213

Commutators of integral operators with afunction in generalized Campanato spaces
新井 龍太郎 (茨城大学 大学院理工学研究科)
Ryutaro Arai (Department of Mathematics, Ibaraki University)

中井 英一 (茨城大学理学部)
Eiichi Nakai (Department of Mathematics, Ibaraki University)
Dedicated to the memory of Professor Yasuj i Takahashi
1Introduction

This is an announcement of [1].
Let \mathbb{R}^{n} be the n ‐dimensional Euclidean space. Let b\in BMO(\mathbb{R}^{n}) and T be a
Calderón‐Zygmund singular integral operator. In 1976 Coifman, Rochberg andWeiss [3] proved that the commutator [b, T]=bT-Tb is bounded on L^{p}(\mathbb{R}^{n})(1<p<\infty) , that is,\Vert[b, T]f\Vert_{L^{p}}=\Vert bTf-T(bf)\Vert_{Lp}\leq C\Vert b\Vert_{BMO} \Vert f\Vert_{L^{p}},where C is a positive constant independent of b and f . For the fractional integral
operator I_{\alpha} , Chanillo [2] proved the boundedness of [b, I_{\alpha}] in 1982. That is,\Vert[b, I_{\alpha}]f\Vert_{Lq}\leq C\Vert b\Vert_{BMO}\Vert f\Vert_{Lp},where \alpha\in(0, n), p, q\in(1, \infty) and -n/p+\alpha=-n/q . These results wereextended to Morrey spaces by Di Fazio and Ragusa [4] in 1991.
In this talk we discuss the boundedness of the commutators [b, T] and [b, I_{\rho}] on

generalized Morrey spaces with variable growth condition, where T is a Calderón‐
Zygmund operator, I_{\rho} is a generalized fractional integral operator and b is afunction in generalized Campanato spaces with variable growth condition.

We denote by B(x, r) the open ball centered at x\in \mathbb{R}^{n} and of radius r , thatis,B(x, r)=\{y\in \mathbb{R}^{n}:|y-x|<r\}.
For a measurable set G\subset \mathbb{R}^{n} , we denote by|G| and \chi_{G} the Lebesgue measure of
For a function f\in L_{1oc}^{1}(\mathbb{R}^{n})

G and the characteristic function of G , respectively.

and a ball B , let

f_{B}= \int_{B}f=\int_{B}f(y) \'{a} y=\frac{1}{|B|}\int_{B}f(y) dy .

214
For a variable growth function \varphi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) and a ball B=B(x, r)
we write\varphi(B)=\varphi(x, r) .

Definition 1.1. For p\in[1, \infty ) and \varphi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) ) let L^{(p,\varphi)}(\mathbb{R}^{n}) bethe sets of all functions f such that the following functional is finite:\Vert f\Vert_{L(p,\varphi)}(\mathb {R}^{n})=\sup_{B}(\frac{1}{\varphi(B)}\int_ {B}|f(y)|^{p}dy)^{1/p},

where the supremum is taken over all balls B in \mathbb{R}^{n}.

Then \Vert f\Vert_{L(p,\varphi)}(\mathbb{R}^{n}) is a norm and L^{(p,\varphi)}(\mathbb{R}^{n}) is a Banach space. If \varphi_{\lambda}(x, r)=r^{\lambda}
with \lambda\in[-n, 0] , then L^{(p,\varphi_{\lambda})}(\mathbb{R}^{n}) is the classical Morrey spaces. If \lambda=-n , thenL^{(p,\varphi_{-n})}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n}) . If \lambda=0 , then L^{(p,\varphi_{0})}(\mathbb{R}^{n})=L^{\infty}(\mathbb{R}^{n}) .

Definition 1.2. For p\in[1, \infty ) and \varphi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) , let \mathcal{L}^{(p,\varphi)}(\mathbb{R}^{n}) bethe sets of all functions f such that the following functional is finite:\Vert f\Vert_{\mathcal{L}(p,\varphi)}(\mathb {R}^{n})=\sup_{B}(\frac{1} {\varphi(B)}\int_{B}|f(y)-f_{B}|^{p}dy)^{1/p},

where the supremum is taken over all balls B in \mathbb{R}^{n}.

Then \Vert f\Vert_{\mathcal{L}(p,\varphi)}(\mathb {R}^{n}) is a norm modulo constant functions and thereby \mathcal{L}^{(p,\varphi)}(\mathbb{R}^{n})
is a Banach space. If p=1 and \varphi\equiv 1 , then \mathcal{L}^{(p,\varphi)}(\mathbb{R}^{n})=BMO(\mathbb{R}^{n}) . If p=1
and \varphi(r)=r^{\alpha}(0<\alpha\leq 1) , then \mathcal{L}^{(p,\varphi)}(\mathbb{R}^{n})=Lip_{\alpha}(\mathbb{R}^{n}) .
A linear operator T from S(\mathbb{R}^{n}) to S'(\mathbb{R}^{n}) is said to be a Calderón‐Zygmund
operator if T is bounded on L^{2}(\mathbb{R}^{n}) and there exists a standard kernel K suchthat, for f\in C_{comp}^{\infty}(\mathbb{R}^{n}) ,Tf(x)= \int_{\mathbb{R}^{n} K(x, y)f(y)dy, x\not\in
It is known that any Calderón‐Zygmund operator

supp f.

T is bounded onL^{p}(\mathbb{R}^{n}) for

1<p<\infty.

For a function \rho : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) , we consider generalized fractional
integral operators I_{\rho} defined by

I_{\rho}f(x)= \int_{\mathbb{R}^{n} \frac{\rho(x,|x-y|)}{|x-y|^{n} f(y)dy,

where we always assume that

\int_{0}^{1}\frac{\rho(x,t)}{t}dt<\infty

for each x\in \mathbb{R}^{n} .

(1.1)

and that there exist positive constants C, K_{1} and K_{2} with K_{1}<K_{2} such that\sup_{r\leq t\leq 2r}\rho(x, t)\leq C\int_{K_{1}r}^{K_{2}r}\frac{\rho(x,t)}{t} dt

for all x\in \mathbb{R}^{n} and

r>0 .

(1.2)

If \rho(x, r)=r^{\alpha} , then I_{\rho} is the usual fractional integral operator I_{\alpha} . It is known
as the Hardy‐Littlewood‐Sobolev theorem that I_{\alpha} is bounded from Ij^{p}(\mathbb{R}^{n}) toL^{q}(\mathbb{R}^{n}) , if \alpha\in(0, n), p, q\in(1, \infty) and -n/p+\alpha=-n/q.

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2Main results

We say that \theta is almost increasing (resp. almost decreasing) if there exists a
positive constant C such that , for all x\in \mathbb{R}^{n} and r, s\in(0, \infty) ,\theta(x, r)\leq C\theta(x, s)(resp. \theta(x, s)\leq C\theta(x, r) ),if r<s.

In this talk we consider the following classes of \varphi :Definition 2.1. (i) Let \mathcal{G}^{dec} be the set of all functions \varphi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty)
such that \varphi is almost decreasing, and that r\mapsto\varphi(x, r)r^{n} is almost increasing.
(ii) Let \mathcal{G}^{inc} be the set of all functions \varphi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) such that \varphi isalmost increasing, and that r\mapsto\varphi(x, r)/r is almost decreasing.
Let\varphi\in \mathcal{G}^{dec} . If \varphi satisfies\lim_{rar ow 0}\varphi(x, \tau)=\infty, \lim_{rar ow\infty} \varphi(x_{\grave{\tau}}\tau)=0 ,(2.1)

then there exists \overline{\varphi}\in \mathcal{G}^{dec} such that \varphi\sim\overline{\varphi} and that \varphi(x, \cdot) is continuous, strictly
decreasing and bijective from (0, \infty) to itself for each x.
We also consider the following conditions:\exists C>0\forall x, y\in \mathbb{R}^{n}\forall r\in(0, \infty),\frac{1}{C}\leq\frac{\theta(x,r)}{\theta(y,r)}\leq C, if |x-y|\leq r .
\exists C>0\forall x\in \mathbb{R}^{n}\forall r\in(0, \infty)

(2.2)

,

\int_{r}^{\infty}\frac{\varphi(x,t)}{t}dt\leq C\varphi(x, r) .

(2.3)

For functions f in Morrey spaces, we define [b, T]f on each ball B by[b, T]f(x)=[\'{o}, T](f \chi_{2B})(x)+\int_{\mathbb{R}^{n}\backslash 2B}(b(x)-b(y) K(x, y)f(y)dy,x\in B.

Then we have the following theorem.
Theorem 2.1. Let

1<p\leq q<\infty and \varphi, \psi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) . Assume

that \varphi\in \mathcal{G}^{dec} and \psi\in \mathcal{G}^{inc} . Let T be a Calderón‐Zygmund operator.

(i) Assume that \psi satisfy (2.2), that \varphi satisfies (2.3), and that there exists a
positive constant C_{0} such that, for all x\in \mathbb{R}^{n} and r\in(0, \infty) ,\psi(x, r)\varphi(x, r)^{1/p}\leq C_{0}\varphi(x, \ovalbox{\t \small REJECT} r) ^{1/q}.
If b\in \mathcal{L}^{(1,\psi)}(\mathbb{R}^{n}) , then [b, T]f is well defined for all f\in L^{(p,\varphi)}(\mathbb{R}^{n}) and there
exists a positive constant C , independent of b and f , such that

\Vert[b, T]f\Vert_{L(q,\varphi^{-})}\leq C\Vert b\Vert_{\mathcal{L}^{(1,\psi)} \Vert f\Vert_{L(p,\varphi)}.

216(ii) Conversely, assume that \varphi satisfies (2.2) and that there exists a positive
constant C_{0} such that, for all x\in \mathbb{R}^{n} and r\in(0, \infty) ,C_{0}\psi(x, r)\varphi(x, r)^{1/p}\geq\varphi(x, r)^{1/q}.
If T is a convolusion type such that

Tf(x)=p.v. \int_{\mathbb{R}^{n}}K(x-y)f(y)dy
with homogeneous kernel K satisfying K(x)=|x|^{-n}K(x/|x|), \int_{S^{n-1}}K=0
and K\in C^{\infty}(S^{n-1}) and K\not\equiv 0 , and if [b, T] is bounded from L^{(p,\varphi)}(\mathbb{R}^{n})to L^{(q,\varphi)}(\mathbb{R}^{n}) , then b\in \mathcal{L}^{(1,\psi)}(\mathbb{R}^{n}) and there exists a positive constant C,
independent of b , such that

\Vert ó \Vert_{\mathcal{L}(1,\psi)}\leq C\Vert[b, T]\Vert_{L}(p,\varphi)_{arrow}L(q, \varphi) ,where \Vert[b, T]\Vert_{L(p,\varphi)_{arrow}L(q,\varphi)} is the opetator norm of [b, T]fr\cdot omL^{(p,\varphi)}(\mathbb{R}^{n}) toL^{(q,\varphi)}(\mathbb{R}^{n}).

In the above theorem, if \psi\equiv 1 and \varphi(x, r)=r^{-n} , then \mathcal{L}^{(1,\psi)}(\mathbb{R}^{n})=BMO(\mathbb{R}^{n})and L^{(p,\varphi)}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n}) . This is the case of the theorem by Coifman, Rochberg
and Weiss.

If \psi(x, r)=r^{\alpha}, 0<\alpha\leq 1 , and \varphi(x, r)=r^{-n} , then \mathcal{L}^{(1,\psi)}(\mathbb{R}^{n})=Lip_{\alpha}(\mathbb{R}^{n}) ,L^{(p,\varphi)}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n}) and L^{(q,\varphi)}(\mathbb{R}^{n})=L^{q}(\mathbb{R}^{n}) with -n/p+\alpha=-n/q . That is,\Vert[b, T]f\Vert_{Lq}\lessapprox\Vert b\Vert_{Lip}.\Vert f\Vert_{L^{p}}.
This is the case of Janson [5, Lemma 12].
Theorem 2.2. Let

1<p<q<\infty and \rho, \varphi, \psi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) . Assume

that \varphi\in \mathcal{G}^{dec} and \psi\in \mathcal{G}^{inc} . Assume also that \rho satisfies (1.1) and (1.2). Let\rho^{*}(x, r)=\int_{0}^{\gamma}\frac{\rho(x,t)}{t}dt.

(i) Assume that \rho, \rho^{*} and \psi satisfy (2.2), that \varphi satisfies (2.3) and that there
exist positive constants \in, C_{\rho}, C_{0}, C_{1} and an exponent \overline{p}\in(p, q] such that,for all x, y\in \mathbb{R}^{n} and r, s\in(0, \infty) ,C_{\rho} \frac{\rho(x,r)}{r^{n-\in} \geq\frac{\rho(x,s)}{s^{n-\epsilon} , Of

r<s

,

(2.4)

| \frac{\rho(x,r)}{r^{n} -\frac{\rho(y,s)}{s^{n} |\leq C_{\rho}(|r-s|+|x-y|) \frac{\rho^{*}(x,r)}{r^{n+1}} ,
if

(2.5)

\frac{1}{2}\leq\frac{r}{s}\leq 2 and |x-y|<r/2_{\dot{r}}\int_{0}^{\Gamma}\frac{\rho(x,t)}{t}dt\varphi(x の 1/p+ \int_{T}^{\infty}\frac{\rho(x,t)\varphi(x,t)^{1/p} {t}dt\leq C_{0} \varphi(x, t)^{1/\overline{p} , (2.6)\psi(x, r)\varphi(x, r)^{1/\overline{p}}\leq C_{1}\varphi(x, r)^{1/q} .

(2.7)

217
If b\in \mathcal{L}^{(1,\psi)}(\mathbb{R}^{n}) , then [b, I_{\rho}]f is well defined for all f\in L^{(p,\varphi)}(\mathbb{R}^{n}) and there
exists a positive constant C , independent of b and f , such that

\Vert[b, I_{\rho}]f\Vert_{L(q,\varphi)}\leq C\Vert b\Vert_{\mathcal{L}^{(1, \psi)} \Vert f\Vert_{L(p,\varphi)}.
(ii) Conversely, a\mathcal{S}sume that \varphi satisfies (2.2), that \rho(x, r)=r^{\alpha}, 0<\alpha<n , andthat

C_{0}\psi (x, r)r^{\alpha}\varphi(x, r)^{1/p}\geq\varphi(x, r)^{1/q} .

(2.8)

If [b, I_{\alpha}] is bounded from L^{(p,\varphi)}(\mathbb{R}^{n}) to L^{(q,\varphi)}(\mathbb{R}^{n}) , then b\in \mathcal{L}^{(1,\psi)}(\mathbb{R}^{n}) andthere exists a positive constant C , vndependent of b , such that

\Vert b\Vert_{\mathcal{L}(1,\psi)}\leq C\Vert[b, I_{\alpha}]\Vert_{L}(p, \varphi)_{arrow}L(q,\varphi),

where \Vert[b, I_{\alpha}]\Vert_{L(p,\varphi)_{arrow}L(q,\varphi)} is the opetator norm of [b, I_{\alpha}] from L^{(p,\varphi)}(\mathbb{R}^{n}) toL^{(q,\varphi)}(\mathbb{R}^{n})3、

Sketch of proof

We give a sketch of the proof of Theorem 2.2. To prove the theorem we use thefollowing inequality and theorem:

M^{\#}([b, I_{\rho}]f)(x)\leq C\Vert b\Vert_{\mathcal{L}(1,\psi)} ( M_{\psi^{\eta} (|I_{\rho}f|^{\eta})(x) ^{1/\eta}+(M_{(\rho\psi)^{\eta} (|f ^{\eta})(x) ^{1/\eta})
where1<\eta<\infty,,\rho^{*}(x, r)=\int_{0}^{r}\rho(x, t)t^{-1}dt andM^{\#}f(x)= \sup_{B\ni x}\int_{B}|f(y)-f_{B}|dy, M_{\rho}f(x)=\sup_{B\ni x}\rho (B)\int_{B}|f(y)|dy.
Theorem 3.1 (Nakai, 2014). Let p\in[1, \infty ) be a constant and \varphi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) . Assume that there exists a positive constant C such that,\varphi (x, r)\geq C\varphi(x, s) for all x\in \mathbb{R}^{n} and r\in(0, s) .

Then the operator M is bounded from L^{(p,\varphi)}(\mathbb{R}^{n}) to vtself if p\in(1, \infty) .

Theorem 3.2 (Nakai, 2014). Let 1<p<q<\infty and \rho, \varphi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) .
Assume that \rho satisfies (1.1) and (1.2) and that \varphi is in \mathcal{G}^{dcc} and satisfies (2.1).
Assume also that there exists a positive constant C such that, for all x\in \mathbb{R}^{n} andr\in(0, \infty),\int_{0}^{r}\frac{\rho(x,t)}{t}dt\varphi(x, r)^{1/p}+\int_{T}^{\infty} \frac{\rho(x,t)\varphi(x,t)^{1/p} {t}dt\leq C\varphi(x, r)^{1/q}.
Then I_{\rho} is bounded from L^{(p,\varphi)}(\mathbb{R}^{n}) to L^{(q,\varphi)}(\mathbb{R}^{n}) .

218
1<p<q<\infty and \rho, \varphi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) .
\mat
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cal
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}
^
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d
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that \varphi is inand satisfies (2.1). Assume also that

Theorem 3.3. Let\rho(x, r)\varphi(x, r)^{1/p}\leq C_{0}\varphi(x, r)^{1/q} .

Assume(3.1)

Then M_{\rho} is bounded from L^{(p,\varphi)}(\mathbb{R}^{n}) to L^{(q,\varphi)}(\mathbb{R}^{n}) .
Proof. We may assume that \varphi(x, \cdot) is continuous, strictly decreasing and bijective
from(0, \infty) to itself for each x\in \mathbb{R}^{n}.

We prove that, for f\in L^{(p,\varphi)}(\mathbb{R}^{n}) with \Vert f\Vert_{L(p,\varphi)}(\mathbb{R}^{n})=1,

M_{\rho}f(x)\leq CMf(x)^{p/q}, x\in \mathbb{R}^{n} ,(3.2)

for some positive constant C independent of f and x . To prove (3.2) we show
that, for any ball B=B(x, r) ,\rho(B)\int_{B}|f|\leq C_{0}Mf(x)^{p/q} .

(3.3)\varphi(x, u)=Mf(x)^{p} . If r\leq u , then \varphi(B)=\varphi(x, r)\geq
Mf(x)^{p} and \varphi(B)^{1/q-1/p}\leq Mf(x)^{p/q-1} . By (3. 1) we have

Chooseu>0 such that\rho(B)\int_{B}|f|\leq C_{0}\varphi(B)^{1/q-1/p}\int_{B}|f|\leq C_{0}Mf(x) ^{p/q}.
If

r>u

have

, then \varphi(B)=\varphi(x, r)<Mf(x)^{p} and \varphi(B)^{1/q}<Mf(x)^{p/q} . By (3.1) we\rho(B)\int_{B}|f|\leq\rho(B)(\int_{B}|f|^{p})^{1/p}\leq\rho(B)\varphi(B) ^{1/p}\leq C_{0}\varphi(B)^{1/q}\leq C_{0}Mf(x)^{p/q}.
Then we have (3.3) and the conclusion.
Proposition 3.4. Let 1\leq p<\infty and \varphi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) .

f\in L_{1oc}^{1}(\mathbb{R}^{n})\squareThen, for,

\Vert f\Vert_{\mathcal{L}(p,\varphi)}\leq C\Vert M\# f\Vert_{L(p,\varphi)} ,(3.4)

where C is a positive constant independent of f.
Corollary 3.5. Let 1\leq p<\infty and \varphi : \mathbb{R}^{n}\cross(0, \infty)arrow(0, \infty) . Assume that\varphi\in \mathcal{G}^{dcc} and that \varphi satisfies (2.3). For f\in L_{1oc}^{1}(\mathbb{R}^{n}), if \dot
1 { \imath} mf_{B(0,r)}rarrow\infty=0 , then\Vert f\Vert_{L(p,\varphi)}\leq C\Vert M\# f\Vert_{L(p,\varphi)} ,(3.5)

where C is a positive constant independent of f.

Lemma 3.6 ([8, Theorem 2.1 and Remark 2.1]). Let p\in[1, \infty ) and \varphi is in \mathcal{G}^{dec}and satisfies (2.3). Then, for every f\in \mathcal{L}^{(p,\varphi)}(\mathbb{R}^{n}), f_{B(0,r)} converges as rarrow\infty
and\Vert f-1\dot{ \imath} mf_{B(0,r)}\Vert_{L(p,\varphi)}\sim rar ow\infty\Vert f\Vert_{\mathcal{L}(p,\varphi)},

219
For any cube Q\subset \mathbb{R}^{n} centered at a\in \mathbb{R}^{n} and with sidelength
denote by \mathcal{Q}^{dy}(Q) the set of all dyadic cubes with respect to Q.
For any cube Q\subset \mathbb{R}^{n} , let

2r>0 ,

we

M_{Q}^{dy}f(x)= \sup_{R\in \mathcal{Q}^{dy}(Q),x\in R\subset Q}\int_{Q}|f(y) |dy_{\mathfrak{i}M_{Q}^{\#,dy}f(x)= \sup_{R\in \mathcal{Q}^{dy}(Q),x\in R\subset Q}\int_{Q}|f(y) -f_{Q}|dy.
Lemma 3.7 (Tsutsui, 2011 Komori, 2015). Let Q be a cube and f\in L^{1}(Q) .
Then, for any 0<\gamma\leq 1 and \lambda>|f|_{Q},|\{x\in Q: M_{Q}^{dy}f(x)>2\lambda, M_{Q}^{\#,dy}f(x)\leq\gamma\lambda\}|\leq 2^{n}\gamma|\{x\in Q:M_{Q}^{dy}f(x)>\lambda\}| .

(3.6)

Lemma 3.8. There exists a positive constant C , for any cube Q and any function

f\in L^{1}(Q),\Vert f-f_{Q}\Vert_{L^{p}(Q)}\leq C\Vert M_{Q}^{\#,dy}f\Vert_{L^{p}(Q)}.

Proof. By the good \lambda inequality (3.6) and the standard argument we have thefollowing boundedness: There exists a positive constant C , for any cube Q andany function f\in L^{1}(Q) ,\Vert M_{Q}^{dy}f\Vert_{L^{p}(Q)}\leq C(\Vert M_{Q}^{\#,dy}f\Vert_{L^{p}(Q)}+ |Q|^{1/p}|f|_{Q})Actually, for any L>2|f|_{Q},\int_{0}^{L}p\lambda^{p-1}|\{x\in Q:M_{Q}^{dy}f(x)>\lambda\}|d\lambda= \int_{0}^{2|f _{Q} p\lambda^{p-1}|\{x\in Q:M_{Q}^{dy}f(x)>\lambda\}|d\lambda+ \int_{2|f _{Q} ^{ゐ}p\lambda^{p-1}|\{x\in Q:M_{Q}^{dy}f(x)>\lambda\}|d\lambda\leq(2|f _{Q})^{p}|Q|+2^{p}\int_{|f _{Q} ^{L/2}p\lambda^{p-1}|\{x\in Q:M_{Q} ^{dy}f(x)>2\lambda\}|d\lambda.

By the good \lambda inequality (3.6) we have

(3.7)

220

2^{P} \int_{|f _{Q} ^{L/2}p\lambda^{p-1}|\{x\in Q: M_{Q}^{dy}f(x)>2\lambda\}|d \lambda\leq 2^{n+p}\gamma\int_{|f_{Q} ^{L/2}p\lambda^{p-1}|\{x\in Q:M_{Q}^{dy}f(x) >\lambda\}|d\lambda+2^{P} \int_{|f_{Q} ^{L/2}p\lambda^{p-1}|\{x\in Q:M_{Q}^{\#.dy}f(x) >\gamma\lambda\}|d\lambda\leq 2^{n+p}\gamma\int_{0}^{L}p\lambda^{p-1}|\{x\in Q:M_{Q}^{dy}f(x)>\lambda\} |d\lambda+2^{p} \gamma^{-p}\int_{0}^{\infty})>\lambda\}|d\lambda.

Then, for small \gamma>0,(1-2^{n+p} \gamma)\int_{0}^{L}p\lambda^{p-1}|\{x\in Q:M_{Q}^{dy}f(x)>\lambda\} |d\lambda\leq(2|f|_{Q})^{p}|Q|+2^{p}\gamma^{-p}\int_{0}^{\infty}p\lambda^{p-1}|\{x\in Q:M_{Q}^{\#,dy}f(x)>\lambda\}|d\lambda.

Letting Larrow\infty , we have (3.7). Substitute f-f_{Q} for f in (3.7). Then

\Vert f-f_{Q}\Vert_{L^{p}(Q)}\leq\Vert M_{Q}^{dy}(f-f_{Q})\Vert_{L^{p}(Q)}\les approx\Vert M_{Q}^{\#,dy}f\Vert_{L^{p}(Q)}+|Q|^{1/p}\int_{Q}|f-f_{Q}|\leq\Vert M_{Q}^{\#,dy}f\Vert_{Lp(Q)}+|Q|^{1/p}\inf_{x\in Q}M_{Q}^{\#,dy}f(x)

.

Since

|Q|^{1/p} \inf_{x\in Q}M_{\dot{Q} ^{\#dy}f(x)=(\int_{Q}[\inf_{x\in Q}M_{Q}^{\#, dy}f(x)]^{p}dy)^{1/p}\leq\Vert M_{Q}^{\#,dy}f\Vert_{Lp(Q)},

we have the conclusion.

\squareProof of Proposition 3.4. For any ball B=B(x, r) , take the cube Q centered atx and with sidelength 2r . Then B\subset Q . By Lemma 3.8 we have

( \frac{1}{\varphi(B)}\int_{B}|f- _{B}|^{p})^{1/p}\leq(\frac{2}{\varphi'(B)} \frac{|Q|}{|B|}\int_{Q}|f- _{Q}|^{p})^{1/p}\les ap rox(\frac{1}{(\varphi(B)}\int_{Q}(M_{Q}^{\#,dy}f)^{p})^{1/p}\les approx\Vert M^{\#}f\Vert_{L(p,\varphi)}(\mathbb{R}^{n})This shows the conclusion.

.

\square221
221

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