The boundedness of multilinear operators on generalized Morrey spaces over the quasi-metric space of non-homogeneous type

Abstract

In this paper we study the multilinear fractional integral operators, the multilinear Calderón-Zygmund operators and the multi-sublinear maximal operators defined on the quasi-metric space with non-doubling measure. We obtain the boundedness of these operators on the generalized Morrey spaces over the quasi-metric space of non-homogeneous type.

MSC:42B20, 42B25, 42B35.

1 Introduction and main results

The boundedness of fractional integral operators on the classical Morrey spaces was studied by Adams [1], Chiarenza and Frasca et al. [2]. In [2], by establishing a pointwise estimate of fractional integrals in terms of the Hardy-Littlewood maximal function, they showed the boundedness of fractional integral operators on the Morrey spaces. In 2005, Sawano and Tanaka [3] gave a natural definition of Morrey spaces for Radon measures which might be non-doubling but satisfied the growth condition, and they investigated the boundedness in these spaces of some classical operators in harmonic analysis. Later on, Sawano [4] defined the generalized Morrey spaces on ${\mathbb{R}}^n$ for non-doubling measure and showed the properties of maximal operators, fractional integral operators and singular operators in this space.

Simultaneously, in 1999, Kenig and Stein [5] gave the boundedness for multilinear fractional integrals on Lebesgue spaces. In 2002, Grafakos and Torres [6] obtained the boundedness for multilinear Calderón-Zygmund operators on Lebesgue spaces. From then on, the theory on multilinear integral operators has attracted much attention as a rapidly developing field in harmonic analysis. Recently, the authors have studied the boundedness of multilinear fractional integrals on Herz-Morrey spaces in [7–10] and the boundedness of multilinear Calderón-Zygmund operators on the Morrey-type spaces in [11–14]. Particularly, the authors [7, 11, 13, 15] established the boundedness for the multilinear operators on Morrey spaces over ${\mathbb{R}}^n$ with non-doubling measures. In this paper, we focus on the multilinear operators on generalized Morrey spaces over quasi-metric space $(X,\rho ,\mu )$ of non-homogeneous type and extend the works in [3–7, 11, 16].

Let $(X,\rho )$ be a quasi-metric space with the quasi-metric function $\rho :X\to [0,\mathrm{\infty })$ satisfying the conditions:

• (1) $\rho (x,y)>0$ for all $x\ne y$, and $\rho (x,x)=0$ for all $x\in X$.

• (2) There exists a constant $a_0\ge 1$ such that $\rho (x,y)\le a_0\rho (y,x)$ for all $x,y\in X$.

• (3) There exists a constant $a_1\ge 1$ such that

  $$\rho (x,y)\le a_1(\rho (x,z)+\rho (z,y))$$

  (1.1)

for all $x,y,z\in X$.

Here we point out that there is no notion of dyadic cubes on the quasi-metric space and so the method for ${\mathbb{R}}^n$ used in [15] does not work on $(X,\rho )$. Recently, Hytönen [17] introduced the notion of geometrically doubling space.

Definition 1.1 The quasi-metric space $(X,\rho )$ is called geometrically doubling if there exists some $N_0\in \mathbb{N}$ such that any ball $B(x,r)\subset X$, where $B(x,r):=\{y\in X:\rho (x,y)<r\}$ with the center x and the radius r, can be covered by at most $N_0$ balls $B(x_i,r/2)$ with $x_i\in B(x,r)$.

Remark 1.2 Similarly as Hytönen showed in Lemma 2.3 in [17], one can deduce that if the quasi-metric space $(X,\rho )$ is geometrically doubling, then, for any $\delta \in (0,1)$, any ball $B(x,r)\subset X$ can be covered by at most $N_0{\delta }^{-n}$ balls $B(x_i,\delta r)$ with $x_i\in B(x,r)$, where $n={log}_2{N}_0$.

Given a Borel measure μ on the quasi-metric space $(X,\rho )$ such that μ is finite on bounded sets, and let $(X,\rho )$ be geometrically doubling, then continuous, boundedly supported functions are dense in $L^p(X,\mu )$ for $p\in [1,\mathrm{\infty })$. See Proposition 3.4 in [17] for details.

The above triple $(X,\rho ,\mu )$ will be called a quasi-metric space of non-homogeneous type if the measure μ satisfies the following growth condition,

$$\mu (B(x,r))\le C_{0}r$$

(1.2)

with the constant $C_0$ independent of the ball $B(x,r)\subset X$. The set of all balls $B\subseteq X$ satisfying $\mu (B)>0$ is denoted by $\mathcal{B}(\mu )$. We know that the analysis on non-homogeneous spaces plays important roles in solving the Painlevé problem as well as the Vitushkin conjecture [18, 19]. For motives of developing analysis on non-homogeneous spaces and more examples, one could see [20].

Now we give the definition of the generalized Morrey space over $(X,\rho ,\mu )$, which is a generalization of the classical Morrey space. Here we remark that Morrey spaces play important roles in the study of partial differential equations.

Definition 1.3 Let $1\le p<\mathrm{\infty }$ and a function $\varphi :(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ be such that $r^{\frac{1}{p}}\varphi (r)$ is non-decreasing. The generalized Morrey space $L^{p,\varphi }(X,k,\mu )$ over X, where $k>a_1$, is defined as

$$L^{p,\varphi }(X,k,\mu ):=\{f\in L_{\mathrm{loc}}^p(\mu ):{\parallel f\parallel }_{L^{p,\varphi }(X,k,\mu )}<\mathrm{\infty }\}$$

with the norm ${\parallel f\parallel }_{L^{p,\varphi }(X,k,\mu )}$ given by

$${\parallel f\parallel }_{L^{p,\varphi }(X,k,\mu )}:=\underset{B\in \mathcal{B}(\mu )}{sup}\frac{1}{\varphi (\mu (kB))}{(\frac{1}{\mu (kB)}{\int }_B{|f(x)|}^{p}d\mu (x))}^{\frac{1}{p}},$$

where kB is the ball with the same center and k times radius of the ball B.

In case $\varphi (r)=r^{-\frac{1}{q}}$, $1\le p<q<\mathrm{\infty }$, the space $L^{p,\varphi }(X,k,\mu )$ becomes the classical Morrey space $L^{p,q}(X,k,\mu )$ over X. Particularly, $L^{p,p}(X,k,\mu )=L^p(X,\mu )$.

Remark 1.4 It is worth to point out that if $k_1,k_2>a_1$, then $L^{p,\varphi }(X,k_1,\mu )$ and $L^{p,\varphi }(X,k_2,\mu )$ coincide as a set and their norms are mutually equivalent. This can be observed by the same arguments used in [4]. For the sake of convenience, we provide the detail. Let $a_1<k_1\le k_2$. Then the inclusion $L^{p,\varphi }(X,k_1,\mu )\subseteq L^{p,\varphi }(X,k_2,\mu )$ is obvious by that fact that $r^{\frac{1}{p}}\varphi (r)$ is non-decreasing. To see the reverse inclusion, let $f\in L^{p,\varphi }(X,k_2,\mu )$ and $B(x,r)\in \mathcal{B}(\mu )$ be fixed. It is sufficient to estimate

$$I=\frac{1}{\varphi (\mu (B(x,k_{1}r)))}{(\frac{1}{\mu (B(x,k_{1}r))}{\int }_{B(x,r)}{|f(x)|}^{p}d\mu (x))}^{\frac{1}{p}}.$$

The geometrically doubling condition shows that the ball $B(x,r)$ can be covered by at most $N=N_0{\delta }^{-n}$ balls $B(x_i,\delta r)$ with $x_i\in B(x,r)$ for any $\delta \in (0,1)$. Moreover, by the quasi-triangle inequality (1.1), we can see that $B(x_i,k_2\delta r)\subseteq B(x,k_{1}r)$ if we choose $0<\delta <(k_1-a_1)/(a_1{k}_2)$. Thus,

$$\begin{array}{rl}{I}^p& \le \sum _{i=1}^N\frac{1}{\varphi {(\mu (B(x,k_{1}r)))}^p\mu (B(x,k_{1}r))}{\int }_{B(x_i,\delta r)}{|f(x)|}^{p}d\mu (x)\\ \le \sum _{i:B(x_i,\delta r)\in \mathcal{B}(\mu )}\frac{1}{\varphi {(\mu (B(x_i,k_2\delta r)))}^p\mu (B(x_i,k_2\delta r))}{\int }_{B(x_i,\delta r)}{|f(x)|}^{p}d\mu (x)\\ \le N{\left({\parallel f\parallel }_{L^{p,\varphi }(X,k_2,\mu )}\right)}^p,\end{array}$$

which implies that $L^{p,\varphi }(X,k_1,\mu )=L^{p,\varphi }(X,k_2,\mu )$ for any $k_1,k_2>a_1$. With this fact in mind, we sometimes omit parameter k in $L^{p,\varphi }(X,k,\mu )$, i.e., write it by $L^{p,\varphi }(X,\mu )$.

In this article, we consider the multilinear fractional integral operator, the multilinear Calderón-Zygmund operator and the multi-sublinear maximal operator. The multilinear fractional integral is defined by

$${\mathcal{I}}_{\alpha ,m}(f_1,\dots ,f_m)(x)={\int }_{X^m}\frac{f_1(y_1)\cdots f_m(y_m)}{{(\rho (x,y_1)+\cdots +\rho (x,y_m))}^{m-\alpha }}d\mu (y_1)\cdots d\mu (y_m),$$

where $0<\alpha <m$. When $m=1$, we denote ${\mathcal{I}}_{\alpha ,m}$ by ${\mathcal{I}}_{\alpha }$.

Let be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values into the space of tempered distributions. Following [6], we say that is an m-linear Calderón-Zygmund operator if it extends to a bounded multilinear operator from $L^{p_1}(X,\mu )\times L^{p_2}(X,\mu )\times \cdots \times L^{p_m}(X,\mu )$ to $L^p(X,\mu )$ for some $1\le p_1,\dots ,p_m<\mathrm{\infty }$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}+\cdots +\frac{1}{p_m}$, and if there exists a kernel function , the so-called multilinear Calderón-Zygmund kernel, defined away from the diagonal $x=y_1=\cdots =y_m$ in $X^{m+1}$, satisfying

$$\mathcal{T}(f_1,\dots ,f_m)(x)={\int }_{X^m}\mathcal{K}(x,y_1,\dots ,y_m)f_1(y_1)\cdots f_m(y_m)d\mu (y_1)\cdots d\mu (y_m)$$

for all $x\notin {\bigcap }_{i=1}^{m}supp{f}_i$, where $f_i$’s are smooth functions with compact support; and the kernel function satisfies the size condition

$$|\mathcal{K}(x,y_1,y_2,\dots ,y_m)|\le C{(\sum _{i=1}^m\rho (x,y_i))}^{-m}$$

(1.3)

and some smoothness conditions; see [6, 16] for details. In fact, as for the m-linear Calderón-Zygmund operator , we assume that, by a similar argument as that in [6, 16] for the case $X={\mathbb{R}}^n$, if $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}+\cdots +\frac{1}{p_m}$, then the m-linear Calderón-Zygmund operator satisfies

$$\mathcal{T}:L^{p_1}(X,\mu )\times L^{p_2}(X,\mu )\times \cdots \times L^{p_m}(X,\mu )\to L^p(X,\mu )$$

for any $1<p_1,p_2,\dots ,p_m<\mathrm{\infty }$.

We will also consider the multi-sublinear maximal operator ${\mathcal{M}}_{\kappa }$, for $\kappa >a_1^2$, defined by

$${\mathcal{M}}_{\kappa }(f_1,\dots ,f_m)(x)=\underset{x\in B\in \mathcal{B}(\mu )}{sup}\prod _{i=1}^m\frac{1}{\mu (\kappa B)}{\int }_B|f_i(y_i)|d\mu (y_i).$$

In case $m=1$, we denote it by $M_{\kappa }$.

The main result of this paper can be stated as follows.

Theorem 1.5 Let $0<\alpha <m$ and $1<p_i<\mathrm{\infty }$, and let $\frac{1}{q}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}-\alpha >0$. For each $i=1,\dots ,m$, let ${\varphi }_i:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ satisfy

$$\frac{1}{C_1}\le \frac{{\varphi }_i(t)}{{\varphi }_i(r)}\le C_1\mathit{\text{if}}1\le \frac{t}{r}\le 2,$$

(1.4)

and

$${\int }_r^{\mathrm{\infty }}{t}^{\alpha /m-1}{\varphi }_i(t)dt\le C_2{r}^{\alpha /m}{\varphi }_i(r)$$

(1.5)

with positive constants $C_1$ and $C_2$ independent of $r>0$. Then there exists a constant C independent of any admissible $f_i$ such that

$${\parallel {\mathcal{I}}_{\alpha ,m}(f_1,\dots ,f_m)\parallel }_{L^{q,\psi }(X,\mu )}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )},$$

where $\psi (t)=t^{\alpha }{\varphi }_1(t){\varphi }_2(t)\cdots {\varphi }_m(t)$.

If we take ${\varphi }_i(t)=t^{-\frac{1}{l_i}}$ and $0<l_i<\frac{m}{\alpha }$, then ${\varphi }_i$ satisfies conditions (1.4) and (1.5). We remark that if condition (1.4) is replaced by

$${\varphi }_i(u)\le C_3{\varphi }_i(v)\text{for }u\ge v$$

(1.6)

with the constant $C_3>0$, then the theorem is also valid. This can be seen from the proof of the theorem in the next section. Theorem 1.5 yields the following corollary.

Corollary 1.6 Let $0<\alpha <m$ and $1<p_i<\mathrm{\infty }$, and let $\frac{1}{q}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}-\alpha >0$. Let $0<l_i<\frac{m}{\alpha }$ and $\frac{1}{h}=\frac{1}{l_1}+\cdots +\frac{1}{l_m}-\alpha$. Then there exists a constant C independent of $f_i$ such that

$${\parallel {\mathcal{I}}_{\alpha ,m}(f_1,\dots ,f_m)\parallel }_{L^{q,h}(X,\mu )}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,l_i}(X,\mu )}.$$

Theorem 1.7 Let $1<p_i<\mathrm{\infty }$ and $\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}<1$. If the functions ${\varphi }_i:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ satisfy condition (1.4) or (1.6), and satisfy

$${\int }_r^{\mathrm{\infty }}{\varphi }_i{(t)}^{\frac{p_i}{p}}\frac{dt}{t}\le C_4{\varphi }_i{(r)}^{\frac{p_i}{p}}$$

(1.7)

with the constant $C_4$ independent of $r>0$, then there exists a constant C independent of any admissible $f_i$ such that

$${\parallel \mathcal{T}(f_1,\dots ,f_m)\parallel }_{L^{p,\varphi }(X,\mu )}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )},$$

(1.8)

where $\varphi (t)={\varphi }_1(t){\varphi }_2(t)\cdots {\varphi }_m(t)$.

Here we point out that if each $f_i\in L^{p_i}(X,\mu )\cap L^{p_i,{\varphi }_i}(X,\mu )$, then the multilinear Calderón-Zygmund operator $\mathcal{T}(f_1,\dots ,f_m)$ is well defined, and we will prove estimate (1.8) with the absolute constant C independent of these admissible functions. More remarks on the admissibility for $f_i\in L^{p_i,{\varphi }_i}(X,\mu )$ will be given in Remark 3.1 in Section 3.

Observe that ${\varphi }_i(t)=t^{-\frac{1}{l_i}}$, for any $0<l_i<\mathrm{\infty }$, satisfies the conditions in the theorem, thus the corollary follows.

Corollary 1.8 Let $1<p_i<\mathrm{\infty }$ and $\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}<1$. Let $0<l_i<\mathrm{\infty }$ and $\frac{1}{l}=\frac{1}{l_1}+\cdots +\frac{1}{l_m}$. Then there exists a constant C independent of $f_i$ such that

$${\parallel \mathcal{T}(f_1,\dots ,f_m)\parallel }_{L^{p,l}(X,\mu )}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,l_i}(X,\mu )}.$$

Theorem 1.9 Assume that ${\mathcal{M}}_{\kappa }$ is a multi-sublinear maximal operator. Let $1<p_i<\mathrm{\infty }$, $\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}<1$, and ${\varphi }_i$ satisfy condition (1.6). Then there exists a constant C independent of any admissible $f_i$ such that

$${\parallel {\mathcal{M}}_{\kappa }(f_1,\dots ,f_m)\parallel }_{L^{p,\varphi }(X,\mu )}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )},$$

where $\varphi (t)={\varphi }_1(t)\cdots {\varphi }_m(t)$.

We notice that the results above are new even for the case of Euclidean spaces. Throughout this paper, the letter C always denotes a positive constant that may vary at each occurrence but is independent of the essential variable.

2 Proof of Theorem 1.5

Let us first give some requisite theorems and lemmas.

Theorem 2.1 [21]

Let $0<\alpha <1$, $1<p<\frac{1}{\alpha }$ and $\frac{1}{q}=\frac{1}{p}-\alpha$, then the operator ${\mathcal{I}}_{\alpha }$ is bounded from $L^p(X,\mu )$ into $L^q(X,\mu )$ if and only if $\mu (B(x,r))\le Cr$, where the constant C is independent of x and r.

Lemma 2.2 [13]

Suppose that μ is a Borel measure on X with the growth condition (1.2). Let $\frac{1}{q}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}-\alpha >0$ with $0<\alpha <m$ and $1\le p_j\le \mathrm{\infty }$. Then

• (a) if each $p_j>1$,

  $${\parallel {\mathcal{I}}_{\alpha ,m}(f_1,\dots ,f_m)\parallel }_{L^q(X,\mu )}\le C\prod _{j=1}^m{\parallel f_j\parallel }_{L^{p_j}(X,\mu )};$$

• (b) if $p_j=1$ for some j,

  $${\parallel {\mathcal{I}}_{\alpha ,m}(f_1,\dots ,f_m)\parallel }_{L^{q,\mathrm{\infty }}(X,\mu )}\le C\prod _{j=1}^m{\parallel f_j\parallel }_{L^{p_j}(X,\mu )}.$$

Proof This lemma can follow the same argument that, for the classical setting, was given by Kenig and Stein [5]. We may assume that all $1\le p_i<\mathrm{\infty }$. One can find $0<{\alpha }_i<1/p_i$ such that $\alpha ={\sum }_{i=1}^m{\alpha }_i$. Set $1/q_i=1/p_i-{\alpha }_i$, since $1/q={\sum }_{i=1}^{m}1/q_i$, $0<{\alpha }_i\le 1$, $1<q_i<\mathrm{\infty }$, and

$$\rho {(x,y_1)}^{1-{\alpha }_1}\rho {(x,y_2)}^{1-{\alpha }_2}\cdots \rho {(x,y_m)}^{1-{\alpha }_m}\le {(\rho (x,y_1)+\cdots +\rho (x,y_m))}^{m-\alpha }.$$

It follows that

$${\mathcal{I}}_{\alpha ,m}(f_1,\dots ,f_m)(x)\le \prod _{i=1}^m{\mathcal{I}}_{{\alpha }_i}(f_i)(x).$$

Then, by the Hölder inequality and Theorem 2.1, we obtain the lemma. In fact, one could also get the lemma from [13] (or Remark 1.3, p.290 in [13]). □

Now we give the proof of Theorem 1.5.

Proof of Theorem 1.5 Let $B=B(x_0,r)$ be the ball in $B\in \mathcal{B}(\mu )$, with center $x_0\in X$ and radius $r>0$, and let $B^{\ast }=B(x_0,2a_{1}r)$. For $f_i\in L^{p_i,{\varphi }_i}(X,\mu )$, we split it as $f_i=f_i^0+f_i^{\mathrm{\infty }}$, where $f_i^0=f_i{\chi }_{B^{\ast }}$ for $i=1,\dots ,m$. Using this decomposition, we get

$$|{\mathcal{I}}_{\alpha ,m}(f_1,\dots ,f_m)(x)|\le |{\mathcal{I}}_{\alpha ,m}(f_1^0,\dots ,f_m^0)(x)|+{\sum }^{\prime }|{\mathcal{I}}_{\alpha ,m}(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})(x)|,$$

where each term in ∑^′ contains at least one ${\tau }_i=\mathrm{\infty }$.

Then it suffices to show

$$\frac{1}{\psi (\mu (k{B}^{\ast }))}{\left(\frac{1}{\mu (k{B}^{\ast })}{\int }_B{\left|{\mathcal{I}}_{\alpha ,m}(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})\right|}^{q}d\mu \right)}^{\frac{1}{q}}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}$$

(2.1)

for some $k>a_1$ and for each ${\tau }_i\in \{0,\mathrm{\infty }\}$.

Let us first estimate for the case ${\tau }_1=\cdots ={\tau }_m=0$. From the definition of $L^{p_i,{\varphi }_i}(X,\mu )$ we have

$${({\int }_{B^{\ast }}{|f_i(x)|}^{p_i}d\mu (x))}^{\frac{1}{p_i}}\le C{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}{\varphi }_i\left(\mu \left(k{B}^{\ast }\right)\right)\mu {\left(k{B}^{\ast }\right)}^{\frac{1}{p_i}}.$$

(2.2)

From this and by the $L^{p_1}(X,\mu )\times \cdots \times L^{p_m}(X,\mu )\to L^q(X,\mu )$ boundedness of ${\mathcal{I}}_{\alpha ,m}$, Lemma 2.2, we have

$$\begin{array}{r}{(\frac{1}{\mu (k{B}^{\ast })}{\int }_B{|{\mathcal{I}}_{\alpha ,m}(f_1^0,\dots ,f_m^0)(x)|}^{q}d\mu (x))}^{\frac{1}{q}}\\ \le \frac{C}{\mu {(k{B}^{\ast })}^{\frac{1}{q}}}\prod _{i=1}^m{\parallel f_i^0\parallel }_{L^{p_i}(X,\mu )}\le C\mu {\left(k{B}^{\ast }\right)}^{\alpha }\varphi \left(\mu \left(k{B}^{\ast }\right)\right)\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\\ \le C\psi \left(\mu \left(k{B}^{\ast }\right)\right)\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )},\end{array}$$

which implies that in the case all ${\tau }_i=0$, inequality (2.1) holds.

To estimate (2.1) for the case ${\tau }_1=\cdots ={\tau }_m=\mathrm{\infty }$, let $x\in B$, then

$$|{\mathcal{I}}_{\alpha ,m}(f_1^{\mathrm{\infty }},\dots ,f_m^{\mathrm{\infty }})(x)|\le {\int }_{{(X\setminus B^{\ast })}^m}\frac{|f_1(y_1)\cdots f_m(y_m)|d\mu (y_1)\cdots d\mu (y_m)}{{(\rho (x,y_1)+\cdots +\rho (x,y_m))}^{m-\alpha }}.$$

Note, for $x\in B$ and $y_i\in X\setminus B^{\ast }$, we get by (1.1) that

$$2a_{1}r<\rho (x_0,y_i)\le a_1(\rho (x_0,x)+\rho (x,y_i))\le a_1(r+\rho (x,y_i)),$$

hence $\rho (x,y_i)\ge r$. This and condition (1.2) of the measure μ imply that

$$\rho (x,y_i)\ge r\ge {(2a_1{C}_{0}k)}^{-1}\mu \left(k{B}^{\ast }\right):=r^{\ast }$$

and so

$${(X\setminus B^{\ast })}^m\subseteq \prod _{i=1}^m\{y_i:\rho (x,y_i)\ge r^{\ast }\}\subseteq \{(y_1,y_2,\dots ,y_m):\sum _{i=1}^m\rho (x,y_i)\ge r^{\ast }\}.$$

Hence we can derive that

$$\begin{array}{c}|{\mathcal{I}}_{\alpha ,m}(f_1^{\mathrm{\infty }},\dots ,f_m^{\mathrm{\infty }})(x)|\hfill \\ \le {\int }_{{\sum }_{i=1}^m\rho (x,y_i)\ge r^{\ast }}\frac{|f_1(y_1)\cdots f_m(y_m)|d\mu (y_1)\cdots d\mu (y_m)}{{(\rho (x,y_1)+\cdots +\rho (x,y_m))}^{m-\alpha }}\hfill \\ =\sum _{j=0}^{\mathrm{\infty }}{\int }_{2^j{r}^{\ast }\le {\sum }_{i=1}^m\rho (x,y_i)<2^{j+1}{r}^{\ast }}\frac{|f_1(y_1)\cdots f_m(y_m)|d\mu (y_1)\cdots d\mu (y_m)}{{(\rho (x,y_1)+\cdots +\rho (x,y_m))}^{m-\alpha }}\hfill \\ \le \sum _{j=0}^{\mathrm{\infty }}\frac{1}{{(2^j{r}^{\ast })}^{m-\alpha }}\prod _{i=1}^m{\int }_{\rho (x,y_i)<2^{j+1}{r}^{\ast }}|f_i(y_i)|d\mu (y_i).\hfill \end{array}$$

Using the Hölder inequality and the inequality similar to (2.2), we can see that the inequality above can be controlled by

$$\begin{array}{r}C\sum _{j=0}^{\mathrm{\infty }}\frac{1}{{(2^j{r}^{\ast })}^{m-\alpha }}\prod _{i=1}^m{({\int }_{B(x,2^{j+1}{r}^{\ast })}{|f_i(y_i)|}^{p_i}d\mu (y_i))}^{\frac{1}{p_i}}{\left(\mu \left(B(x,2^{j+1}{r}^{\ast })\right)\right)}^{1-\frac{1}{p_i}}\\ \le C\sum _{j=0}^{\mathrm{\infty }}\frac{1}{{(2^j{r}^{\ast })}^{m-\alpha }}\prod _{i=1}^m{\varphi }_i\left(\mu \left(B(x,k{2}^{j+1}{r}^{\ast })\right)\right)\mu \left(B(x,k{2}^{j+1}{r}^{\ast })\right){\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\\ \le C\sum _{j=0}^{\mathrm{\infty }}\frac{1}{{(2^j{r}^{\ast })}^{m-\alpha }}\prod _{i=1}^m{\varphi }_i\left(2^j\mu \left(k{B}^{\ast }\right)\right)2^j\mu \left(k{B}^{\ast }\right){\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\\ \le C\prod _{i=1}^m\left[\sum _{j=0}^{\mathrm{\infty }}{\left(2^j\mu \left(k{B}^{\ast }\right)\right)}^{\alpha /m}{\varphi }_i\left(2^j\mu \left(k{B}^{\ast }\right)\right)\right]{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )},\end{array}$$

where, in the second inequality, we have utilized the non-decreasing of function $r^{\frac{1}{p_i}}{\varphi }_i(r)$ and the fact $\mu (B(x,k{2}^{j+1}{r}^{\ast }))\le C_{0}k{2}^{j+1}{r}^{\ast }\le 2^j\mu (k{B}^{\ast })$. Recall conditions (1.4) (or (1.6)) and (1.5) for the function ${\varphi }_i$, one sees that

$$\begin{array}{rl}\sum _{j=0}^{\mathrm{\infty }}{\left(2^j\mu \left(k{B}^{\ast }\right)\right)}^{\alpha /m}{\varphi }_i\left(2^j\mu \left(k{B}^{\ast }\right)\right)& \le C\sum _{j=0}^{\mathrm{\infty }}{\int }_{2^j\mu (k{B}^{\ast })}^{2^{j+1}\mu (k{B}^{\ast })}{t}^{\alpha /m-1}{\varphi }_i(t)dt\\ \le C{\int }_{\mu (B^{\ast })}^{\mathrm{\infty }}{t}^{\alpha /m-1}{\varphi }_i(t)dt\le C\mu {\left(k{B}^{\ast }\right)}^{\alpha /m}{\varphi }_i\left(\mu \left(k{B}^{\ast }\right)\right).\end{array}$$

Hence we obtain the pointwise estimate

$$|{\mathcal{I}}_{\alpha ,m}(f_1^{\mathrm{\infty }},\dots ,f_m^{\mathrm{\infty }})(x)|\le C\psi \left(\mu \left(k{B}^{\ast }\right)\right)\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}$$

for $x\in B$, which follows from inequality (2.1) for the case all ${\tau }_i=\mathrm{\infty }$.

It is left to consider the case ${\tau }_{i_1}=\cdots ={\tau }_{i_l}=0$ for some $\{i_1,\dots ,i_l\}\subset \{1,\dots ,m\}$ and $1\le l<m$. For this case, we can write for $x\in B$ that

$$\begin{array}{rl}|{\mathcal{I}}_{\alpha ,m}(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})(x)|& \le {\int }_{X^m}\frac{|f_1(y_1)\cdots f_m(y_m)|d\mu (y_1)\cdots d\mu (y_m)}{{(\rho (x,y_1)+\cdots +\rho (x,y_m))}^{m-\alpha }}\\ \le {\int }_{{(B^{\ast })}^l}\prod _{i\in \{i_1,\dots ,i_l\}}|f_i(y_i)|d\mu (y_i)\cdot {\int }_{{(X\setminus B^{\ast })}^{m-l}}\frac{{\prod }_{i\notin \{i_1,\dots ,i_l\}}|f_i(y_i)|d\mu (y_i)}{{({\sum }_{i\notin \{i_1,\dots ,i_l\}}\rho (x,y_i))}^{m-\alpha }}\\ :=A_1(x)\cdot A_2(x).\end{array}$$

To estimate $A_1(x)$, we use the Hölder inequality to give that, for any $x\in B$,

$$\begin{array}{rl}{A}_1(x)& =\prod _{i\in \{i_1,\dots ,i_l\}}{\int }_{B^{\ast }}|f_i(y_i)|d\mu (y_i)\\ \le C\prod _{i\in \{i_1,\dots ,i_l\}}{({\int }_{B^{\ast }}{|f_i|}^{p_i}d\mu (y_i))}^{\frac{1}{p_i}}\mu {\left(B^{\ast }\right)}^{1-\frac{1}{p_i}}\\ \le C\mu {\left(k{B}^{\ast }\right)}^l\prod _{i\in \{i_1,\dots ,i_l\}}\left({\varphi }_i\left(\mu \left(k{B}^{\ast }\right)\right){\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\right).\end{array}$$

Estimating $A_2(x)$, by the same idea used for the case all ${\tau }_i=\mathrm{\infty }$ above, we get for any $x\in B$ that

$$\begin{array}{rl}{A}_2(x)& \le \sum _{j=0}^{\mathrm{\infty }}\frac{1}{{(2^j{r}^{\ast })}^{m-\alpha }}\prod _{i\notin \{i_1,\dots ,i_l\}}{\int }_{B(x,2^{j+1}{r}^{\ast })}|f_i(y_i)|d\mu (y_i)\\ \le C\sum _{j=0}^{\mathrm{\infty }}\frac{1}{{(2^j{r}^{\ast })}^{m-\alpha }}\prod _{i\notin \{i_1,\dots ,i_l\}}{\varphi }_i\left(2^j\mu \left(k{B}^{\ast }\right)\right)2^j\mu \left(k{B}^{\ast }\right){\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\\ \le C\sum _{j=0}^{\mathrm{\infty }}\frac{1}{{(2^j\mu (k{B}^{\ast }))}^{l-\alpha l/m}}\prod _{i\notin \{i_1,\dots ,i_l\}}{\varphi }_i\left(2^j\mu \left(k{B}^{\ast }\right)\right){\left(2^j\mu \left(k{B}^{\ast }\right)\right)}^{\alpha /m}{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}.\end{array}$$

Noting $l-l\alpha /m>0$ and using condition (1.5), we have

$$\begin{array}{rl}{A}_2(x)& \le C{\left(\mu \left(k{B}^{\ast }\right)\right)}^{\alpha l/m-l}\prod _{i\notin \{i_1,\dots ,i_l\}}{\varphi }_i\left(\mu \left(k{B}^{\ast }\right)\right)\mu {\left(k{B}^{\ast }\right)}^{\alpha /m}{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\\ \le C{\left(\mu \left(k{B}^{\ast }\right)\right)}^{\alpha -l}\prod _{i\notin \{i_1,\dots ,i_l\}}{\varphi }_i\left(\mu \left(k{B}^{\ast }\right)\right){\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}.\end{array}$$

Therefore, for $x\in B$, we have

$$\begin{array}{rl}|{\mathcal{I}}_{\alpha ,m}(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})(x)|& \le A_1(x)\cdot A_2(x)\\ \le C\mu {\left(k{B}^{\ast }\right)}^{\alpha }\prod _{i=1}^m{\varphi }_i\left(\mu \left(k{B}^{\ast }\right)\right){\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\\ \le C\psi \left(\mu \left(k{B}^{\ast }\right)\right)\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}.\end{array}$$

Hence we obtain the desired inequality (2.1) for any cases. The proof of the theorem is complete. □

3 Proof of Theorems 1.7 and 1.9

In this section we first investigate the boundedness of the m-linear Calderón-Zygmund operator on the product of spaces $L^{p_i,{\varphi }_i}(X,\mu )$ for $i=1,2,\dots ,m$.

Proof of Theorem 1.7 We also let $B=B(x_0,r)$ be the ball in $\mathcal{B}(\mu )$, with center $x_0\in X$ and radius $r>0$, and let $B^{\ast }=B(x_0,2a_{1}r)$. For the admissible $f_i\in L^{p_i,{\varphi }_i}(X,\mu )$, without loss of generality, we may initially assume that $f_i$ are all smooth boundedly supported functions, which are dense in $L^{p_i}(X,\mu )$, and let $f_i\in L^{p_i}(X,\mu )\cap L^{p_i,{\varphi }_i}(X,\mu )$, then $\mathcal{T}(f_1,\dots ,f_m)$ is a well-defined function belonging to $L^p(X,\mu )$. If we split each $f_i$ as $f_i=f_i^0+f_i^{\mathrm{\infty }}$, where $f_i^0=f_i{\chi }_{B^{\ast }}$ for $i=1,\dots ,m$, and utilize the multi-linearity of , we have the following decomposition,

$$|\mathcal{T}(f_1,\dots ,f_m)(x)|\le |\mathcal{T}(f_1^0,\dots ,f_m^0)(x)|+{\sum }^{\prime }|\mathcal{T}(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})(x)|,$$

where each term in ∑^′ contains at least one ${\tau }_i=\mathrm{\infty }$.

Noting that is bounded from $L^{p_1}(X,\mu )\times \cdots \times L^{p_m}(X,\mu )\to L^p(X,\mu )$, we have

$$\begin{array}{r}{(\frac{1}{\mu (k{B}^{\ast })}{\int }_B{|\mathcal{T}(f_1^0,\dots ,f_m^0)(x)|}^{p}d\mu (x))}^{\frac{1}{p}}\\ \le \frac{C}{\mu {(k{B}^{\ast })}^{\frac{1}{p}}}\prod _{i=1}^m{\parallel f_i^0\parallel }_{L^{p_i}(X,\mu )}\\ \le C\varphi \left(\mu \left(k{B}^{\ast }\right)\right)\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}.\end{array}$$

(3.1)

For the case ${\tau }_1=\cdots ={\tau }_m=\mathrm{\infty }$, we note that for $x\in B$ and $y_i\in X\setminus B^{\ast }$, one can deduce from the properties of the quasi-metric ρ that

$$\frac{1}{2a_1}\rho (x_0,y_i)\le \rho (x,y_i)\le (\frac{a_0}{2}+a_1)\rho (x_0,y_i).$$

Thus we can observe that

$$\frac{1}{{({\sum }_{i=1}^m\rho (x,y_i))}^m}\simeq \frac{1}{{({\sum }_{i=1}^m\rho (x_0,y_i))}^m}=m{\int }_{\rho (x_0,y_1)+\cdots +\rho (x_0,y_m)}^{\mathrm{\infty }}\frac{dl}{l^{m+1}}.$$

This, together with the Fubini theorem, we have, for $x\in B$,

$$\begin{array}{rl}|\mathcal{T}(f_1^{\mathrm{\infty }},\dots ,f_m^{\mathrm{\infty }})(x)|& \le {\int }_{{(X\setminus B^{\ast })}^m}\frac{{\prod }_{i=1}^m|f_i(y_i)|d\mu (y_i)}{{({\sum }_{i=1}^m\rho (x,y_i))}^m}\\ \le C{\int }_{{\sum }_{i=1}^m\rho (x_0,y_i)\ge 2a_{1}r}\left({\int }_{\rho (x_0,y_1)+\cdots +\rho (x_0,y_m)}^{\mathrm{\infty }}\frac{dl}{l^{m+1}}\right)\prod _{i=1}^m|f_i(y_i)|d\mu (y_i)\\ \le C{\int }_{2a_{1}r}^{\mathrm{\infty }}\frac{1}{l^{m+1}}({\int }_{{\sum }_{i=1}^m\rho (x_0,y_i)<l}\prod _{i=1}^m|f_i(y_i)|d\mu (y_i))dl\\ \le C{\int }_{2a_{1}r}^{\mathrm{\infty }}\frac{1}{l^{m+1}}(\prod _{i=1}^m{\int }_{\rho (x_0,y_i)<l}|f_i(y_i)|d\mu (y_i))dl.\end{array}$$

Noting $\mu (k{B}^{\ast })\le C_{0}2k{a}_{1}r$ and applying the Hölder inequality, we see that the inequality above is bounded by

$$\begin{array}{r}C{\int }_{\mu (k{B}^{\ast })/C_{0}k}^{\mathrm{\infty }}\frac{1}{l^{m+1}}\prod _{i=1}^m{({\int }_{B(x_0,l)}{|f_i(y_i)|}^{p_i}d\mu (y_i))}^{\frac{1}{p_i}}\mu {(B(x_0,l))}^{1-\frac{1}{p_i}}dl\\ \le C{\int }_{\mu (k{B}^{\ast })/C_{0}k}^{\mathrm{\infty }}\frac{1}{l^{m+1}}\prod _{i=1}^m\left({\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}{\varphi }_i\left(\mu (kB(x_0,l))\right)\mu (kB(x_0,l))\right)dl\end{array}$$

which, by using the non-decreasing of function $r{\varphi }_i(r)$, is controlled by

$$\begin{array}{r}C{\int }_{\mu (k{B}^{\ast })}^{\mathrm{\infty }}(\prod _{i=1}^m{\varphi }_i(l))\frac{dl}{l}\cdot \prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\\ \le C{\left(\prod _{i=1}^m{\int }_{\mu (k{B}^{\ast })}^{\mathrm{\infty }}{\varphi }_i{(l)}^{\frac{p_i}{p}}\frac{dl}{l}\right)}^{\frac{p}{p_i}}\cdot \prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\\ \le C\varphi \left(\mu \left(k{B}^{\ast }\right)\right)\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}.\end{array}$$

Therefore, we get for $x\in B$ that

$$|\mathcal{T}(f_1^{\mathrm{\infty }},\dots ,f_m^{\mathrm{\infty }})(x)|\le C\varphi \left(\mu \left(k{B}^{\ast }\right)\right)\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}.$$

(3.2)

It is left to consider the case that there is $1\le l<m$ and $\{i_1,\dots ,i_l\}\subset \{1,\dots ,m\}$ such that ${\tau }_i=0$ if $i\in \{i_1,\dots ,i_l\}$, and ${\tau }_i=\mathrm{\infty }$ if $i\notin \{i_1,\dots ,i_l\}$. For $x\in B$, we can write that

$$\begin{array}{r}|\mathcal{T}(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})(x)|\\ \le {\int }_{{(B^{\ast })}^l}\prod _{i\in \{i_1,\dots ,i_l\}}|f_i(y_i)|d\mu (y_i){\int }_{{(X\setminus B^{\ast })}^{m-l}}\frac{{\prod }_{i\notin \{i_1,\dots ,i_l\}}|f_i(y_i)|d\mu (y_i)}{{({\sum }_{i\notin \{i_1,\dots ,i_l\}}\rho (x,y_i))}^m}\\ :=E_1(x)\cdot E_2(x).\end{array}$$

With the same argument as $A_1(x)$ we have

$$E_1(x)\le C\mu {\left(k{B}^{\ast }\right)}^l\prod _{i\in \{i_1,\dots ,i_l\}}{\varphi }_i\left(\mu \left(k{B}^{\ast }\right)\right){\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}.$$

Using a similar argument as that for the estimate of $\mathcal{T}(f_1^{\mathrm{\infty }},\dots ,f_m^{\mathrm{\infty }})(x)$, we can deduce that

$$\begin{array}{rl}{E}_2(x)& \le C{\int }_{\mu (k{B}^{\ast })}^{\mathrm{\infty }}(\prod _{i\notin \{i_1,\dots ,i_l\}}{\varphi }_i(t))\frac{dt}{t^{l+1}}\cdot \prod _{i\notin \{i_1,\dots ,i_l\}}{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\\ \le C\mu {\left(k{B}^{\ast }\right)}^{-l}\left[\prod _{i\notin \{i_1,\dots ,i_l\}}{\left({\int }_{\mu (k{B}^{\ast })}^{\mathrm{\infty }}{\varphi }_i{(t)}^{\frac{p_i}{p}}\frac{dt}{t}\right)}^{\frac{p}{p_i}}\right]\prod _{i\notin \{i_1,\dots ,i_l\}}{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\\ \le C\mu {\left(k{B}^{\ast }\right)}^{-l}\left[\prod _{i\notin \{i_1,\dots ,i_l\}}{\varphi }_i\left(\mu \left(k{B}^{\ast }\right)\right)\right]\prod _{i\notin \{i_1,\dots ,i_l\}}{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}.\end{array}$$

Hence we obtain that

$$|\mathcal{T}(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})(x)|\le E_1(x)\cdot E_2(x)\le C\varphi \left(\mu \left(k{B}^{\ast }\right)\right)\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}.$$

(3.3)

Therefore, combining inequalities (3.1), (3.2) and (3.3), we have

$${(\frac{1}{\mu (k{B}^{\ast })}{\int }_B{|\mathcal{T}(f_1,\dots ,f_m)(x)|}^{p}d\mu (x))}^{\frac{1}{p}}\le C\varphi \left(\mu \left(k{B}^{\ast }\right)\right)\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )},$$

which completes the proof of Theorem 1.7. □

Remark 3.1 We have actually proved Theorem 1.7 in the case $f_i\in L^{p_i}(X,\mu )\cap L^{p_i,{\varphi }_i}(X,\mu )$. Here we need give some remarks about the definition and boundedness of $\mathcal{T}(f_1,\dots ,f_m)$ with $f_i\in L^{p_i,{\varphi }_i}(X,\mu )$ for $i=1,2,\dots ,m$. Fix any $x_0\in X$ and $R>0$, and use the same notations $f_i^{{\tau }_i}=f_i{\chi }_{B(x_0,2a_{1}R)}$ if ${\tau }_i=0$, and $f_i^{{\tau }_i}=f_i-f_i^0$ if ${\tau }_i=\mathrm{\infty }$. Using a similar argument as (3.2) and (3.3), we have, if some ${\tau }_i=\mathrm{\infty }$,

$$\begin{array}{r}{\int }_{X^m}|\mathcal{K}(x,y_1,\dots ,y_m)f_1^{{\tau }_1}(y_1)\cdots f_m^{{\tau }_m}(y_m)|d\mu (y_1)\cdots d\mu (y_m)\\ \le C\varphi \left(\mu \left(B(x_0,3a_1^{2}R)\right)\right)\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}\end{array}$$

with the constant C independent of R, for all $f_i\in L^{p_i,{\varphi }_i}(X,\mu )$, $i=1,\dots ,m$, and $x\in B(x_0,R)\subset X$.

In view of this fact, and if ${lim}_{R\to \mathrm{\infty }}\varphi (\mu (B(x_0,3a_1^{2}R)))=0$, then we can extend the definition of for $f_i\in L^{p_i,{\varphi }_i}(X,\mu )$ by

$$\begin{array}{r}\mathcal{T}(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})(x)\\ =\underset{R\to \mathrm{\infty }}{lim}(\mathcal{T}(f_1^0,\dots ,f_m^0)(x)\\ +\sum _{\mathrm{some}{\tau }_i=\mathrm{\infty }}{\int }_{X^m}|\mathcal{K}(x,y_1,\dots ,y_m)f_1^{{\tau }_1}(y_1)\cdots f_m^{{\tau }_m}(y_m)|d\mu (y_1)\cdots d\mu (y_m)).\end{array}$$

By the definition, it is easy to see that the following properties hold.

• (1) If $\rho (x_0,x)\le R_0$, then the terms in the brackets on the right-hand side of the equation above do not depend on R as long as $R>R_0$.

• (2) Suppose that $1<p_1,\dots ,p_m<\mathrm{\infty }$, and if $f_i\in L^{p_i}(X,\mu )\cap L^{p_i,{\varphi }_i}(X,\mu )$, then the definitions of $\mathcal{T}(f_1,\dots ,f_m)$ for $f_i\in L^{p_i}(X,\mu )$ and for $f_i\in L^{p_i,{\varphi }_i}(X,\mu )$ coincide.

• (3) Theorem 1.7 holds for any admissible $f_i\in L^{p_i,{\varphi }_i}(X,\mu )$, $i=1,\dots ,m$.

Finally, we consider the multi-sublinear maximal function ${\mathcal{M}}_{\kappa }(f_1,\dots ,f_m)(x)$, which is strictly smaller than the m-fold produce of the maximal function $M_{\kappa }(f_i)(x)$. Hence we have the following lemma.

Lemma 3.2 If $\kappa >a_1^2$ and $p,p_i>1$, and $\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}$, then there exists a constant C independent of $f_i$ such that

$${\parallel {\mathcal{M}}_{\kappa }(f_1,\dots ,f_m)\parallel }_{L^p(X,\mu )}\le C\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i}(X,\mu )}.$$

Proof of Theorem 1.9 With the same notions, we decompose each $f_i\in L^{p_i,{\varphi }_i}(X,\mu )$ according to the ball $B^{\ast }:=B(x_0,a_1(1+\lambda )r)$ as $f_i=f_i^0+f_i^{\mathrm{\infty }}$, where λ is a large positive constant that will be determined later. We have

$$|{\mathcal{M}}_{\kappa }(f_1,\dots ,f_m)(x)|\le |{\mathcal{M}}_{\kappa }(f_1^0,\dots ,f_m^0)(x)|+{\sum }^{\prime }|{\mathcal{M}}_{\kappa }(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})(x)|,$$

where each term in ∑^′ contains at least one ${\tau }_i\ne 0$.

It follows from Lemma 3.2 that for any $k>a_1$ and $\kappa >a_1^2$,

$$\begin{array}{r}{(\frac{1}{\mu (k{B}^{\ast })}{\int }_{B(x_0,r)}{|{\mathcal{M}}_{\kappa }(f_1^0,\dots ,f_m^0)(x)|}^{p}d\mu (x))}^{\frac{1}{p}}\\ \le C\varphi \left(\mu \left(k{B}^{\ast }\right)\right)\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}.\end{array}$$

It is left to study the case ${\tau }_{i_1}=\cdots ={\tau }_{i_l}=0$ and ${\tau }_{i_{l+1}}=\cdots ={\tau }_{i_m}=\mathrm{\infty }$ for some $1\le l<m$. Hence for $x\in B(x_0,r)$ we have

$$\begin{array}{rl}{\mathcal{M}}_{\kappa }(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})(x)=& \underset{x\in D\in \mathcal{B}(\mu )}{sup}\prod _{i=1}^m\frac{1}{\mu (\kappa D)}{\int }_D|f_i^{{\tau }_i}(y_i)|d\mu (y_i)\\ \le & \underset{x\in D\in \mathcal{B}(\mu )}{sup}\prod _{i\in \{i_1,\dots ,i_l\}}\frac{1}{\mu (\kappa D)}{\int }_D|f_i^0(y_i)|d\mu (y_i)\\ \cdot \prod _{i\notin \{i_1,\dots ,i_l\}}\frac{1}{\mu (\kappa D)}{\int }_D|f_i^{\mathrm{\infty }}(y_i)|d\mu (y_i).\end{array}$$

Let $r_D$ be the radius of the ball D and $c_D$ be the center of D. We note that on the right-hand side of the inequality above, the balls D in the integrals must satisfy that $x\in D\cap B(x_0,r)$ and some $y_{i_m}\in D\cap (X\setminus B^{\ast })$, which implies

$$\begin{array}{c}{a}_1\rho (x,y_{i_m})\ge \rho (x_0,y_{i_m})-a_1\rho (x_0,x)\ge a_1\lambda r,\hfill \\ \rho (x,y_{i_m})\le a_1{a}_0\rho (c_D,x)+a_1\rho (c_D,y_{i_m})\le a_1(1+a_0)r_D.\hfill \end{array}$$

Further, a simple calculus yields

$$B:=B(x_0,r)\subset (a_1+a_1^3{(1+a_0)}^2{\lambda }^{-1})D\subset \frac{\kappa +a_1^2}{2a_1}D$$

as long as we take λ big enough, because of $\kappa >a_1^2$. Thus,

$${\mathcal{M}}_{\kappa }(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})(x)\le \underset{B\subset D\in \mathcal{B}(\mu )}{sup}\prod _{i=1}^m\frac{1}{\mu (\frac{2a_1\kappa }{\kappa +a_1^2}D)}{\int }_D|f_i^{{\tau }_i}(y_i)|d\mu (y_i).$$

If let $k=2a_1\kappa /(\kappa +a_1^2)$, then $k>a_1$, and we can get from the Hölder inequality and condition (1.6) on ${\varphi }_i$ that

$$\begin{array}{rl}{\mathcal{M}}_{\kappa }(f_1^{{\tau }_1},\dots ,f_m^{{\tau }_m})(x)& \le \underset{B\subset D\in \mathcal{B}(\mu )}{sup}\prod _{i=1}^m\frac{1}{\mu (kD)}{\int }_D|f_i^{{\tau }_i}(y_i)|d\mu (y_i)\\ \le C\underset{B\subset D\in \mathcal{B}(\mu )}{sup}\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}{\varphi }_i(\mu (kD))\\ \le C\varphi (\mu (kB))\prod _{i=1}^m{\parallel f_i\parallel }_{L^{p_i,{\varphi }_i}(X,\mu )}.\end{array}$$

The theorem is proved. □