Sharp bounds for Hardy-type operators on mixed radial-angular central Morrey spaces

Abstract

By using the rotation method, a sharp bound for an n-dimensional Hardy operator on mixed radial-angular central Morrey spaces is obtained. Furthermore, a sharp weak-type estimate for an n-dimensional Hardy operator on mixed radial-angular central Morrey spaces is established. In addition, we also calculate the sharp constant for an n-dimensional m-linear Hardy operator from product mixed radial-angular central Morrey spaces to mixed radial-angular weak central Morrey spaces. Since mixed radial-angular central Morrey spaces are generalizations of both central Morrey spaces and mixed radial-angular spaces, the main theorems in this paper extend various existing results to a more general setting.

1 Introduction

Let f be a nonnegative integrable function on $\mathbb{R}$. The classical Hardy operator is defined by

$$ H(f) (x) := \frac{1}{x} \int _{0}^{x}f(t)\,dt , $$

where $x\neq 0$.

As we know, the classical Hardy operator, initially introduced by Hardy [14], yields the following Hardy inequality:

$$ \bigl\Vert H(f) \bigr\Vert _{L^{p}}\le \frac{p}{p-1} \Vert f \Vert _{L^{p}}, $$

where the constant $\frac{p}{p-1}$ is the best possible.

Hardy-type operators in higher dimension were introduced by Faris [9]. Later, Christ and Grafakos [4] gave an equivalent expression of an n-dimensional Hardy operator

$$ {\mathcal{H}}(f) (x) := \frac{1}{\Omega _{n} \vert x \vert ^{n}} \int _{ \vert y \vert < \vert x \vert }f(y)\,dy , \quad x\in \mathbb{R}^{n} \backslash \{0\}, $$

(1.1)

where f is a nonnegative measurable function on $\mathbb{R}^{n}$ and $\Omega _{n}=\frac{\pi ^{n/2}}{\Gamma (1+n/2)}$ is the volume of the unit ball in $\mathbb{R}^{n}$.

In the same paper [4], Christ and Grafakos proved that the operator norm of $\mathcal{H}$ on $L^{p}(\mathbb{R}^{n})$ $(1< p<\infty )$ is also $\frac{p}{p-1}$. Moreover, a sharp weak $(p,p)$ bound for $\mathcal{H}$ was obtained by Zhao et al. [46]: namely, for $1\leq p\leq \infty$, we have

$$ \bigl\Vert \mathcal{H}(f) \bigr\Vert _{L^{p,\infty}}\leq 1 \cdot \Vert f \Vert _{L^{p}}, $$

where the constant 1 is the best possible. Recently, much attention has been paid to the sharp bounds for Hardy-type operators. We refer the readers to [12, 13, 20, 36, 42, 45–47] for more studies of sharp bounds for Hardy-type operators on different function spaces.

In 2012, the Hardy operator was extended to the multilinear setting by Fu et al. [11]. Let $m\in \mathbb{N}^{+}$, $f_{1}, f_{2},\dots , f_{m}$ be nonnegative locally integrable functions on $\mathbb{R}^{n}$. The m-linear Hardy operator is defined by

$$\begin{aligned} &\mathcal{H}_{m}(f_{1},\dots ,f_{m}) (x) \\ &\quad :=\frac{1}{\Omega _{mn} \vert x \vert ^{mn}} \int _{ \vert (y_{1},y_{2},\dots ,y_{m}) \vert < \vert x \vert }f_{1}(y_{1})f_{2}(y_{2}) \cdots f_{m}(y_{m})\,dy _{1}\,dy _{2} \cdots \,dy _{m}. \end{aligned}$$

(1.2)

Two other variants of m-linear Hardy operators were also introduced and studied by Bényi and Oh [2]. Sharp bounds for the m-linear Hardy operator from product Lebesgue spaces to Lebesgue spaces and from product central Morrey spaces to central Morrey spaces were obtained in [11]. After that, many researchers studied sharp constants for n-dimensional m-linear Hardy operators and their variants on product Lebesgue spaces and product Morrey-type spaces. The readers are referred to [10, 28, 40] for more details. For some earlier development of Hardy-type inequalities, one can refer to the book [15]. We also refer the readers to [22, 26] for some recent progress on Hardy-type inequalities and related topics.

Note that recently the mixed radial-angular spaces have been used to improve some classical results in partial differential equations, see [3, 5, 8, 29, 39]. Later, many integral operators in harmonic analysis were proven to be bounded on these spaces. For instance, Duoandikoetxea and Oruetxebarria [7] established the extrapolation theorems on mixed radial-angular spaces to study the boundedness of operators on mixed radial-angular spaces which are weighted bounded. Moreover, the boundedness of some operators with rough kernels on mixed radial-angular spaces were also studied by Liu et al. [23–25]. Now we recall the definition of mixed radial-angular spaces.

Definition 1.1

For $n\geq 2$ and $1\leq p,\bar{p}\leq \infty$, the mixed radial-angular space $L^{p}_{|x|}L_{\theta}^{\bar{p}}(\mathbb{R}^{n})$ consists of all functions f in $\mathbb{R}^{n}$ for which

$$ \Vert f \Vert _{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}}}:= \biggl( \int _{0}^{\infty} \biggl( \int _{\mathbb{S}^{n-1}} \bigl\vert f(r,\theta ) \bigr\vert ^{\bar{p}}\,d\theta \biggr)^{p/\bar{p}}r^{n-1}\,dr \biggr)^{1/p}< \infty , $$

where $\mathbb{S}^{n-1}$ denotes the unit sphere in $\mathbb{R}^{n}$. If $p=\infty$ or $\bar{p}=\infty$, then we have to make appropriate modifications.

Similarly, we can define the weak mixed radial-angular spaces.

Definition 1.2

For $n\geq 2$ and $1\leq p,\bar{p}\leq \infty$, the weak mixed radial-angular space $wL^{p}_{|x|}L_{\theta}^{\bar{p}}(\mathbb{R}^{n})$ consists of all functions f in $\mathbb{R}^{n}$ for which

$$ \Vert f \Vert _{wL^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}}}:=\sup_{t>0}t \Vert \chi _{\{x \in \mathbb{R}^{n}: \vert f(x) \vert >t\}} \Vert _{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}}}< \infty , $$

where $\chi _{\{x\in \mathbb{R}^{n}:|f(x)|>t\}}$ denotes the characteristic function of the set $\{x\in \mathbb{R}^{n}:|f(x)|>t\}$.

Obviously, the mixed radial-angular spaces are particular cases of mixed-norm Lebesgue spaces studied by Benedek and Panzone [1]. We refer the readers to [6, 18, 19, 32–34] for more studies on mixed-norm Lebesgue spaces.

In order to study the local properties of solutions to partial differential equations, Morrey [31] introduced the classical Morrey space $M^{\lambda}_{p}(\mathbb{R}^{n})$, which consists of all functions f with finite norm

$$ \Vert f \Vert _{M^{\lambda}_{p}}:=\sup_{x\in \mathbb{R}^{n},R>0} \frac{1}{ \vert B(x,R) \vert ^{1/p+\lambda}} \Vert f \Vert _{L^{p}(B(x,R))}, $$

where $1\leq p<\infty$, $-1/p\leq \lambda <0$ and $B(x,R)$ denotes the ball in $\mathbb{R}^{n}$ centered at x with radius R. By restricting the balls centered at the origin, we obtain the central Morrrey space $\dot{M}^{\lambda}_{p}(\mathbb{R}^{n})$, which consists of all functions f with finite norm

$$ \Vert f \Vert _{\dot{M}^{\lambda}_{p}}:=\sup_{R>0} \frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert f \Vert _{L^{p}(B(0,R))}, $$

for $1\leq p<\infty$, $-1/p\leq \lambda <0$.

Nowadays, Morrey-type spaces have become some of the most important function spaces in the theory of function spaces, see the book [37] for a comprehensive theory of Morrey-type spaces. By combining Morrey spaces with different function spaces, we can construct some new function spaces, for instance, Morrey–Herz spaces, Besov–Morrey spaces, Hardy–Morrey spaces, Morrey–Lorentz spaces, and so on. See [16, 17, 21, 27, 30, 35, 38, 44] for more details.

Naturally, by combining the definitions of central Morrey spaces and mixed radial-angular spaces, we can define mixed radial-angular central Morrey spaces, see [41] for a more general version of these spaces.

Definition 1.3

For $n\geq 2$, $1\leq p<\infty$, $-1/p\leq \lambda <0$, and $1\leq \bar{p}<\infty$, the mixed radial-angular central Morrey space $\dot{M}^{\lambda}_{p,\bar{p}}(\mathbb{R}^{n})$ consists of all functions f in $\mathbb{R}^{n}$ for which

$$ \Vert f \Vert _{\dot{M}^{\lambda}_{p,\bar{p}}}:=\sup_{R>0} \frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert f\chi _{B(0,R)} \Vert _{{L^{p}_{ \vert x \vert }L_{ \theta}^{\bar{p}}}}< \infty , $$

where $\chi _{B(0,R)}$ denotes the characteristic function of the ball $B(0,R)$.

Similarly, the mixed radial-angular weak central Morrey spaces can be defined as follows:

Definition 1.4

For $n\geq 2$, $1\leq p<\infty$, $-1/p\leq \lambda <0$, and $1\leq \bar{p}<\infty$, the mixed radial-angular weak central Morrey space $w\dot{M}^{\lambda}_{p,\bar{p}}(\mathbb{R}^{n})$ consists of all functions f in $\mathbb{R}^{n}$ for which

$$ \Vert f \Vert _{w\dot{M}^{\lambda}_{p,\bar{p}}}:=\sup_{R>0} \frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert f\chi _{B(0,R)} \Vert _{{wL^{p}_{ \vert x \vert }L_{ \theta}^{\bar{p}}}}< \infty . $$

Obviously, we recover the spaces $L^{p}_{|x|}L_{\theta}^{\bar{p}}(\mathbb{R}^{n})$ and $wL^{p}_{|x|}L_{\theta}^{\bar{p}}(\mathbb{R}^{n})$ if $\lambda =-1/p$ in Definitions 1.3 and 1.4, respectively.

Very recently, the sharp constant for an n-dimensional Hardy operator $\mathcal{H}$ from $L^{p}_{|x|}L_{\theta}^{\bar{p}_{1}}(\mathbb{R}^{n})$ to $L^{p}_{|x|}L_{\theta}^{\bar{p}_{2}}(\mathbb{R}^{n})$ was obtained by Wei and Yan [43]. In addition, a sharp weak type estimate was also considered in [43]. Inspired by [11, 12, 43, 46], we will consider the sharp constants for an n-dimensional Hardy operator $\mathcal{H}$ and n-dimensional m-linear Hardy operator $\mathcal{H}_{m}$ on mixed radial-angular central Morrey spaces in this paper. Moreover, we will also give a sharp weak estimate for $\mathcal{H}$ from $\dot{M}^{\lambda}_{p,\bar{p}_{1}}(\mathbb{R}^{n})$ to $w\dot{M}^{\lambda}_{p,\bar{p}_{2}}(\mathbb{R}^{n})$.

This paper is organized as follows. The sharp bound for n-dimensional Hardy operator from $\dot{M}^{\lambda}_{p,\bar{p}_{1}}(\mathbb{R}^{n})$ to $\dot{M}^{\lambda}_{p,\bar{p}_{2}}(\mathbb{R}^{n})$ is obtained in Sect. 2. We calculate the operator norm of $\mathcal{H}$ from $\dot{M}^{\lambda}_{p,\bar{p}_{1}}(\mathbb{R}^{n})$ to $w\dot{M}^{\lambda}_{p,\bar{p}_{2}}(\mathbb{R}^{n})$ in Sect. 3. In addition, we also calculate the operator norm for an n-dimensional m-linear Hardy operator from $\dot{M}^{\lambda _{1}}_{p_{1},\bar{p}_{1}}(\mathbb{R}^{n})\times \dot{M}^{ \lambda _{2}}_{p_{2},\bar{p}_{2}}(\mathbb{R}^{n})\times \cdots \times \dot{M}^{\lambda _{m}}_{p_{m},\bar{p}_{m}}(\mathbb{R}^{n})$ to $\dot{M}^{\lambda}_{p,\bar{p}}(\mathbb{R}^{n})$ in Sect. 4.

2 Sharp bound for $\mathcal{H}$ on $\dot{M}^{\lambda}_{p,\bar{p}}(\mathbb{R}^{n})$

The main result in this section is the following:

Theorem 2.1

Let $n\geq 2$, $1< p,\bar{p}_{1},\bar{p}_{2}<\infty$, and $-1/p\leq \lambda <0$. Then the n-dimensional Hardy operator $\mathcal{H}$ defined in (1.1) is bounded from $\dot{M}^{\lambda}_{p,\bar{p}_{1}}(\mathbb{R}^{n})$ to $\dot{M}^{\lambda}_{p,\bar{p}_{2}}(\mathbb{R}^{n})$. Moreover,

$$ \Vert \mathcal{H} \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}\rightarrow \dot{M}^{ \lambda}_{p,\bar{p}_{2}}}=\frac{1}{1+\lambda}w_{n}^{1/\bar{p}_{2}-1/ \bar{p}_{1}}, $$

where $w_{n}=2\pi ^{n/2}/\Gamma (n/2)$ is the measure of $\mathbb{S}^{n-1}$.

Proof

We borrow some ideas from [4, 11], where a rotation method was used. Set

$$ g(x)=\frac{1}{w_{n}} \int _{\mathbb{S}^{n-1}}f\bigl( \vert x \vert \theta \bigr)\,d\theta , \quad x\in \mathbb{R}^{n}. $$

(2.1)

Obviously, $g(x)$ is a radial function. Moreover, for any $R>0$ and locally integrable function f on $\mathbb{R}^{n}$, we have

$$\begin{aligned} &\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert f\chi _{B(0,R)} \Vert _{{L^{p}_{ \vert x \vert }L_{ \theta}^{\bar{p}_{1}}}} \\ &\quad =\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R} \biggl( \int _{ \mathbb{S}^{n-1}} \bigl\vert g(r,\theta ) \bigr\vert ^{\bar{p}_{1}}\,d\theta \biggr)^{p/ \bar{p}_{1}}r^{n-1}\,dr \biggr)^{1/p} \\ &\quad =w^{1/\bar{p}_{1}}_{n}\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R} \bigl\vert g(r) \bigr\vert ^{p}r^{n-1}\,dr \biggr)^{1/p}, \end{aligned}$$

(2.2)

where $g(r)$ should be recognized as $g(r)=g(x)$ for any $x\in \mathbb{R}^{n}$ with $|x|=r$, since g is a radial function.

In view of (2.1) and (2.2), by using Hölder’s inequality, we get

$$\begin{aligned} &\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert f\chi _{B(0,R)} \Vert _{{L^{p}_{ \vert x \vert }L_{ \theta}^{\bar{p}_{1}}}} \\ &\quad =w^{1/\bar{p}_{1}}_{n}\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R} \biggl\vert \frac{1}{w_{n}} \int _{\mathbb{S}^{n-1}}f(r\theta )\,d\theta \biggr\vert ^{p}r^{n-1}\,dr \biggr)^{1/p} \\ &\quad =w^{1/\bar{p}_{1}-1}_{n}\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R} \biggl\vert \int _{\mathbb{S}^{n-1}}f(r\theta )\,d\theta \biggr\vert ^{p}r^{n-1}\,dr \biggr)^{1/p} \\ &\quad \leq w^{1/\bar{p}_{1}-1}_{n}\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R} \biggl( \int _{\mathbb{S}^{n-1}} \bigl\vert f(r\theta ) \bigr\vert ^{\bar{p}_{1}}\,d\theta \biggr)^{p/\bar{p}_{1}} \biggl( \int _{\mathbb{S}^{n-1}}\,d\theta \biggr)^{p/\bar{p}'_{1}}r^{n-1}\,dr \biggr)^{1/p} \\ &\quad =\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R} \biggl( \int _{ \mathbb{S}^{n-1}} \bigl\vert f(r\theta ) \bigr\vert ^{\bar{p}_{1}}\,d\theta \biggr)^{p/ \bar{p}_{1}}r^{n-1}\,dr \biggr)^{1/p} \\ &\quad \leq \Vert f \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}}, \end{aligned}$$

where $\bar{p}'_{1}$ satisfies $1/\bar{p}_{1}+1/\bar{p}'_{1}=1$. By taking the supremum over all $R>0$, we have

$$ \Vert g \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}}\leq \Vert f \Vert _{\dot{M}^{\lambda}_{p, \bar{p}_{1}}}. $$

Moreover, Fu et al. [11, Proof of Theorem 1] showed that $\mathcal{H}(g)$ is equal to $\mathcal{H}(f)$. Therefore, we arrive at

$$ \frac{ \Vert \mathcal{H}(f) \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{2}}}}{ \Vert f \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}}} \leq \frac{ \Vert \mathcal{H}(g) \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{2}}}}{ \Vert g \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}}}. $$

Consequently, the operator $\mathcal{H}$ and its restriction to radial functions have the same operator norm on $\dot{M}^{\lambda}_{p,\bar{p}_{1}}(\mathbb{R}^{n})$. Without loss of generality, we can assume that f is a radial function.

For a radial function f, $\mathcal{H}(f)(x)$ is also a radial function. Moreover, we have

$$\begin{aligned} &\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \bigl\Vert \mathcal{H}(f)\cdot \chi _{B(0,R)} \bigr\Vert _{{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{2}}}} \\ &\quad =\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R} \biggl( \int _{ \mathbb{S}^{n-1}} \bigl\vert \mathcal{H}(f) (r,\theta ) \bigr\vert ^{\bar{p}_{2}}\,d\theta \biggr)^{p/\bar{p}_{2}}r^{n-1}\,dr \biggr)^{1/p} \\ &\quad =\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R} \biggl( \int _{ \mathbb{S}^{n-1}} \bigl\vert \mathcal{H}(f) (r) \bigr\vert ^{\bar{p}_{2}}\,d\theta \biggr)^{p/ \bar{p}_{2}}r^{n-1}\,dr \biggr)^{1/p} \\ &\quad =w^{1/\bar{p}_{2}}_{n}\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R} \bigl\vert \mathcal{H}(f) (r) \bigr\vert ^{p}r^{n-1}\,dr \biggr)^{1/p}, \end{aligned}$$

(2.3)

where $\mathcal{H}(f)(r)$ should be recognized as $\mathcal{H}(f)(r)=\mathcal{H}(f)(x)$ for any $x\in \mathbb{R}^{n}$ with $|x|=r$ noting that $\mathcal{H}(f)$ is a radial function.

By changing variables, there holds

$$ \mathcal{H}(f) (r)=\frac{1}{\Omega _{n}} \int _{B(0,1)}f(ry)\,dy . $$

(2.4)

Combining (2.3) with (2.4), and using Minkowski’s inequality, we get

$$\begin{aligned} &\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \bigl\Vert \mathcal{H}(f)\cdot \chi _{B(0,R)} \bigr\Vert _{{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{2}}}} \\ &\quad =\frac{w_{n}^{1/\bar{p}_{2}}}{\Omega _{n}} \frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R} \biggl\vert \int _{B(0,1)}f(ry)\,dy \biggr\vert ^{p}r^{n-1}\,dr \biggr)^{1/p} \\ &\quad \leq \frac{w_{n}^{1/\bar{p}_{2}}}{\Omega _{n}} \int _{B(0,1)} \biggl( \frac{1}{ \vert B(0,R) \vert ^{1+p\lambda}} \int _{0}^{R} \bigl\vert f(ry) \bigr\vert ^{p}r^{n-1}\,dr \biggr)^{1/p} \,dy \\ &\quad =\frac{w_{n}^{1/\bar{p}_{2}}}{\Omega _{n}} \int _{B(0,1)} \biggl( \frac{1}{ \vert B(0,R) \vert ^{1+p\lambda}} \int _{0}^{R} \bigl\vert f\bigl(r \vert y \vert \bigr) \bigr\vert ^{p}r^{n-1}\,dr \biggr)^{1/p} \,dy \\ &\quad =\frac{w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}}}{\Omega _{n}} \int _{B(0,1)} \frac{1}{ \vert B(0,R \vert y \vert ) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R \vert y \vert } w_{n}^{p/ \bar{p}_{1}} \bigl\vert f(r) \bigr\vert ^{p}r^{n-1}\,dr \biggr)^{1/p} \vert y \vert ^{n\lambda}\,dy \\ &\quad \leq \frac{w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}}}{\Omega _{n}} \int _{B(0,1)} \vert y \vert ^{n\lambda}\,dy \Vert f \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}} \\ &\quad =w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}}\frac{w_{n}}{\Omega _{n}} \frac{1}{n(1+\lambda )} \Vert f \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}}= \frac{1}{1+\lambda}w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}} \Vert f \Vert _{\dot{M}^{ \lambda}_{p,\bar{p}_{1}}}, \end{aligned}$$

where we have used the identity $w_{n}=n\Omega _{n}$.

By taking the supremum over all $R>0$, we obtain

$$ \bigl\Vert \mathcal{H}(f) \bigr\Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{2}}}\leq \frac{1}{1+\lambda}w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}} \Vert f \Vert _{\dot{M}^{ \lambda}_{p,\bar{p}_{1}}}. $$

To prove that the constant $\frac{1}{1+\lambda}w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}}$ is optimal, we only need to consider the case $-1/p<\lambda <0$, since the case $\lambda =-1/p$ has been studied in [43, Theorem 2.1]. For $-1/p<\lambda <0$, we take $f_{0}(x)=|x|^{n\lambda}$. Obviously, $f_{0}(x)=|x|^{n\lambda}$ satisfies $\mathcal{H}(f_{0})(x)=\frac{1}{1+\lambda}f_{0}(x)$. Moreover,

$$\begin{aligned} \frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert f_{0} \chi _{B(0,R)} \Vert _{{L^{p}_{ \vert x \vert }L_{ \theta}^{\bar{p}_{1}}}} &= \frac{w_{n}^{1/\bar{p}_{1}}}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R}r^{np \lambda}r^{n-1}\,dr \biggr)^{1/p} \\ &= \frac{w_{n}^{1/\bar{p}_{1}}}{\Omega _{n}^{1/p+\lambda}[n(1+p\lambda )]^{1/p}}< \infty , \end{aligned}$$

(2.5)

which yields that $f_{0}$ lies in $\dot{M}^{\lambda}_{p,\bar{p}_{1}}(\mathbb{R}^{n})$, and

$$\begin{aligned} \Vert f_{0} \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}}&=\sup _{R>0} \frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert f_{0} \chi _{B(0,R)} \Vert _{{L^{p}_{ \vert x \vert }L_{ \theta}^{\bar{p}_{1}}}} \\ &= \frac{w_{n}^{1/\bar{p}_{1}}}{\Omega _{n}^{1/p+\lambda}[n(1+p\lambda )]^{1/p}}. \end{aligned}$$

Similarly,

$$\begin{aligned} \bigl\Vert \mathcal{H}(f_{0}) \bigr\Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{2}}}&= \frac{1}{1+\lambda}\sup_{R>0} \frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert f_{0} \chi _{B(0,R)} \Vert _{{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{2}}}} \\ &=\frac{1}{1+\lambda} \frac{w_{n}^{1/\bar{p}_{2}}}{\Omega _{n}^{1/p+\lambda}[n(1+p\lambda )]^{1/p}}. \end{aligned}$$

Consequently,

$$ \bigl\Vert \mathcal{H}(f_{0}) \bigr\Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{2}}}= \frac{1}{1+\lambda}w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}} \Vert f_{0} \Vert _{ \dot{M}^{\lambda}_{p,\bar{p}_{1}}}, $$

which finishes the proof. □

Remark 2.1

When $p=\bar{p}_{1}=\bar{p}_{2}\in (1,\infty )$ in Theorem 2.1, we recover the result of [11, Theorem 2] in the linear situation.

3 Sharp weak bound for $\mathcal{H}$ from $\dot{M}^{\lambda}_{p,\bar{p}_{1}}(\mathbb{R}^{n})$ to $w\dot{M}^{\lambda}_{p,\bar{p}_{2}}(\mathbb{R}^{n})$

This section establishes a sharp weak estimate for an n-dimensional Hardy operator from mixed radial-angular central Morrey spaces to mixed radial-angular weak central Morrey spaces. Our main result is as follows.

Theorem 3.1

Let $n\geq 2$, $1\leq p,\bar{p}_{1},\bar{p}_{2}<\infty$, and $-1/p\leq \lambda <0$. Then the n-dimensional Hardy operator $\mathcal{H}$ defined in (1.1) is bounded from $\dot{M}^{\lambda}_{p,\bar{p}_{1}}(\mathbb{R}^{n})$ to $w\dot{M}^{\lambda}_{p,\bar{p}_{2}}(\mathbb{R}^{n})$. Moreover,

$$ \Vert \mathcal{H} \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}\rightarrow w \dot{M}^{\lambda}_{p,\bar{p}_{2}}}=w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}}. $$

Proof

We only give the proof of the case $1< p,\bar{p}_{1},\bar{p}_{2}<\infty$, with the usual modifications made when $p=1$ or $\bar{p}_{i}=1$, $i=1,2$. By using Hölder’s inequality on mixed-norm Lebesgue spaces (see [1]), we have

$$\begin{aligned} \bigl\vert \mathcal{H}(f) (x) \bigr\vert &\leq \frac{1}{\Omega _{n} \vert x \vert ^{n}} \Vert f\chi _{B(0, \vert x \vert )} \Vert _{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{1}}} \Vert \chi _{B(0, \vert x \vert )} \Vert _{L^{p'}_{ \vert x \vert }L_{ \theta}^{\bar{p}'_{1}}} \\ &\leq \frac{1}{\Omega _{n} \vert x \vert ^{n}} \Vert f\chi _{B(0, \vert x \vert )} \Vert _{L^{p}_{ \vert x \vert }L_{ \theta}^{\bar{p}_{1}}} \Vert \chi _{B(0, \vert x \vert )} \Vert _{L^{p'}_{ \vert x \vert }L_{\theta}^{ \bar{p}'_{1}}} \\ &= w_{n}^{1/\bar{p}_{1}-1/p'}\Omega _{n}^{\lambda} \vert x \vert ^{n\lambda} \frac{1}{ \vert B(0, \vert x \vert ) \vert ^{1/p+\lambda}} \Vert f\chi _{B(0, \vert x \vert )} \Vert _{L^{p}_{ \vert x \vert }L_{ \theta}^{\bar{p}_{1}}} \\ &\leq w_{n}^{1/\bar{p}'_{1}-1/p'}\Omega _{n}^{\lambda} \vert x \vert ^{n\lambda} \Vert f \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}}, \end{aligned}$$

where $p'$ and $\bar{p}'_{1}$ satisfy $1/p+1/p'=1$ and $1/\bar{p}_{1}+1/\bar{p}'_{1}=1$.

Denote by $A=w_{n}^{1/\bar{p}'_{1}-1/p'}\Omega _{n}^{\lambda}\|f\|_{\dot{M}^{ \lambda}_{p,\bar{p}_{1}}}$. Noting that $\lambda <0$, we have

$$\begin{aligned} \bigl\Vert \mathcal{H}(f) \bigr\Vert _{w\dot{M}^{\lambda}_{p,\bar{p}_{2}}}&\leq \sup _{R>0} \sup_{t>0}\frac{t}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert \chi _{ \{x\in B(0,R): A \vert x \vert ^{n\lambda}>t \}} \Vert _{{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{2}}}} \\ &= \sup_{R>0}\sup_{t>0} \frac{t}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert \chi _{ \{x\in B(0,R): \vert x \vert < (t/A)^{\frac{1}{n\lambda}} \}} \Vert _{{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{2}}}}. \end{aligned}$$

Now we divide our discussion into two cases:

(i) If $0< R\leq (t/A)^{\frac{1}{n\lambda}}$, then we have

$$\begin{aligned} \begin{aligned} &\sup_{t>0}\sup_{0< R\leq (t/A)^{\frac{1}{n\lambda}}} \frac{t}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert \chi _{ \{x\in B(0,R): \vert x \vert < (t/A)^{ \frac{1}{n\lambda}} \}} \Vert _{{L^{p}_{ \vert x \vert }L_{\theta}^{ \bar{p}_{2}}}} \\ &\quad =\frac{w_{n}^{1/\bar{p}_{2}-1/p}}{\Omega _{n}^{\lambda}}\sup_{t>0} \sup_{0< R\leq (t/A)^{\frac{1}{n\lambda}}} tR^{-n\lambda} \\ &\quad = w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}} \Vert f \Vert _{\dot{M}^{\lambda}_{p, \bar{p}_{1}}}, \end{aligned} \end{aligned}$$

since $\lambda <0$.

(ii) If $R>(t/A)^{\frac{1}{n\lambda}}$, then we have

$$\begin{aligned} &\sup_{t>0}\sup_{R>(t/A)^{\frac{1}{n\lambda}}} \frac{t}{ \vert B(0,R) \vert ^{1/p+\lambda}} \Vert \chi _{ \{x\in B(0,R): \vert x \vert < (t/A)^{ \frac{1}{n\lambda}} \}} \Vert _{{L^{p}_{ \vert x \vert }L_{\theta}^{ \bar{p}_{2}}}} \\ &\quad \leq \frac{w_{n}^{1/\bar{p}_{2}-1/p}}{\Omega _{n}^{\lambda}}\sup_{t>0} \sup _{R>(t/A)^{\frac{1}{n\lambda}}} tR^{-n(1/p+\lambda )}(t/A)^{ \frac{1}{{p\lambda}}} \\ &\quad = w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}} \Vert f \Vert _{\dot{M}^{\lambda}_{p, \bar{p}_{1}}}, \end{aligned}$$

since $\lambda +1/p\geq 0$.

Combining the estimates of (i) and (ii), we obtain

$$ \bigl\Vert \mathcal{H}(f) \bigr\Vert _{w\dot{M}^{\lambda}_{p,\bar{p}_{2}}}\leq w_{n}^{1/ \bar{p}_{2}-1/\bar{p}_{1}} \Vert f \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}}. $$

On the other hand, we need to show that the constant $w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}}$ is the best possible. Note that we only need to consider the case $-1/p<\lambda <0$, since the case $\lambda =-1/p$ has been studied in [43, Theorem 4.1]. Taking $f_{0}(x)=\chi _{[0,1]}(|x|)$, $x\in \mathbb{R}^{n}$, for $-1/p<\lambda <0$, we have $f_{0}\in \dot{M}^{\lambda}_{p,\bar{p}_{1}}(\mathbb{R}^{n})$. In fact, when $0< R\leq 1$,

$$ \Vert f_{0}\chi _{B(0,R)} \Vert _{{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{1}}}}= \frac{w_{n}^{1/\bar{p}_{1}}}{n^{1/p}}R^{n/p}. $$

When $R>1$,

$$ \Vert f_{0}\chi _{B(0,R)} \Vert _{{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{1}}}}= \frac{w_{n}^{1/\bar{p}_{1}}}{n^{1/p}}. $$

Denoting by $E=\|f_{0}\chi _{B(0,R)}\|_{{L^{p}_{|x|}L_{\theta}^{\bar{p}_{1}}}}$, we have

$$\begin{aligned} \Vert f_{0} \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}}&=\max \biggl\{ \sup _{0< R \leq 1}\frac{E}{ \vert B(0,R) \vert ^{1/p+\lambda}},\sup_{R>1} \frac{E}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggr\} \\ &=\max \Bigl\{ \sup_{0< R\leq 1}w_{n}^{1/\bar{p}_{1}-1/p} \bigl(\Omega _{n}R^{n}\bigr)^{- \lambda},\sup _{R>1}w_{n}^{1/\bar{p}_{1}-1/p} \Omega _{n}^{-\lambda}R^{-n(1/p+ \lambda )} \Bigr\} \\ &=w_{n}^{1/\bar{p}_{1}-1/p} \Omega _{n}^{-\lambda}, \end{aligned}$$

since $-1/p\leq \lambda <0$. Moreover,

$$ \mathcal{H}(f_{0}) (x)=\textstyle\begin{cases} \vert x \vert ^{-n}, & \vert x \vert > 1, \\ 1,& \vert x \vert \leq 1. \end{cases} $$

Therefore, $\mathcal{H}(f_{0})(x)\leq 1$. On the other hand, when $0< R\leq 1$, we have

$$ \bigl\Vert \mathcal{H}(f_{0})\chi _{B(0,R)} \bigr\Vert _{wL^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{2}}}= \sup_{0< t\leq 1}t \Vert \chi _{\{x\in B(0,R): 1>t\}} \Vert _{L^{p}_{ \vert x \vert }L_{ \theta}^{\bar{p}_{2}}} =\frac{w_{n}^{1/\bar{p}_{2}}}{n^{1/p}}R^{n/p}. $$

(3.1)

When $R>1$, we have

$$ \bigl\Vert \mathcal{H}(f_{0})\chi _{B(0,R)} \bigr\Vert _{wL^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{2}}}= \sup_{0< t\leq 1}t \Vert \chi _{\{x\in B(0,1): 1>t\}\cup \{1\leq \vert x \vert < R: \vert x \vert ^{-n}>t \} } \Vert _{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{2}}}. $$

If $1< R\leq t^{-1/n}$, then

$$\begin{aligned} &t \Vert \chi _{\{x\in B(0,1): 1>t\}\cup \{1\leq \vert x \vert < R: \vert x \vert ^{-n}>t\} } \Vert _{L^{p}_{ \vert x \vert }L_{ \theta}^{\bar{p}_{2}}} \\ &\quad \leq \sup_{0< t\leq \frac{1}{R^{n}}}t w_{n}^{1/\bar{p}_{2}} \biggl( \frac{R^{n}}{n} \biggr)^{1/p}= \frac{w_{n}^{1/\bar{p}_{2}}}{R^{n/p'}n^{1/p}}. \end{aligned}$$

If $1< t^{-1/n}< R$, then

$$\begin{aligned} & t \Vert \chi _{\{x\in B(0,1): 1>t\}\cup \{1\leq \vert x \vert < R: \vert x \vert ^{-n}>t\} } \Vert _{L^{p}_{ \vert x \vert }L_{ \theta}^{\bar{p}_{2}}} \\ &\quad \leq \sup_{\frac{1}{R^{n}}< t< 1} \frac{w_{n}^{1/\bar{p}_{2}}}{n^{1/p}}t^{1-1/p}= \frac{w_{n}^{1/\bar{p}_{2}}}{n^{1/p}}. \end{aligned}$$

Consequently, for $R>1$, we have

$$ \bigl\Vert \mathcal{H}(f_{0})\chi _{B(0,R)} \bigr\Vert _{wL^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}_{2}}}= \frac{w_{n}^{1/\bar{p}_{2}}}{n^{1/p}}. $$

(3.2)

It follows from (3.1) and (3.2) that

$$ \begin{aligned} & \bigl\Vert \mathcal{H}(f_{0}) \bigr\Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{2}}} \\ &\quad = \max \biggl\{ \sup_{0< R\leq 1} \bigl\vert B(0,R) \bigr\vert ^{-1/p-\lambda}\cdot \frac{w_{n}^{1/\bar{p}_{2}}}{n^{1/p}}R^{n/p}, \sup _{R>1} \bigl\vert B(0,R) \bigr\vert ^{-1/p- \lambda} \cdot \frac{w_{n}^{1/\bar{p}_{2}}}{n^{1/p}} \biggr\} \\ &\quad =w_{n}^{1/\bar{p}_{2}-1/p} \Omega _{n}^{-\lambda}=w_{n}^{1/\bar{p}_{2}-1/ \bar{p}_{1}} \Vert f_{0} \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}}, \end{aligned} $$

since $1\leq p<\infty$ and $-1/p< \lambda <0$.

Therefore,

$$ \Vert \mathcal{H} \Vert _{\dot{M}^{\lambda}_{p,\bar{p}_{1}}\rightarrow w \dot{M}^{\lambda}_{p,\bar{p}_{2}}}=w_{n}^{1/\bar{p}_{2}-1/\bar{p}_{1}}. $$

 □

Obviously, Theorem 3.1 improves the result of [12, Theorem 2.1].

4 Sharp bound for $\mathcal{H}_{m}$ from $\dot{M}^{\lambda _{1}}_{p_{1},\bar{p}_{1}}(\mathbb{R}^{n})\times \dot{M}^{\lambda _{2}}_{p_{2},\bar{p}_{2}}(\mathbb{R}^{n})\times \cdots \times \dot{M}^{\lambda _{m}}_{p_{m},\bar{p}_{m}}(\mathbb{R}^{n})$ to $\dot{M}^{\lambda}_{p,\bar{p}}(\mathbb{R}^{n})$

This section is devoted to calculating the operator norm of $\mathcal{H}_{m}$ from $\dot{M}^{\lambda _{1}}_{p_{1},\bar{p}_{1}}(\mathbb{R}^{n})\times \dot{M}^{ \lambda _{2}}_{p_{2},\bar{p}_{2}}(\mathbb{R}^{n})\times \cdots \times \dot{M}^{\lambda _{m}}_{p_{m},\bar{p}_{m}}(\mathbb{R}^{n})$ to $\dot{M}^{\lambda}_{p,\bar{p}}(\mathbb{R}^{n})$. To state our main result, we need the Beta and Gamma functions. Let $z_{1}$ and $z_{2}$ be complex numbers with positive real parts. Then the Beta function $B(z_{1},z_{2})$ and the Gamma function $\Gamma (z_{1})$ are defined by

$$ B(z_{1},z_{2}):= \int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt $$

and

$$ \Gamma (z_{1}):= \int _{0}^{\infty }t^{z_{1}-1}e^{-t}\,dt , $$

respectively.

Our main result in this section can be read as follows.

Theorem 4.1

Let $m,n\geq 2$, $1< p_{i},\bar{p_{i}}<\infty$, $-1/p_{i}<\lambda _{i}<0$, $i=1,2,\dots ,m$ and $1< p,\bar{p}<\infty$, $-1/p<\lambda <0$ such that $1/p=1/p_{1}+1/p_{2}+\cdots +1/p_{m}$, $\lambda =\lambda _{1}+\lambda _{2}+\cdots +\lambda _{m}$, and $p_{i}\lambda _{i}=p\lambda$, $i=1,2,\dots ,m$. Then the n-dimensional m-linear Hardy operator $\mathcal{H}_{m}$ defined in (1.2) maps $\dot{M}^{\lambda _{1}}_{p_{1},\bar{p}_{1}}(\mathbb{R}^{n})\times \dot{M}^{ \lambda _{2}}_{p_{2},\bar{p}_{2}}(\mathbb{R}^{n})\times \cdots \times \dot{M}^{\lambda _{m}}_{p_{m},\bar{p}_{m}}(\mathbb{R}^{n})$ to $\dot{M}^{\lambda}_{p,\bar{p}}(\mathbb{R}^{n})$ with operator norm

$$\begin{aligned} & \Vert \mathcal{H}_{m} \Vert _{\dot{M}^{\lambda _{1}}_{p_{1},\bar{p}_{1}} \times \dot{M}^{\lambda _{2}}_{p_{2},\bar{p}_{2}}\times \cdots \times \dot{M}^{\lambda _{m}}_{p_{m},\bar{p}_{m}}\rightarrow \dot{M}^{ \lambda}_{p,\bar{p}}} \\ &\quad =w_{n}^{1/\bar{p}-\sum _{i=1}^{m}1/\bar{p}_{i}} \frac{w_{n}^{m}}{w_{mn}}\frac{m}{\lambda +m} \frac{1}{2^{m-1}} \frac{\Pi _{i=1}^{m}\Gamma (\frac{n}{2}(\lambda _{i}+1))}{\Gamma (\frac{n}{2}(m+\lambda ))}. \end{aligned}$$

Proof

We only prove the case $m=2$ since the cases $m\geq 3$ can be proven in a similar way.

A similar process as in the proof of Theorem 2.1 yields that the norm of the operator $\mathcal{H}_{2}$ from $\dot{M}^{\lambda _{1}}_{p_{1},\bar{p}_{1}}(\mathbb{R}^{n})\times \dot{M}^{ \lambda _{2}}_{p_{2},\bar{p}_{2}}(\mathbb{R}^{n})$ to $\dot{M}^{\lambda}_{p,\bar{p}}(\mathbb{R}^{n})$ is equal to the norm of $\mathcal{H}_{2}$ acting on radial functions. For two radial functions $f_{1}$ ad $f_{2}$, we then write, for $r=|x|$,

$$\begin{aligned} \mathcal{H}_{2}(f_{1},f_{2}) (x)&= \frac{1}{\Omega _{2n}} \int _{ \vert (y_{1},y_{2}) \vert < 1} f_{1}\bigl( \vert x \vert y_{1}\bigr)f_{2}\bigl( \vert x \vert y_{2}\bigr)\,dy _{1}\,dy _{2} \\ &=\frac{1}{\Omega _{2n}} \int _{ \vert (y_{1},y_{2}) \vert < 1} f_{1}(ry_{1})f_{2}(ry_{2})\,dy _{1}\,dy _{2}. \end{aligned}$$

By using Minkowski’s and Hölder’s inequalities, we get

$$\begin{aligned} &\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \bigl\Vert \mathcal{H}_{2}(f_{1},f_{2}) \cdot \chi _{B(0,R)} \bigr\Vert _{{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}}}} \\ &\quad =\frac{w_{n}^{1/\bar{p}}}{\Omega _{2n}} \frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \biggl( \int _{0}^{R} \biggl\vert \int _{ \vert (y_{1},y_{2}) \vert < 1} f_{1}(ry_{1})f_{2}(ry_{2})\,dy _{1}\,dy _{2} \biggr\vert ^{p}r^{n-1}\,dr \biggr)^{1/p} \\ &\quad \leq \frac{w_{n}^{1/\bar{p}}}{\Omega _{2n}} \int _{ \vert (y_{1},y_{2}) \vert < 1} \biggl(\frac{1}{ \vert B(0,R) \vert ^{1+p\lambda}} \int _{0}^{R} \bigl\vert f_{1}(ry_{1})f_{2}(ry_{2}) \bigr\vert ^{p}r^{n-1}\,dr \biggr)^{1/p} \,dy _{1}\,dy _{2} \\ &\quad \leq \frac{w_{n}^{1/\bar{p}}}{\Omega _{2n}} \int _{ \vert (y_{1},y_{2}) \vert < 1} \biggl(\frac{1}{ \vert B(0,R) \vert ^{1+p_{1}\lambda _{1}}} \int _{0}^{R} \bigl\vert f_{1} \bigl(r \vert y_{1} \vert \bigr) \bigr\vert ^{p_{1}}r^{n-1}\,dr \biggr)^{1/p_{1}} \\ &\quad\quad {}\times \biggl(\frac{1}{ \vert B(0,R) \vert ^{1+p_{2}\lambda _{2}}} \int _{0}^{R} \bigl\vert f_{2} \bigl(r \vert y_{2} \vert \bigr) \bigr\vert ^{p_{2}}r^{n-1}\,dr \biggr)^{1/p_{2}} \,dy _{1}\,dy _{2} \\ &\quad =\frac{w_{n}^{1/\bar{p}-1/\bar{p}_{1}-1/\bar{p}_{2}}}{\Omega _{2n}} \int _{ \vert (y_{1},y_{2}) \vert < 1} \frac{1}{ \vert B(0,R \vert y_{1} \vert ) \vert ^{1/p_{1}+\lambda _{1}}} \biggl( \int _{0}^{R \vert y_{1} \vert } w_{n}^{p_{1}/\bar{p}_{1}} \bigl\vert f_{1}(r) \bigr\vert ^{p_{1}}r^{n-1}\,dr \biggr)^{1/p_{1}} \\ &\quad\quad {}\times \frac{1}{ \vert B(0,R \vert y_{2} \vert ) \vert ^{1/p_{2}+\lambda _{2}}} \biggl( \int _{0}^{R \vert y_{2} \vert } w_{n}^{p_{2}/\bar{p}_{2}} \bigl\vert f_{2}(r) \bigr\vert ^{p_{2}}r^{n-1}\,dr \biggr)^{1/p_{2}} \vert y_{1} \vert ^{n\lambda _{1}} \vert y_{2} \vert ^{n\lambda _{2}}\,dy _{1}\,dy _{2} \\ &\quad \leq \frac{w_{n}^{1/\bar{p}-1/\bar{p}_{1}-1/\bar{p}_{2}}}{\Omega _{2n}} \int _{ \vert (y_{1},y_{2}) \vert < 1} \vert y_{1} \vert ^{n\lambda _{1}} \vert y_{2} \vert ^{n\lambda _{2}}\,dy _{1}\,dy _{2} \Vert f_{1} \Vert _{\dot{M}^{\lambda _{1}}_{p_{1},\bar{p}_{1}}} \Vert f_{2} \Vert _{ \dot{M}^{\lambda _{2}}_{p_{2},\bar{p}_{2}}}. \end{aligned}$$

It was proven by Fu et al. [11, Proof of Theorem 2] that

$$ \int _{ \vert (y_{1},y_{2}) \vert < 1} \vert y_{1} \vert ^{n\lambda _{1}} \vert y_{2} \vert ^{n\lambda _{2}}\,dy _{1}\,dy _{2}= \frac{w_{n}^{2}}{2n}\frac{1}{\lambda +2} B \biggl(\frac{n}{2}(1+ \lambda _{1}),\frac{n}{2}(1+\lambda _{2}) \biggr). $$

By using the identity $w_{2n}=2n\Omega _{2n}$ and

$$ B \biggl(\frac{n}{2}(1+\lambda _{1}), \frac{n}{2}(1+\lambda _{2}) \biggr)= \frac{\Gamma (\frac{n}{2}(1+\lambda _{1}))\Gamma (\frac{n}{2}(1+\lambda _{2}))}{\Gamma (\frac{n}{2}(2+\lambda ))}, $$

(4.1)

we obtain

$$\begin{aligned} & \bigl\Vert \mathcal{H}_{2}(f_{1},f_{2}) \bigr\Vert _{\dot{M}^{\lambda}_{p,\bar{p}}} \\ &\quad =\sup_{R>0}\frac{1}{ \vert B(0,R) \vert ^{1/p+\lambda}} \bigl\Vert \mathcal{H}_{2}(f_{1},f_{2}) \cdot \chi _{B(0,R)} \bigr\Vert _{{L^{p}_{ \vert x \vert }L_{\theta}^{\bar{p}}}} \\ &\quad \leq w_{n}^{1/\bar{p}-1/\bar{p}_{1}-1/\bar{p}_{2}} \frac{w_{n}^{2}}{w_{2n}} \frac{1}{\lambda +2} \frac{\Gamma (\frac{n}{2}(1+\lambda _{1}))\Gamma (\frac{n}{2}(1+\lambda _{2}))}{\Gamma (\frac{n}{2}(2+\lambda ))} \Vert f_{1} \Vert _{\dot{M}^{\lambda _{1}}_{p_{1},\bar{p}_{1}}} \Vert f_{2} \Vert _{ \dot{M}^{\lambda _{2}}_{p_{2},\bar{p}_{2}}}, \end{aligned}$$

and this proves one direction in the claimed identity.

On the other hand, taking $\bar{f}_{i}(x)=|x_{i}|^{n\lambda _{i}}$ for $x_{i}\in \mathbb{R}^{n}$, $i=1,2$, in view of (2.5), we have $\bar{f}_{i}\in \dot{M}^{\lambda _{i}}_{p_{i},\bar{p_{i}}}(\mathbb{R}^{n})$ and, moreover,

$$ \Vert \bar{f_{i}} \Vert _{\dot{M}^{\lambda _{i}}_{p_{i},\bar{p_{i}}}} = \frac{w_{n}^{1/\bar{p_{i}}}}{\Omega _{n}^{1/p_{i}+\lambda _{i}}[n(1+p_{i}\lambda _{i})]^{1/p_{i}}}. $$

(4.2)

By a simple calculation, we can see that

$$ \mathcal{H}_{2}(\bar{f}_{1},\bar{f}_{2}) (x) =\bar{f}_{1}(x)\bar{f}_{2}(x) \frac{1}{\Omega _{2n}} \int _{ \vert (y_{1},y_{2}) \vert < 1} \vert y_{1} \vert ^{n\lambda _{1}} \vert y_{2} \vert ^{n \lambda _{2}}\,dy _{1}\,dy _{2}. $$

This observation, combined with (4.1), (4.2) and the fact $p_{i}\lambda _{i}=p\lambda$, $i=1,2$, finishes the proof. □

By taking $\bar{p}=p$ and $\bar{p}_{i}=p_{i}$, $i=1,2,\dots ,m$, we recover the result of Fu et al. [11, Theorem 2].