersson-Carlesson $[3, 4]$ , Krantz-Li [11] and so on. Especially, Andersson-norm estimates of the corona solutions which are constructed by aCarlsson [4] proved theconcrete integral representation formula. In this paper, we give some Orlicz space versions for inter-norm estimates of the corona solutions
polation theorems of Marcinkiewicz tyPe and prove the. Moreover we also show that the A c0nditi0n is reasonable in asense.
for$H^{p}$$H^{\mathrm{p}}$$H_{\phi}$$\phi$$\in\Delta_{2}\cap\nabla_{2}$$2^{-}$

1Introduction
In this paper, we consider candidate of holomorphic space, in which we discuss the corona tyPe problem.
The corona problem was conjectured by S.Kakutani as early as 1941 and was solved affirmatively by
L.Carleson in 1962. Here, the corona problem is meant to be aproblem about the structure of the. That is, open unit disc $D$ is dense in with respect to the Gelfand
maximal ideal spaceof?
topology This question is equivalent to the existence problem as follows. For any /1, , $f_{m}\in H^{\infty}(D)$such that $\inf_{z\in D}\sum_{k=1}^{m}|f_{k}(z)|\geq\delta>0$ , is there exist , , $g_{m}\in H^{\infty}(D)$ such that $\sum_{k=1}^{m}f_{k}(z)g_{k}(z)=1$?
,,
and , ,
are refered to as the corona data and the corona solutions respectively. Let$X$ be aholomorphic space. We consider the question whether the mapping defined by$H^{\infty}(D)$$\mathcal{M}$$\mathcal{M}$$\cdots$$\cdots$$g_{1}$$f_{1}$$\cdots$$f_{m}$$g_{1}$$\cdots$$g_{m}$$X\mathrm{x}$$\cdots \mathrm{x}X\ni(.g_{1}, \cdots,g_{m})\mapsto\sum_{k=1}^{m}f_{k}g_{k}\in X$

is surjective. We say that $X$ has the $X$ -corona solution (for the corona data
is serjective. Then, let: $Xarrow X$ , $(k=1, \cdots, m)$ be an operator such that

$f1$

,

$\cdots$,$f_{m}$) if this mapping$T_{k}$$h(z)= \sum_{k=1}^{m}f_{k}(z)\cdot T_{k}h(z)$,$(h\in X, z\in\Omega)$, $(k=$
if $X$ has the $X$-corona solution for the corona data $f1$ , , . In particular we refer to,, $m$ ) as the $X$ -corona solution if
is bounded on $X$ in such sense as $||T_{k}h||x\leq C||h||x$ .
Then the corona theorem asserts thathas the-corona solutions for any corona data.
On the other hand, the corona problem in several complex variables has not been solved yet. In somestudies of the corona problem in several complex variables so far, the-corona type problem has beensolved affirmatively. That is, it is shown thathas the-corona solution. (For details, see Amar [1],Andersson [2], Andersson-Carlsson $[3, 4]$ , Krantz-Li [11] and so on.)Now, we are motivated by the question whether
can be approximated by some holomorphic, which is aspaces $X$ having the $X$-corona solution. And we consider the Hardy-Orlicz space$\cdots$$1$$\cdots$$T_{k}h$$f_{m}$$T_{k}$$H^{\infty}(D)$$H^{\infty}(D)$$H^{p}$$H^{p}$$H^{p}$$H^{\infty}$$H_{\phi}(\Omega)$

57
generalization of Hardy spaces, as acandidate of such space. In what follows, we letbe abounded strictly pseudoconvex domain with asmooth boundary of class .
At first, we review some convex functions. We refer to aconvex function : $Rarrow R+\cup\{\infty\}$ as aYoung function if (1) $\phi(x)=\varphi(-x)$ , (2) $0(0)=0$ and (3) $\lim_{xarrow\infty}\phi(x)=\infty$ . Moreover, acontinuous
Young ftinction is called an $N$ function if (1) $\phi(x)=0$ iff $x=0$ and (2),. Then, we introduce two classifications for convex functions which play an important role below. A
Young function : $Rarrow R_{+}$ satisfies theif there exists apositive constant $K$-condition
such that$(x\geq 0)$ .
$\phi(2x)\leq K\phi(x)$ ,$H^{p}$$\Omega\subset C^{n}$$C^{3}$$\phi$$\lim_{xarrow 0}\frac{\phi(x)}{x}=0$$\varphi$$\lim_{xarrow\infty}\frac{\phi(x)}{x}=$$\infty$$(\varphi\in\Delta_{2})$$\Delta_{2}$$\phi$

And aYoung function : ffconstant $a>1$ such that$\phi$

the$arrow R+\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{s}$$\nabla_{2}$$\phi(x)\leq\frac{1}{2a}\phi(ax)$Let be an $N$ -function satisfying thedefined as follows.
$\phi$$\Delta_{2}$and$H_{\phi}(\Omega)=\{f\in \mathcal{O}(\Omega)$

Since$\zeta\in\partial\Omega$:$\nabla_{2}$,-condition$(x\geq 0)$$(\phi\in\nabla_{2})$if there exists apositive.

-condition. Then, the Hardy-Orlicz space$\lim_{\epsilonarrow}\sup_{0}\int_{\partial\Omega_{e}}\phi(|f|)d\sigma_{\epsilon}<\infty\}$is.

belongs to the Nevanlinna class, has the nontangential limit. From now on, we identify
with afunction space on the boundary$f\in H\psi(\Omega)$$H_{\phi}(\Omega)$$f$$f(\zeta)$an.

$H_{\phi}(\Omega)$

at almost every

2Main results
We use the real variable methods such as an Orlicz space version of the interpolation theorem ofMarcinkiewicz tyPe, Hardy-Littlewood maximal operator, nontangential maximal operator and Orlicz
space theory to characterize the Hardy-Orlicz space. Our main results are as follows.
Theorem 1Suppose that
. Then every function in Hardy-Orlicz spacecan beapproimated by some functions holomorphic up to the boundary with respect to Luxemberg norm:$\phi\in\Delta_{2}\cap\nabla_{2}$$H_{\phi}(\Omega)$$H_{\phi}(\Omega)$$\cong$$[A(\partial\Omega)]_{L_{\phi}(\partial\Omega)}$,

where we recall thatis the restriction of
to the boundary
as the closure ofwith respect to the Luxemberg norm.
$C(\overline{\Omega})\cap \mathcal{O}(\Omega)$$A(\partial\Omega)$$[A(\partial\Omega)]_{L_{\phi}(\partial\Omega)}$an and we mean$\mathrm{A}(\mathrm{d}\mathrm{Q})$

Theorem 2Suppose that
S coincides utith Hardy-Orlicz space$\phi\in\Delta_{2}\cap\nabla_{2}$. Then the image of Orlicz space, that is,$L_{\phi}(\partial\Omega)$by the Szeg\"o projection$H_{\phi}(\Omega)$$SL_{\phi}(\partial\Omega)$$=$$H_{\phi}(\Omega)$.

By combining the theorem above and an Orlicz space version of the interpolation theorem of Marcinkiewicz

tyPe, we obtain an interpolation theorem for Hardy-Orlicz spaces

58
Theorem 3Let ,be satisfying that
, where andare the leftderivatives of andrespectively. We suppose that a sublinear operator $B$ defined on
andis of weak type $(1, 1)$ and of weak typerespectively. Then $B$ is defined on
andthe following holds:$\phi_{2}\in\Delta_{2}\cap\nabla_{2}$$\acute{\varphi}$$\sup_{\lambda>0}\frac{\varphi(\lambda)\phi_{2}(\lambda)}{\phi(\lambda)\varphi_{2}(\lambda)}<1$$\varphi$$\acute{\varphi}$$H^{1}(\Omega)$$H_{\phi_{2}}(\Omega)$$(\phi_{2}, \phi_{2})$$\int_{\partial\Omega}\acute{\varphi}(|Bf|)\ \ovalbox{\tt\small REJECT}$

where$\varphi_{2}$$\acute{\varphi}_{2}$$S$$C \inf\{\int_{\partial\Omega}\phi(|g|)d\sigma$$H_{\phi}(\Omega)$: $g\in L_{\phi}(\partial\Omega)s.t.f=Sg\}$ ,is the Szeg\"o projection.

Before the corona type decomposition of Hardy-Orlicz spaces, we review the corona typedecomposition of Hardy spaces
as follows. Andersson-Carlsson [4] shows that an explicit integral
formula due to Berndtsson [5] provides the-corona solutions.
$H_{\phi}(\Omega)$$H^{p}(\Omega)$$H^{p}$

Theorem 4(Andersson-Carlsson [4])Let $1\leq p<\infty$ . If $f1$ , ,, then there
satisfies that $\sum_{i=1}^{m}|f_{i}(z)|\geq\delta>0$ for all
integral operators$(i=1, \cdots, m)$ such that $\sum_{i=1}^{m}f_{i}(z)T_{i}h(z)=h(z)$ , $(z\in\Omega)$ and:for a positive constant $C$ .
$\cdots$$T_{i}$$f_{m}\in H^{\infty}(\Omega)$$z\in\Omega$$e$$\dot{m}t$$\mathrm{H}\mathrm{I}(\mathrm{Q})arrow \mathrm{H}\mathrm{I}(\mathrm{Q} )$$||T_{i}h||_{p}\leq C||h||_{p}$By combining the theorems above, we can show that this integral formula due to Berndtsson [5]admits-estimates if.
$\phi\in\Delta_{2}\cap\nabla_{2}$$H_{\phi}$

Corollary 1Let

. If

$f1$ ,,are corona data, that is, they satisfy that$(i=1, \cdots, m)$, then there exist integral operators : Hlsuch that $\sum_{i=1}^{m}f_{i}(z)T_{i}h(z)=h(z)$ , $(z\in\Omega)$ . $h\hslash hemore$ it follows that there exists a positive constant$C$such that$\phi\in\Delta_{2}\cap\nabla_{2}$for all$\sum_{i=1}^{m}|f_{i}(z)|\geq\delta>0$$\cdots$$f_{m}\in H^{\infty}(\Omega)$$z\in\Omega$$T_{i}$$\int_{\partial\Omega}\phi(|T_{i}h|)d\sigma\leq C\inf\{\int_{\partial\Omega}\phi(|g|)d\sigma:g\in L_{\phi}(\partial\Omega)$

where

$S$

$(\mathrm{Q})arrow \mathrm{H}1(\mathrm{Q})$such that $h=Sg\}$ ,is the Szeg\"o projection.

From the theorems above, we may say that the Hardy-Orlicz spacewith amoderate growth
condition (i.e.
) has the-corona solution. On the other hand, aquestion whether thecondition thatis too strong occurs. Then we investigate the relation between the boundednessof the Szeg\"o projection and the operators constructing the corona solutions and the gorwthness of the$N$ -functionin order to find areasonable condition with respect to the growthness of .
Theorem 5Let be an N-function. We suppose that is the Szeg\"o projection on 0. If is of weaktype:
$H_{\phi}(\Omega)$$\acute{\varphi}\in\Delta_{2}\cap\nabla_{2}$$H_{\phi}(\Omega)$$\phi\in\Delta_{2}$$\varphi$$\phi$$S$$\phi$$S$$(\phi, \phi)$$\phi(\lambda)\sigma(\{|Sf|>\lambda\})\leq C_{1}\int_{\partial\Omega}\phi(C_{2}|f|)d\sigma$then$\phi$

satisfies the$\Delta_{2}$Theorem $6Let$ /1,
that:. If every operator,$f_{m}\in H^{\infty}(\Omega)$be the corona data satisfying that

$T_{i}$$\phi(\lambda)\sigma(\{|T_{i}h|>\lambda\})\leq C$$\phi$

satisfies the$\Delta_{2}$,$\sum_{\dot{l}=1}^{m}||f_{\dot{l}}||_{\infty}<1$. We suppose, $(i=1, \cdots,m)$ is a linear operator such that $h(z)= \sum_{\dot{|}=1}^{m}fi(z)Tih(z)$ , $(z\in$satisfies that$\mathrm{H}1(\mathrm{Q})arrow H^{1}(\Omega)$$\Omega)$then$(\lambda>0, f\in L_{\phi}(\partial\Omega))$-condition.

$\cdots$$T_{i}$,-condition.

an$\phi(|h|)d\sigma$,$(\lambda>0, h\in H_{\phi}(\Omega))$,59
3Preliminaries
Most main theorems are obtained as applications of an Oriicz space version of the interpolation theorem
to improve theof Marcinkiewicz type. At first, we give adefinition of weak type inequality ininterpolation theorem in Gallardo [7], where $X$ is aspace of homogeneous type. We denote the quasidistance over $X$ by and the Borel regular measure on $X$ with doubling condition by . Let us recall
that an operator is said to be quasi-additive if $|T(f+g)|\leq C(|Tf|+|Tg|)$ for aconstant $C>0$ . If
$C=1$ here, thenis called sublinear.
$L_{\phi}(X)$

$d$

$\mu$

$T$

$T$

Definition 1A sublinear operator defined on an Oriicz spacesuch thatandpositive constants$T$$e$ $\dot{m}ts$$\phi(\lambda)\mu(\{x\in X : |Tf|>\lambda\})\leq C_{1}\int_{X}\phi(C_{2}|f|)d\mu$Lemma 1Let ,$\phi$$\phi_{1}$and$L_{\phi}(X)$is of weak type$(\phi, \phi)$

if there$C_{2}$$C_{1}$$\phi_{2}$,$(f\in L_{\phi}(X), \lambda>0)$.

be three $N$-functions satisfying the follow ing grow th conditions:$\sup_{\lambda>0}\frac{\varphi(\lambda)\phi_{1}(\lambda)}{\phi(\lambda)\varphi_{1}(\lambda)}$

inf

$<$

1,

$>$

1,

$\underline{\varphi(\lambda)\phi_{2}(\lambda)}$$\lambda>0\varphi(\lambda)\varphi_{2}(\lambda)$

whereconstants$\varphi,’\varphi_{1}$$C_{1}$

andandare the left der ivatives of ,$\phi$$\varphi_{2}$$C_{2}$

and$\varphi_{1}$respectively.

$\varphi_{2}$Then, there eist positive

such that$\int_{u}^{\infty}\frac{\varphi(t)}{\phi_{1}(t)}dt$$\leq$$C_{1} \frac{\phi(u)}{\phi_{1}(u)}$$\int_{0}^{u}\frac{\varphi(t)}{\phi_{2}(t)}dt$$\leq$$C_{2} \frac{\phi(u)}{\varphi_{2}(u)}$Proof: We may take apositive number$r$,$(u>0)$ ,,$(u>0)$ .

such that.

$\sup_{\lambda>0}\frac{\varphi(\lambda)\varphi_{1}(\lambda)}{\phi(\lambda)\varphi_{1}(\lambda)}<r<1$

Then it follows that$\frac{\varphi(\lambda)}{\phi_{1}(\lambda)}<r\phi(\lambda)\frac{\varphi_{1}(\lambda)}{\varphi_{1}(\lambda)^{2}}=-r\phi(\lambda)\frac{d}{d\lambda}(\frac{1}{\phi_{1}(\lambda)})’$On the other hand, for any$\lambda_{0}>0$, the following hold.

$(\lambda>0)$.

$\mathrm{s}$$\log\frac{\varphi(\lambda)}{\phi(\lambda_{0})}=\int_{\lambda_{\mathrm{O}}}^{\lambda}\frac{\varphi(t)}{\phi(t)}dt\leq r\int_{\lambda_{0}}^{\lambda}\frac{\varphi_{1}(t)}{\varphi_{1}(t)}dt=\log(\frac{\varphi_{1}(\lambda)}{\phi_{1}(\lambda_{0})})^{r}$(A$\geq\lambda_{0}$).

60Hence we obtain that$\int_{u}^{\infty}\frac{\varphi(\lambda)}{\varphi_{1}(\lambda)}d\lambda\leq-r[\frac{\phi(\lambda)}{\phi_{1}(\lambda)}.]_{u}^{\infty}+r\int_{u}^{\infty}\frac{\varphi(\lambda)}{\phi_{1}(\lambda)}d\lambda=r,\frac{\phi(u)}{\varphi_{1}(u)}+r\int_{u}^{\infty}\frac{\varphi(\lambda)}{\phi_{1}(\lambda)}d\lambda$

since

,

$\frac{\phi(\lambda)}{\phi_{1}(\lambda)}\leq\frac{\phi(\lambda_{0})}{\phi_{1}(\lambda_{\mathrm{O}})^{r}}\varphi_{1}^{l}(\lambda)^{\mathrm{r}-1}=C\varphi_{1}^{l}(\lambda)^{\mathrm{r}-1}arrow 0$$(\lambdaarrow\infty)$,$(u>0)$ ,. Thus we conclude that$\int_{u}^{\infty}\frac{\varphi(\lambda)}{\phi(\lambda)}d\lambda\leq\frac{r}{1-r}\frac{\phi(u)}{\phi_{1}(u)}$$(u>0)$ .

,We can show the another inequality in the same way as above.

$\square$Using Lemma 1, we can improve the interpolation theorem in Gallardo [7] to prove the next theorem.

Theorem 7Let ,andoperator is of weak type:$\phi$$\phi_{1}$$T$$\phi_{2}$be as in the lemma above and ,and of weak type. Then. We suppose that a sublinear
is bounded on the Orlicz space$\phi_{2}\in\Delta_{2}$$\phi_{1}$$(\phi_{1}, \phi_{1})$$(\varphi_{2}^{l}, \phi_{2})$$T$$L_{\phi}(X)$$\int_{X}\phi(|Tf|)d\mu\leq C_{1}\int_{X}\phi(C_{2}|f|)d\mu$Moreover we can obtain the same conclusion if$T$

is of tyPe,$(f\in L_{\phi}(X))$.

and of weak type$(\infty, \infty)$$(\phi_{2}, \phi_{2})$.

Proof. From the weak type inequality and the sublinearity in the hypothesis, we can assume that

For any f$\in L_{\phi}(X)$$|T(f+g)|$$\leq$$|Tf|+|Tg|$ ,$\phi_{i}(\lambda)\nu(|Tf|>\lambda)$$\leq$$C_{i} \int\phi:(|f|)d\mu$

and any $\lambda>0$ , we take$f_{\lambda}$$f^{\lambda}$

Then since$f_{\lambda}$

andas follows:$f^{\lambda}$$=$$f\chi_{\mathrm{t}1f1>_{T}^{\lambda}\}}$$=$$f-f_{\lambda}$

It may be noted that.

.

$\nu(|Tf|>\lambda)\leq\nu(|Tf_{\lambda}|>\frac{\lambda}{2})+\nu(|Tf^{\lambda}|>\frac{\lambda}{2})$$\int\phi(|Tf|)d\nu$$(i=1, 2)$ .

,, the following holds.

$=$$\int_{0}^{\infty}\varphi(\lambda)\nu(|Tf|>\lambda)d\lambda$$\leq$$\int_{0}^{\infty}\varphi(\lambda)\nu(|Tf_{\lambda}|>\frac{\lambda}{2})d\lambda+\int_{0}^{\infty}\varphi(\lambda)\nu(|Tf^{\lambda}|>\frac{\lambda}{2})d\lambda$$f_{\lambda}\in L_{\phi_{2}}$and$f^{\lambda}\in L_{\phi_{1}}$. In fact,$\acute{\varphi}_{2}(x)\leq C_{R}\phi(x)$,, it follows that
andweak type inequality, the first term in the right hand side above is less than$\varphi_{1}(x)\leq C_{R}’\phi(x)$$(x \leq R=\frac{\lambda}{2})$$\phi_{2}(|f_{\lambda}|)\leq C_{R}\varphi^{l}(|f|)$$\int_{0}^{\infty}\varphi(\lambda)d\lambda\int C_{2}\frac{\phi_{2}(|f_{\lambda}|)}{\phi_{2}(\frac{\lambda}{2})}d\mu$$\leq$.

, $( \frac{\lambda}{2}=R\leq x)$ and. From the$\phi_{1}(|f^{\lambda}|)\leq C_{R}’\phi(|f|)$.

$C_{2} \int\phi_{2}(|f|)d\mu\int_{0}^{2|f|},\frac{\varphi(\lambda)}{\varphi_{2}(\frac{\lambda}{2})}d\lambda$

61
We note that there exists $K>0$ such thatve obtain thatsince$Kh( \frac{\lambda}{2})\geq\phi_{2}(\lambda)$. Then, by using Lerc

$K \int_{0}^{2|f|}\frac{\varphi(\lambda)}{\varphi_{2}(\lambda)}d\lambda$$\leq$$\int_{0}^{2|f|}\frac{\varphi(\lambda)}{\varphi_{2}(\frac{\lambda}{2})}d\lambda$$\acute{\varphi}_{2}\in\Delta_{2}$$\leq$$K’ \frac{\varphi(2|f|)}{\phi_{2}(2|f|)}$$\leq$$K’ \frac{\phi(2|f|)}{\phi_{2}(|f|)}$.

Hence the following holds.
$\int_{0}^{\infty}\varphi(\lambda)\nu(|Tf_{\lambda}|>\frac{\lambda}{2})d\lambda$$\leq$$C_{2}K’ \int\phi_{2}(|f|)\frac{\phi(2|f|)}{\phi_{2}(|f|)}d\mu$$\leq$$C_{2}K’ \int\phi(2|f|)d\mu$.

In asimilar way as above, we can obtain that.

$\int_{0}^{\infty}\varphi(\lambda)\nu(|Tf^{\lambda}|>\frac{\lambda}{2})d\lambda\leq C_{1}K’\int\phi(2|f|)d\mu$

In the case that$T$

is of typeFor any $f\in L_{\phi}(X)$ and any$(\infty, \infty)$, we may assume that$\leq$$C_{1}||f||_{\infty}$$\phi_{2}(\lambda)\nu(|Tf|>\lambda)$$\leq$$C_{2} \int\phi_{2}(|f|)d\mu$$\lambda>0$, we take$f_{\lambda}$$f^{\lambda}$

We note that.

$||Tf||_{\infty}$$\nu(|Tf^{\lambda}|>\frac{\lambda}{2})=0$

since$f_{\lambda}$

and$f^{\lambda}$.

as follows:$=$$f\chi_{\{1f\mathrm{I}>\frac{\lambda}{2C_{1}}\}}$$=$$f-f_{\lambda}$.

.
. Thus we obtain that$||Tf^{\lambda}||_{\infty} \leq C_{1}||f^{\lambda}||_{\infty}\leq C_{1}\frac{\lambda}{2C_{1}}=\frac{\lambda}{2}$$\nu(|Tf|>\lambda)\leq\nu(|Tf_{\lambda}|>\frac{\lambda}{2})+\nu(|Tf^{\lambda}|>\frac{\lambda}{2})=\nu(|Tf_{\lambda}|>\frac{\lambda}{2})$Therefore it follows that$\int\phi(|f|)d\nu$$=$$\int_{0}^{\infty}\varphi(\lambda)\nu(|Tf|>\lambda)d\lambda$$\leq$$\int_{0}^{\infty}\varphi(\lambda)\nu(|Tf_{\lambda}|>\frac{\lambda}{2})d\lambda$$\leq$$C_{2} \int_{0}^{\infty}\varphi(\lambda)d\lambda\frac{\int\varphi_{2}(|f_{\lambda}|)d\mu}{\phi_{2}(\frac{\lambda}{2})}$$\leq$$C_{2} \int\phi_{2}(|f|)d\mu\int_{0}^{2C_{1}|f|}\frac{\varphi(\lambda)}{\phi_{2}(\frac{\lambda}{2})}d\lambda$

.

.

62
Since
holds.

)$q_{2}$$\in\Delta_{2}$, there exists K $>0$ such that

$C_{2} \int\varphi_{2}(|f|)d\mu\int_{0}^{2C_{1}|f1}\frac{\varphi(\lambda)}{\phi_{2}(\frac{\lambda}{2})}d\lambda$$K \phi(\frac{\lambda}{2})\geq\varphi_{2}(\lambda)$. Then, using Lemma 3, the following$\leq$$C_{2}K \int\phi_{2}(|f|)d\mu\int_{0}^{2C_{1}|f\mathrm{I}}\frac{\varphi(\lambda)}{\phi_{2}(\lambda)}d\lambda$$\leq$$C_{2}K \int\acute{\varphi}_{2}(|f|)\frac{\phi(2C_{1}|f|)}{\phi_{2}(2C_{1}|f|)}d\mu$.

Now we should note that $\phi_{2}(|f|)\leq\phi_{2}(2C_{1}|f|)$ if $2C_{1}\geq 1$ and that $\phi_{2}(|f|)\leq L\phi_{2}(2C_{1}|f|)$ for an$L>0$ if $2C_{1}<1$ since. Hence we obtain that$\phi_{2}\in\Delta_{2}$$C_{2}K \int\acute{\varphi}_{2}(|f|)\frac{\phi(2C_{1}|f|)}{\phi_{2}(2C_{1}|f|)}d\mu\leq C_{2}KL\int\phi(2C_{1}|f|)d\mu$This completes the proof.

.

$\square$Furthermore, asmall modification of the proof in Coifman-Weiss [6] leads us to the following.

Theorem 8Letandbe an $N$-function. We suppose thatand thata sublinear operator $B:H_{Re}^{1}(X)+L_{\phi_{2}}(X)arrow M(X)$ is of weak typeand of weak type,where $M(X)$ is the set of all measurable functions on X. If $X$ is bounded, then the following holds:$\phi\in\Delta_{2}\cap\nabla_{2}$$\phi_{2}$$\sup_{\lambda>0}\frac{\varphi(\lambda)\phi_{2}(\lambda)}{\phi(\lambda)\varphi_{2}(\lambda)}<1$$(H_{Re}^{1}, 1)$$\int_{X}\phi(|Bf|)d\mu\leq C\int_{X}\phi(|f|)d\mu$

If

$X$

,

$(f\in L_{\phi}(X))$$(\phi_{2}, \phi_{2})$.

is unbounded , then the following holds:$|\mathrm{I}|Bf||_{(\phi)}\leq C||f||_{(\phi)}$,$(f\in L_{\phi}(X))$.

4Proofs
Proof of Theorem 1. We give asketch of the proof here. Details are left to Imai [8]. Firstly we let. Then we can take asequencesuch that $||f-f_{n}||_{(\phi)}arrow 0$ , (yz).
Using the Poisson kernel, we define afunction $F$ by$f\in[A(\partial\Omega)]_{L_{\phi}(\partial\Omega)}$$f_{n}\in A(\partial\Omega)$$arrow\infty$$P(z, \langle)$,$F(z)= \int_{\partial\Omega}P(z, \zeta)f(\zeta)d\sigma(\zeta)$$(z\in\Omega)$.

In the same way as is shown in Imai [8], we know that $F$ is holomorphic in Q. Moreover it follows that$|F_{\epsilon}(\zeta)|\leq CM_{HL}f(\zeta)$,$(a.e.\zeta\in\partial\Omega)$in Stein [15]. Since the Hardy-Littlewood maximal operator $M_{HL}$ is of weak type $(1, 1)$ and of, it follows thatis integrable from Theorem 7. And, sinceconverges topointwisely at almost every
by means of the well-known property of the Poisson integral, the$\eta \mathrm{p}\mathrm{e}$$(\infty, \infty)$$\acute{\varphi}(M_{HL}f)$$\zeta\in\partial\Omega$$F_{\epsilon}(\zeta)$$f(\zeta)$

63,. Therefore
Lebesgue dominated convergence theorem shows that
that[14].)
This
shows
details,
see RaO-Ren,. (For
we have that
.
and a
of
. And we choose afinite open covering
Conversely, we letfor every $j=1$ , , . If $1=\gamma_{1}+\cdots+\gamma_{q}$ is apartition of unity subordinate to the open
point
by, we define
covering$(\epsilonarrow 0)$$\int_{\partial\Omega}\phi(|F_{\epsilon}|)d\sigmaarrow\int_{\partial\Omega}\varphi(|f|)d\sigma$$(\epsilonarrow 0)$$||F_{\epsilon}-f||_{(\phi)}arrow 0$$[A(\partial\Omega)]_{L_{\phi}^{*}(\partial\Omega)}\subset$$H_{\phi}(\Omega)$$\mathcal{U}=\{U_{1}, \cdots, U_{q}\}$$f\in H_{\phi}(\Omega)$$p_{j}\in U_{j}$$\cdots$an$q$$\mathcal{U}=\{U_{1}, \cdots, U_{q}\}$$f_{j}$,$f_{j}(z)= \int_{\partial\Omega}\frac{K(\zeta,z)}{\Phi(\zeta,z)^{n}}f(\zeta)\gamma_{j}(\zeta)d\sigma(\zeta)$$(z\in\Omega)$,is the Henkin-Ramirez reproducing kernel. Then it is trivial that
where
. Moreover we may write thatneighborhood of$\frac{K(\zeta,z)}{\Phi(\zeta,z)^{n}}$$f_{j}$

is holomorphic in a$\Omega\cup(\partial\Omega\backslash U_{j})$$f_{j}(z)$–$\int_{\partial\Omega}f(()\{\gamma_{j}(\zeta)-\gamma_{j}(z)\}\frac{K(\zeta,z)}{\Phi(\zeta,z)^{n}}d\sigma(\zeta)+f(z)\gamma_{j}(z)$$=$$T_{j}f(z)+f(z)\gamma_{j}(z)$.

Since it is proved that the operator is of type $(1, 1)$ and of typefor sufficiently small $\in>0$ by Stout [18], Theorem 7shows that$(\infty, \infty)\sim$$T_{j}$

when$T_{j}f$is resticted to$\partial\Omega_{\epsilon}$.

$\lim_{\epsilonarrow}\sup_{0}\int_{\partial\Omega}\phi(|(T_{j}f)_{\epsilon}|)d\sigma\leq C\int_{\partial\Omega}\phi(|f|)d\sigma$.
Hence it follows that
Now, for any sufficient small $>0$ , we suppose that$f_{j}\in H\phi(\Omega)$$\epsilon$$f_{j}^{(\epsilon)}(\zeta)=f_{j}(\zeta-\epsilon\nu_{j})$where$\nu_{j}$is the outer unit vector transversal to,an at the point . Then$f_{j}^{(\epsilon)}\in O(\overline{\Omega})$$p_{j}$

and we know that$|f_{j}^{(\epsilon)}(\zeta)|\leq C+CM_{HL}f_{j}(\zeta)$

in the same way as is shown in Imai [8]. Since. Hence it follows that$f_{j}\in L_{\phi}(\partial\Omega)$, Theorem 7shows that $C+CNI_{HL}fj\in$,from the Lebesgue dominated. (For details, see Rao,$\int_{\partial\Omega}\phi(|f_{j}^{(\epsilon)}|)d\sigmaarrow\int_{\partial\Omega}\phi(|f_{j}|)d\sigma$$\mathrm{L};(\mathrm{d}\mathrm{n})$$(\epsilonarrow 0)$

convergence theorem. From this convergence we have.since $f=f1+\cdots+f_{q}$ .
Ren [14].) This shows that$||f_{j}^{(\epsilon)}-f_{j}||arrow \mathrm{O}$$(\epsilonarrow 0)$$\square$$f\in[A(\partial\Omega)]_{L_{\phi}(\partial\Omega)}$

Proof of Theorem 2. Since$\phi\in \mathrm{A}{}_{2}\mathrm{H}\mathrm{V}_{2}$, there exist

and$\phi_{1}$$\phi_{2}\in \mathrm{A}2\mathrm{H}\mathrm{V}2$

such that$\varphi(\lambda)\phi_{1}(\lambda)$$\sup_{\lambda>0\overline{\phi(\lambda)\varphi_{1}(\lambda)}}<$. (For details, see Gallardo [7] and Rao Ren [14].) Hence we can apply Theoandrem 7to the Szeg\"o projection in order to complete the proof.
$1$$\inf_{\lambda>0}\frac{\varphi(\lambda)\phi_{2}(\lambda)}{\phi(\lambda)\varphi_{2}(\lambda)}>1$$\square$$S$

Proof of Theorem 3. We consider the composition operators $A=B\circ S$ of asublinear operators $B$
and on an Orlicz
and the Szego projection . Then, since is bounded on real Hardy space, we can apply Theorem 8to the operator in order to show that
space
$S$

$L_{\phi_{2}}(\partial\Omega)$$A$$H_{Re}^{1}(\partial\Omega)$$A$$\int_{\partial\Omega}\phi(|Ag|)d\sigma\leq C\int_{\partial\Omega}\phi(|g|)d\sigma$,$(g\in L_{\phi}(\partial\Omega))$

.

64
Since

f

$H_{\phi}(\Omega)=SL_{\phi}(\partial\Omega)$$\in H_{\phi}(\Omega)$as shown in Theorem 2, we can take any gsuch that f $=Sg$ for$\in L_{\phi}(\partial\Omega)$

to obtain that.

$\int_{\partial\Omega}\acute{\varphi}(|Bf|)d\sigma=\int_{\partial\Omega}\phi(|Ag|)d\sigma\leq C\int_{\partial\Omega}\phi(|g|)d\sigma$

Since

$g$

is arbitrary function in$L_{\phi}(\partial\Omega)$such that $f=Sg$ , we can conclude that$\int_{\partial\Omega}\phi(|Bf|)\ \leq C\inf\{\int_{\partial\Omega}\phi(|g|)d\sigma$: $g\in L_{\phi}(\partial\Omega)s.t.f=Sg\}$ .

$\square$

Proof of Corolary 1. Sinceandrem 7to operators$1$, there exist

andsuch that. (For Details, see Gallardo [7] and Rao Ren [14].) Hence we can apply Theoin Theorem 4in order to complete the proof. Cl$\phi\in \mathrm{A}2\mathrm{H}\mathrm{V}2$$\phi_{2}\in\Delta_{2}\cap\nabla_{2}$$\phi_{1}$$\sup_{\lambda>0}\frac{\varphi(\lambda)\phi_{1}(\lambda)}{\phi(\lambda)\varphi_{1}(\lambda)}<$$\inf_{\lambda>0}\frac{\varphi(\lambda)\phi_{2}(\lambda)}{\phi(\lambda)\varphi_{2}(\lambda)}>1$$T_{\dot{l}}$Before giving the proofe of Theorem 5and 6, we show alemma as follows.
Lemma 2Let
, that is,$\acute{\varphi}$be an $N$ -function. We suppose that a sublinear operator

$T$

on

$L_{\phi}(\partial\Omega)$

is of weak type$(\phi, \phi)$$\phi(\lambda)\sigma(|Tf|>\lambda)\leq C_{1}$If, then$\sup_{11f\mathrm{I}|_{\infty}\leq 1}||Tf||_{\infty}>C_{2}$$\phi$

ansatisfies the$\phi(C_{2}|f|)d\sigma$$\Delta_{2}$,$(f\in L_{\phi}(\partial\Omega), \lambda>0)$-condition.

Proof. From the hypothesis, there exist $r>1$ and$||f||_{\infty}\leq 1$such that$K=\sigma(\{|Tf|>rC_{2}\})>0$

Then, for any $\lambda>0$ , we define afunction.

$g\in L_{\phi}(\partial\Omega)$.

by$g( \zeta)=\frac{\lambda}{rC_{2}}f(\zeta)$.

By applying the inequality of weak type to , we obtain that$g$.

$\phi(\lambda)\sigma\{|Tg|>\lambda\}\leq C_{1}\int_{\partial\Omega}\phi(C_{2}|g|)d\sigma$

Since $\{|Tg|>\lambda\}=\{|Tf|>rC_{2}\}$ , we have that $\sigma(\{|Tg|>\lambda\})=\sigma(\{|Tf|>rC_{2}\})=K>0$ . Therefore,
we have that$\acute{\varphi}(\lambda)$$\leq$$\sigma(\{|Tf|>rC_{2}\})^{-1}C_{1}\int_{\partial\Omega}\acute{\varphi}(C_{2}\frac{\lambda}{rC_{2}}||f||_{\infty})$$\leq$$C_{1}K^{-1}|| \sigma||\cdot\phi(\frac{\lambda}{r})$

.

(&

65
This inequality shows that$\phi$

satisfies the-condition.

$\Delta_{2}$Now we are ready to prove Theorem 5and 6.
Proof of Theorem 5. Since $SL^{\infty}(\partial\Omega)=BMOA$$\sup${$||Sf||_{\infty}$:$\square$$\supset H^{\infty}$

such that$f\in L^{\infty}$, it follows that$||f||_{\infty}\leq 1$

Therefore we can apply Lemma 2to the Szeg\"o projection S.

$\cdots$$=\infty$.

$\square$:
Proof of Theorem 6. We suppose that $\sup$ {function
holomorphic, $m$ . Now we choose abounded$i=1$ ,. Then we have that$||T_{i}f||_{\infty}$}$f\in H^{\infty}$

such that$h\in H^{\infty}(\Omega)$$||f||_{\infty}\leq 1$such that}$\leq 1$for every$\sum_{i=1}^{m}||f_{i}||_{\infty}<$$||h||_{\infty}\leq 1$$||h||_{\infty}$$\leq$$\sum_{i=1}^{m}||f_{i}||_{\infty}||T_{i}h||_{\infty}$$\leq$$\sum_{i=1}^{m}||f_{i}||_{\infty}$$<$$||h||_{\infty}$.

This is acontradiction. Therefore there exist acertain $k\in\{1, \cdots, m\}$ such that$\sup${$||T_{k}f||_{\infty}$:$f\in H^{\infty}$

Then we can apPly Lemma 2to the operator$T_{k}$.

such that$||f||_{\infty}\leq 1$} $>1$ .

$\square$Acknowledgement. The author would like to thank Professor Hitoshi Arai for valuable discussions.

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