Let f be a non-negative integrable function on ${{\mathbb{R}}^{n}} $. The fractional Hardy operator on ${{\mathbb{R}}^{n}} $ is defined by

$ {\mathbb{H}_\beta }f\left( x \right): = \frac{1}{{{{\left| {B\left( {0,\left| x \right|} \right)} \right|}^{1 - \frac{\beta }{n}}}}}\int_{\left| y \right| < \left| x \right|} {f\left( y \right){\text{d}}y} , $

(1)

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

For weak and strong operator norms of fractional Hardy operator, the following Theorem due to Lu, Zhao, and Yan^[1] is well-known.

Theorem A Suppose 0 < β < n, 1 < p < q < $\frac{n}{\beta } $, and $\frac{1}{p}-\frac{1}{q}=\frac{\beta}{n} $.

(ⅰ) If $f \in L^{p}\left(\mathbb{R}^{n}\right)$, we have

$ {\mathbb{H}_\beta }f\left\| {_{{L^q}\left( {{\mathbb{R}^n}} \right)}} \right. \leqslant C{\left\| f \right\|_{{L^p}\left( {{\mathbb{R}^n}} \right)}}, $

where

$ \begin{array}{*{20}{c}} {{{\left( {\frac{p}{q}} \right)}^{\frac{1}{q}}}{{\left( {\frac{p}{{p - 1}}} \right)}^{\frac{1}{q}}}{{\left( {\frac{q}{{q - 1}}} \right)}^{1 - \frac{1}{q}}}{{\left( {1 - \frac{p}{q}} \right)}^{\frac{1}{p} - \frac{1}{q}}}} \\ { \leqslant C \leqslant {{\left( {\frac{p}{{p - 1}}} \right)}^{\frac{p}{q}}}.} \end{array} $

(ⅱ) If $f \in L^{1}\left(\mathbb{R}^{n}\right) $, then for any λ>0,

$ \left| {x \in {\mathbb{R}^n}:\left| {{\mathbb{H}_\beta }\left( f \right)\left( x \right)} \right| > \lambda } \right| \leqslant {\left( {\frac{1}{\lambda }{{\left\| f \right\|}_{{L^1}}}} \right)^{\frac{n}{{n - \beta }}}}. $

Moreover,

$ {\left\| {{\mathbb{H}_\beta }} \right\|_{{L^1}\left( {{\mathbb{R}^n}} \right) \to {L^{\frac{n}{{n - \beta }},\infty }}\left( {{\mathbb{R}^n}} \right)}} = 1. $

Based on the previous work^[1], Zhao and Lu^[2] obtained the operator norm of fractional Hardy operators on ${{\mathbb{R}}^{n}} $.

Theorem B Suppose that 0 < β < n, 1 < p < q < ∞, and $\frac{1}{p}-\frac{1}{q}=\frac{\beta}{n} $. If $f\in {{L}^{p}}\left( {{\mathbb{R}}^{n}} \right)$, we have

$ {\left\| {{\mathbb{H}_\beta }f} \right\|_{{L^q}\left( {{\mathbb{R}^n}} \right)}} \leqslant A{\left\| f \right\|_{{L^p}\left( {{\mathbb{R}^n}} \right)}}. $

Moreover,

$ {\left\| {{\mathbb{H}_\beta }} \right\|_{{L^q}\left( {{\mathbb{R}^n}} \right) \to {L^q}\left( {{\mathbb{R}^n}} \right)}} = A, $

where

$ A = \left( {\frac{{p'}}{q}} \right){\left( {\frac{n}{{q\beta }} \cdot B\left( {\frac{n}{{q\beta }},\frac{n}{{q'\beta }}} \right)} \right)^{\frac{{ - \beta }}{n}}}. $

In recent years, Hardy operator on product space has received much concern. In 2012, Wang, et al^[3] introduced Hardy operator on product space and obtained the operator norm. Subsequently, Lu, et al^[1] generalized Wang's work to higher-dimensional product space. Very recently, He, et al^[4] investigated Hardy type operators on higher-dimensional product spaces and obtained the sharp bounds. Inspired by Refs.[1-4], we will compute the operator norm of fractional Hardy operator on higher-dimensional product space in the present paper.

Now we are in a position to introduce fractional Hardy operator on higher-dimensional product space. Let f be a non-negative integrable function on $ \mathbb{R}^{n_{1}} \times \ldots \times \mathbb{R}^{n_{m}}$ and then the fractional Hardy operator on product space is defined by

$ \begin{gathered} {\mathbb{H}_{{\beta _1}, \cdots ,{\beta _m}}}f\left( x \right): = \frac{1}{{{{\left| {B\left( {0,\left| {{x_1}} \right|} \right)} \right|}^{1 - \frac{{{\beta _1}}}{{{n_1}}}}}}} \cdots \hfill \\ \frac{1}{{{{\left| {B\left( {0,\left| {{x_m}} \right|} \right)} \right|}^{1 - \frac{{{\beta _m}}}{{{n_m}}}}}}} \times \int_{\left| {{y_1}} \right| < \left| {{x_1}} \right|} \cdots \hfill \\ \int_{\left| {{y_m}} \right| < \left| {{x_m}} \right|} {f\left( {{y_1}, \cdots ,{y_m}} \right){\text{d}}{y_1} \cdots {\text{d}}{y_m}} \hfill \\ \end{gathered} $

(2)

for $ x=\left(x_{1}, x_{2}, \cdots, x_{m}\right) \in \mathbb{R}^{n_{1}} \times \mathbb{R}^{n_{2}} \times \cdots \times \mathbb{R}^{n_{m}}$ and $0 <\beta_{i} <n_{i}, i=1, \cdots, m $.

1 Preliminaries

Before we present the main results, some useful lemmas and definitions are needed.

The mixed norm space was first defined in Ref.[5] by Benedek and Panzone and received much concern (See Refs.[6-10]). In 2018, Wei and Yan^[11] defined a more general mixed norm space which is called weak and strong mixed-norm space. We list its definition for completeness.

Definition 1.1 Let (X_i, S_i, μ_i), for 1≤i≤n, be n given, totally σ-finite measure spaces and P=(p₁, p₂, …, p_n) a given n-tuple with 1≤p_i≤∞. The set I satisfies $ I \subset\{1, \cdots, n\}$. A function f(x₁, x₂, …, x_n) measurable in the product spaces (X, S, μ)=$ \left( \prod\limits_{i=1}^{n}{{{X}_{i}}}, \prod\limits_{i=1}^{n}{{{S}_{i}}}, \prod\limits_{i=1}^{n}{{{\mu }_{i}}} \right)$ is said to belong to the space L^P_I(X) if the number obtained after subsequently taking successfully the mixed norm where we take p_i-norm for i∈I while we take weak p_j-norm for j∈{1, …, n}\I and in natural order is finite. The number so obtained, finite or not, will be denoted by ‖f‖_PI.

We give some necessary remarks for the space L^P_I(X). For more properties, we refer readers to Ref.[11].

(ⅰ) If I={1, …, n}, we call L^P_I(X) strong mixed norm space, which is also denoted by L^P(X) or L^{p₁, …, p_n}(X).

(ⅱ) If the set I is empty, we call L^P_I(X) weak mixed norm space, which is also denoted by wL^P(X) or wL^{p₁, …, p_n}(X).

(ⅲ) The space L^P_I(X) is a quasi-normed space for P≥1.

For the mixed norm, we have a basic lemma which plays an important role in the proof of our main theorems.

Lemma 1.1 Let (X, S, μ) be defined as in the above definitions. If p_n≥…≥p₁≥1 and f∈L^{p_n, …, p₁}(X), then f∈L^{p₁, …, p_n}(X) and there holds

$ {\left\| f \right\|_{{L^{{p_1}, \cdots ,{p_n}}}\left( X \right)}} \leqslant {\left\| f \right\|_{{L^{{p_n}, \cdots ,{p_1}}}\left( X \right)}}. $

Lemma 1.1 is a direct generation of the Minkowski's inequality. For the proof of the lemma, readers are referred to Ref.[12]. It is not hard to see Fubini's theorem is a special case of Lemma 1.1.

In the rest part of our paper, we always consider the spaces on Euclidean space.

2 Main results and proof

Now we formulate our main theorems. We first give the boundedness of Hardy operator on product space from $ L^{1}\left(\mathbb{R}^{n_{1}} \times \cdots \times \mathbb{R}^{n_{m}}\right)$ to $ w L^{\mathit{\boldsymbol{Q}}}\left(\mathbb{R}^{n_{1}} \times \cdots \times \mathbb{R}^{n_{m}}\right)$.

Theorem 2.1 Let 0 < β_i < n_i, Q= $ \left(\frac{n_{1}}{n_{1}-\beta_{2}}, \cdots, \frac{n_{m}}{n_{m}-\beta_{m}}\right), i=1, \cdots, m$. Then the operator $ \mathbb{H}{_{{\beta _1}, \cdots , {\beta _n}}}$ defined by (2) is bounded from $L^{1}\left(\mathbb{R}^{n_{1}} \times \cdots \times \mathbb{R}^{n_{m}}\right) $ to $ w L^{\mathit{\boldsymbol{Q}}}\left(\mathbb{R}^{n_{1}} \times \cdots \times \mathbb{R}^{n_{m}}\right)$. Furthermore,

$ {\left\| {{\mathbb{H}_{{\beta _1}, \cdots ,{\beta _m}}}} \right\|_{{L^1}\left( {{\mathbb{R}^{{n_1}}} \times \cdots \times {\mathbb{R}^{{n_m}}}} \right) \to w{L^Q}\left( {{\mathbb{R}^{{n_1}}} \times \cdots \times {\mathbb{R}^{{n_m}}}} \right)}} = 1. $

(3)

Proof To make the arguments more easily understood, we prove the boundedness of the case m=2 first, and then the case m≥3 is just a repetition of the case m=2.

For m=2, the operator ${{\mathbb{H}}_{{{\beta }_{1}}, {{\beta }_{2}}}} $ can be written as

$ \begin{array}{*{20}{c}} {\left( {{\mathbb{H}_{{\beta _1},{\beta _2}}}f} \right)\left( {{x_1},{x_2}} \right) = \frac{1}{{{{\left| {B\left( {0,\left| {{x_1}} \right|} \right)} \right|}^{1 - \frac{{{\beta _1}}}{{{n_1}}}}}}}} \\ {\frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_1}} \right| < \left| {{x_1}} \right|} {\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {f\left( {{y_1},{y_2}} \right){\text{d}}{y_1}{\text{d}}{y_2}} } .} \end{array} $

When $f \in L^{1}\left(\mathbb{R}^{n_{1}} \times \mathbb{R}^{n_{2}}\right) $, we have

$ \frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {f\left( { \cdot ,{y_2}} \right){\text{d}}{y_2}} \in {L^1}\left( {{\mathbb{R}^{{n_1}}}} \right) $

for all $ x_{2} \in \mathbb{R}^{n_{2}}$. Then Theorem A yields that

$ \begin{array}{*{20}{c}} {{{\left\| {\left( {{\mathbb{H}_{{\beta _1},{\beta _2}}}f} \right)\left( { \cdot ,{x_2}} \right)} \right\|}_{{L^{\frac{{{n_1}}}{{{n_1} - {\beta _1}}},\infty }}\left( {{R^{{n_1}}}} \right)}} = } \\ {\mathop {\sup }\limits_{{\lambda _1} > 0} {\lambda _1}{{\left| {\left\{ {{x_1}:\left( {{\mathbb{H}_{{\beta _1},{\beta _2}}}f} \right)\left( {{x_1},{x_2}} \right) > {\lambda _1}} \right\}} \right|}^{\frac{{{n_1}}}{{{n_1} - {\beta _1}}}}} \leqslant } \\ {1 \cdot {{\left\| {\frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {f\left( { \cdot ,{y_2}} \right){\text{d}}{y_2}} } \right\|}_{{L^1}\left( {{\mathbb{R}^{{n_1}}}} \right)}}.} \end{array} $

(4)

Using Lemma 1.1 (or Fubini's theorem), it is not hard for us to get

$ \begin{gathered} {\left\| {\frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {f\left( { \cdot ,{y_2}} \right){\text{d}}{y_2}} } \right\|_{{L^1}\left( {{\mathbb{R}^{{n_1}}}} \right)}} \hfill \\ = \frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{{\mathbb{R}^{{n_1}}}} {\left| {\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {f\left( {{y_1},{y_2}} \right){\text{d}}{y_2}} } \right|{\text{d}}{y_1}} \hfill \\ \leqslant \frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {\int_{{\mathbb{R}^{{n_1}}}} {\left| {f\left( {{y_1},{y_2}} \right)} \right|{\text{d}}{y_1}{\text{d}}{y_2}} } . \hfill \\ \end{gathered} $

(5)

Obviously, $ \int_{\mathbb{R}^{n_{1}}}\left|f\left(y_{1}, y_{2}\right)\right| \mathrm{d} y_{1} \in L^{1}\left(\mathbb{R}^{n_{2}}\right)$ if $ f\in {{L}^{1}}\left( {{\mathbb{R}}^{{{n}_{1}}}}\times {{\mathbb{R}}^{{{n}_{2}}}} \right)$. Then, applying Lemma 1.1 again, we obtain

$ \begin{array}{*{20}{c}} {\left\| {\frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {} } \right.} \\ {{{\left. {\left( {\int_{{\mathbb{R}^{{n_1}}}} {\left| {f\left( {{y_1},{y_2}} \right)} \right|{\text{d}}{y_1}} } \right){\text{d}}{y_2}} \right\|}_{{L^{\frac{{{n_2}}}{{{n_2} - {\beta _2}}},\infty }}\left( {{\mathbb{R}^{{n_2}}}} \right)}}} \\ { = \mathop {\sup }\limits_{{\lambda _2} > 0} {\lambda _2}\left| {\left\{ {{x_2}:\frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {} } \right.} \right.} \\ {{{\left. {\left. {\left( {\int_{{\mathbb{R}^{{n_1}}}} {\left| {f\left( {{y_1},{y_2}} \right)} \right|{\text{d}}{y_1}} } \right){\text{d}}{y_2} > {\lambda _2}} \right\}} \right|}^{\frac{{{n_2}}}{{{n_2} - {\beta _2}}}}}} \\ { \leqslant 1 \cdot \int_{{\mathbb{R}^{{n_2}}}} {\int_{{\mathbb{R}^{{n_1}}}} {\left| {f\left( {{y_1},{y_2}} \right)} \right|} } {\text{d}}{y_1}{\text{d}}{y_2}} \\ { = {{\left\| f \right\|}_{{L^1}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}}.} \end{array} $

(6)

Combining (4), (5), and (6), we obtain

$ \begin{array}{*{20}{c}} {{{\left\| {{\mathbb{H}_{{\beta _1},{\beta _2}}}f} \right\|}_{w{L^{\frac{{{n_2}}}{{{n_1} - {\beta _2}}},\frac{{{n_2}}}{{{n_2} - {\beta _2}}}}}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}} \leqslant } \\ {1 \cdot {{\left\| f \right\|}_{{L^1}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}},} \end{array} $

for all $ f\in {{L}^{1}}\left( {{\mathbb{R}}^{{{n}_{1}}}}\times {{\mathbb{R}}^{{{n}_{2}}}} \right)$.

On the other hand, we will show that the constant 1 is the best possible. Denote χ_{[0, 1]} by χ₁. Takingf₀(r)=χ₁(r), r>0 and choosing F(x₁, x₂)=f₀(|x₁|)f₀(|x₂|), where $ x_{1} \in \mathbb{R}^{n_{1}}, x_{2} \in \mathbb{R}^{n_{2}}$, we get from the definition of $ {{\mathbb{H}}_{{{\beta }_{1}}, {{\beta }_{2}}}}$ that

$ \begin{array}{*{20}{c}} {{\mathbb{H}_{{\beta _1}{\beta _2}}}F\left( {{x_1},{x_2}} \right) = \frac{1}{{{{\left| {B\left( {0,\left| {{x_1}} \right|} \right)} \right|}^{1 - \frac{{{\beta _1}}}{{{n_1}}}}}{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}} \\ {\int_{\left| {{y_1}} \right| < \left| {{x_1}} \right|} {\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {{f_0}} } \left( {\left| {{y_1}} \right|} \right){f_0}\left( {\left| {{y_2}} \right|} \right){\text{d}}{y_1}{\text{d}}{y_2}} \\ { = \frac{1}{{{{\left| {B\left( {0,\left| {{x_1}} \right|} \right)} \right|}^{1 - \frac{{{\beta _1}}}{{{n_1}}}}}}}\int_{\left| {{y_1}} \right| < \left| {{x_1}} \right|} {{f_0}\left( {\left| {{y_1}} \right|} \right){\text{d}}{y_1}} } \\ {\frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {{f_0}\left( {\left| {{y_2}} \right|} \right){\text{d}}{y_2}} } \\ { = {\mathbb{H}_{{\beta _{\text{1}}}}}{f_0}\left( {\left| {{x_1}} \right|} \right){\mathbb{H}_{{\beta _2}}}{f_0}\left( {\left| {{x_2}} \right|} \right).} \end{array} $

For $ 0 <{{\lambda }_{1}} <\frac{1}{{{\left| B\left( 0, \left| {{x}_{2}} \right| \right) \right|}^{1-\frac{{{\beta }_{2}}}{{{n}_{2}}}}}}\int_{{|{y}_{2}|} <\left| {{x}_{2}} \right|}{{}}{{f}_{0}}\left( \left| {{y}_{2}} \right| \right)\text{d}{{y}_{2}}={{\mathbb{H}}_{{{\beta }_{2}}}}{{f}_{0}}\left( \left| {{x}_{2}} \right| \right)$, we divide x₁ into two cases:

(ⅰ) when |x₁| < 1,

$ \begin{gathered} \left| {{\mathbb{H}_{{\beta _{\text{1}}}}}{f_0}\left( {\left| {{x_1}} \right|} \right)} \right| = \frac{1}{{{{\left| {B\left( {0,\left| {{x_1}} \right|} \right)} \right|}^{1 - \frac{{{\beta _1}}}{{{n_1}}}}}}}\int_{\left| {{y_1}} \right| < \left| {{x_1}} \right|} {{f_0}\left( {\left| {{y_1}} \right|} \right){\text{d}}{y_1}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{1}{{{{\left| {B\left( {0,\left| {{x_1}} \right|} \right)} \right|}^{1 - \frac{{{\beta _1}}}{{{n_1}}}}}}}\int_{\left| {{y_1}} \right| < \left| {{x_1}} \right|} {{\chi _1}\left( {\left| {{y_1}} \right|} \right){\text{d}}{y_1}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{{{v_{{n_1}}}{{\left| {{x_1}} \right|}^{{n_1}}}}}{{v_{{n_1}}^{1 - \frac{{{\beta _1}}}{{{n_1}}}}{{\left| {{x_1}} \right|}^{{n_1} - {\beta _1}}}}}; \hfill \\ \end{gathered} $

(ⅱ) when |x₁|≥1,

$ \begin{gathered} \left| {{\mathbb{H}_{{\beta _{\text{1}}}}}{f_0}\left( {\left| {{x_1}} \right|} \right)} \right| = \frac{1}{{{{\left| {B\left( {0,\left| {{x_1}} \right|} \right)} \right|}^{1 - \frac{{{\beta _1}}}{{{n_1}}}}}}}\int_{\left| {{y_1}} \right| < \left| {{x_1}} \right|} {{\chi _0}\left( {\left| {{y_1}} \right|} \right){\text{d}}{y_1}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{1}{{{{\left| {B\left( {0,\left| {{x_1}} \right|} \right)} \right|}^{1 - \frac{{{\beta _1}}}{{{n_1}}}}}}}\int_{\left| {{y_1}} \right| < 1} {{\text{d}}{y_1}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{{{v_{{n_1}}}}}{{v_{{n_1}}^{1 - \frac{{{\beta _1}}}{{{n_1}}}}{{\left| {{x_1}} \right|}^{{n_1} - {\beta _1}}}}}. \hfill \\ \end{gathered} $

Then, combining both the cases we obtain

$ \begin{array}{*{20}{c}} {\left| {\left\{ {{x_1} \in {\mathbb{R}^{{n_1}}}:\left| {{\mathbb{H}_{{\beta _1},{\beta _2}}}F\left( {{x_1},{x_2}} \right)} \right| > {\lambda _1}} \right\}} \right| = } \\ {\left| {B\left( {0,1} \right) \cap \left\{ {{x_1} \in {\mathbb{R}^{{n_1}}}:} \right.} \right.} \\ {\left. {\left. {v_{{n_1}}^{\frac{{{\beta _1}}}{{{n_1}}}}{{\left| {{x_1}} \right|}^{{\beta _1}}}{\mathbb{H}_{{\beta _2}}}{f_0}\left( {\left| {{x_2}} \right|} \right) > {\lambda _1}} \right\}} \right| + } \\ {\left| {B{{\left( {0,1} \right)}^c} \cap \left\{ {{x_1} \in {\mathbb{R}^{{n_1}}}:} \right.} \right.} \\ {\left. {\left. {v_{{n_1}}^{\frac{{{\beta _1}}}{{{n_1}}}}{{\left| {{x_1}} \right|}^{{\beta _1} - {n_1}}}{\mathbb{H}_{{\beta _2}}}{f_0}\left( {\left| {{x_2}} \right|} \right) > {\lambda _1}} \right\}} \right|} \\ { = \left| {\left\{ {{x_1} \in {\mathbb{R}^{{n_1}}}:\frac{{\lambda _1^{\frac{1}{{{\beta _1}}}}}}{{v_{{n_1}}^{\frac{1}{{{n_1}}}}\mathbb{H}_{{\beta _2}}^{\frac{1}{{{\beta _1}}}}{f_0}\left( {\left| {{x_2}} \right|} \right)}} < \left| {{x_1}} \right| < 1} \right\}} \right| + } \\ {\left| {\left\{ {{x_1} \in {\mathbb{R}^{{n_1}}}:1 \leqslant \left| {{x_1}} \right| < {{\left( {\frac{{v_{{n_1}}^{\frac{{{\beta _1}}}{{{n_1}}}}{\mathbb{H}_{{\beta _2}}}{f_0}\left( {\left| {{x_2}} \right|} \right)}}{{{\lambda _1}}}} \right)}^{\frac{1}{{{n_1} - {\beta _1}}}}}} \right\}} \right|} \\ { = {{\left( {\frac{{{v_{{n_1}}}{\mathbb{H}_{{\beta _2}}}{f_0}\left( {\left| {{x_2}} \right|} \right)}}{{{\lambda _1}}}} \right)}^{\frac{{{n_1}}}{{{n_1} - {\beta _1}}}}} - {{\left( {\frac{{{\lambda _1}}}{{{\mathbb{H}_{{\beta _2}}}{f_0}\left( {\left| {{x_2}} \right|} \right)}}} \right)}^{\frac{{{n_1}}}{{{\beta _1}}}}}.} \end{array} $

Based on the above results, let λ₁→0⁺ and we have the estimate

$ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{{\lambda _1} \to 0} {\lambda _1}{{\left| {\left\{ {{x_1} \in \mathbb{R}:\left| {{\mathbb{H}_{{\beta _1},{\beta _2}}}F\left( {{x_1},{x_2}} \right)} \right| > {\lambda _1}} \right\}} \right|}^{\frac{{{n_1} - {\beta _1}}}{{{n_1}}}}}} \\ { = {v_{{n_1}}}{\mathbb{H}_{{\beta _2}}}{f_0}\left( {\left| {{x_2}} \right|} \right),} \end{array} $

(7)

which implies that

$ \begin{array}{*{20}{c}} {\mathop {\sup }\limits_{0 < {\lambda _1} < {\mathbb{H}_{{\beta _2}}}{f_0}\left( {\left| {{x_2}} \right|} \right)} {\lambda _1}\left| {\left\{ {{x_1}:\left| {{\mathbb{H}_{{\beta _1},{\beta _2}}}F\left( {{x_1},{x_2}} \right)} \right|} \right.} \right.} \\ {{{\left. {\left. { > {\lambda _1}} \right\}} \right|}^{\frac{{{n_1}}}{{{n_1} - {\beta _1}}}}} = {v_{{n_1}}}{\mathbb{H}_{{\beta _2}}}{f_0}\left( {\left| {{x_2}} \right|} \right).} \end{array} $

(8)

For $0 <\lambda_{2} <v_{n_{1}} $, we also divide x₂ into two cases: |x₂| < 1 and |x₂|≥1. As above, we obtain

$ \begin{array}{*{20}{c}} {\mathop {\sup }\limits_{0 < {\lambda _2} < {v_{{n_1}}}} {\lambda _2}\left| {\left\{ {{x_2} \in {\mathbb{R}^{{n_2}}}:{v_{{n_1}}}{\mathbb{H}_{{\beta _2}}}{f_0}\left( {\left| {{x_2}} \right|} \right)} \right.} \right.} \\ {{{\left. {\left. { > {\lambda _2}} \right\}} \right|}^{\frac{{{n_2}}}{{{n_2} - {\beta _2}}}}} = {v_{{n_1}}}{v_{{n_2}}}.} \end{array} $

(9)

Since $\|F{{\|}_{{{L}^{1}}\left( {{\mathbb{R}}^{{{n}_{1}}\times {{n}_{2}}}} \right)}}={{v}_{{{n}_{1}}}}{{v}_{{{n}_{2}}}} $, by combining (8) with (9) we obtain

$ \begin{array}{*{20}{c}} {{{\left\| {{\mathbb{H}_{{\beta _1},{\beta _2}}}F} \right\|}_{w{L^{\frac{{{n_1}}}{{{n_1} - {\beta _1}}},\frac{{{n_2}}}{{{n_2} - {\beta _2}}}}}}}_{\left( {{\mathbb{R}^{{n_1} \times {n_2}}}} \right)}} \\ { = 1 \cdot {{\left\| F \right\|}_{{L^1}\left( {{\mathbb{R}^{{n_1} \times {n_2}}}} \right)}}.} \end{array} $

This finishes the proof of Theorem 2.1.

At last, we prove a more general result involved the weak and strong mixed-norm space.

Theorem 2.2 Suppose P=(p₁, …, p_m)≥1, Q=(q₁, …, q_m)≥1, and for all i=1, …, m, 0 < $ \beta_{i} <n, \frac{1}{p_{i}}-\frac{1}{q_{i}}=\frac{\beta_{i}}{n_{i}}$ . Let $ I \subset\{1, \cdots, m\}$, and we further suppose 1 < p_i < ∞ if i∈I. Then the operator $ {{\mathbb{H}}_{{{\beta }_{1}}, \cdots , {{\beta }_{m}}}}$ defined by (2) is bounded from ${{L}^{\mathit{\boldsymbol{P}}}}\left( {{\mathbb{R}}^{{{n}_{1}}}}\times \cdots \times {{\mathbb{R}}^{{{n}_{m}}}} \right) $ to $ {{L}^{{{\mathit{\boldsymbol{Q}}}_{I}}}}\left( {{\mathbb{R}}^{{{n}_{1}}}}\times \cdots \times {{\mathbb{R}}^{{{n}_{m}}}} \right)$. Moreover, there holds

$ {\left\| {{H_{{\beta _1}, \cdots ,{\beta _m}}}} \right\|_{{L^\mathit{\boldsymbol{p}}}\left( {{\mathbb{R}^{{n_1}}} \times \cdots \times {\mathbb{R}^{{n_m}}}} \right) \to {L^{{\mathit{\boldsymbol{Q}}_I}}}\left( {{\mathbb{R}^{{n_1}}} \times \cdots \times {\mathbb{R}^{{n_m}}}} \right)}} = \prod\limits_{i \in I} {{C_i}} , $

(10)

where

$ {C_i} = {\left( {\frac{{{{p'}_i}}}{{{q_i}}}} \right)^{\frac{1}{{{q_i}}}}}{\left( {\frac{{{n_i}}}{{{q_i}{\beta _i}}} \cdot B\left( {\frac{{{n_i}}}{{{q_i}{\beta _i}}},\frac{{{n_i}}}{{{{q'}_i}{\beta _i}}}} \right)} \right)^{ - \frac{{{\beta _i}}}{{{n_i}}}}}. $

Proof We only prove the case m=2 and I={1} for simplification, that is to say, for p₁>1 and p₂≥1 we need to verify

$ {\left\| {{\mathbb{H}_{{\beta _1},{\beta _2}}}f} \right\|_{{L^{{\mathit{\boldsymbol{Q}}_I}}}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}} \leqslant {C_1} \cdot {\left\| f \right\|_{{L^{{p_1} \cdot {p_2}}}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}}, $

(11)

where the constant C₁ is sharp.

Noting that, when $ f \in L^{p_{1}, p_{2}}\left(\mathbb{R}^{n_{1}} \times \mathbb{R}^{n_{2}}\right)$, $\frac{1}{{{\left| B\left( 0,\left| {{x}_{2}} \right| \right) \right|}^{1-\frac{{{\beta }_{2}}}{{{n}_{2}}}}}}\int_{|{{y}_{2}}| <\left| {{x}_{2}} \right|}{{}} f\left(\cdot, y_{2}\right) \mathrm{d} y_{2} \in L^{p_{1}}\left(\mathbb{R}^{n_{1}}\right)$, for all $x_{2} \in \mathbb{R}^{n_{2}} $, we get from Theorem B that

$ \begin{array}{*{20}{c}} {{{\left\| {\left( {{\mathbb{H}_{{\beta _1},{\beta _2}}}f} \right)\left( { \cdot ,{x_2}} \right)} \right\|}_{{L^{{q_1}}}\left( {{\mathbb{R}^{{n_1}}}} \right)}} \leqslant {C_1} \times } \\ {{{\left\| {\frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {f\left( { \cdot ,{y_2}} \right){\text{d}}{y_2}} } \right\|}_{{L^{{p_1}}}\left( {{\mathbb{R}^{{n_1}}}} \right)}}.} \end{array} $

(12)

Using Lemma 1.1 (or Fubini's theorem), it is not hard for us to get

$ \begin{array}{*{20}{c}} {{{\left\| {\frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} f \left( { \cdot ,{y_2}} \right){\text{d}}{y_2}} \right\|}_{{L^{{p_1}}}\left( {{\mathbb{R}^{{n_1}}}} \right)}}} \\ { = \frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}{{\left[ {\int_{{\mathbb{R}^{{n_1}}}} {{{\left| {\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} f \left( {{y_1},{y_2}} \right){\text{d}}{y_2}} \right|}^{{p_1}}}} {\text{d}}{y_1}} \right]}^{\frac{1}{{{p_1}}}}}} \\ { \leqslant \frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {{{\left[ {\int_{{\mathbb{R}^{{n_1}}}} {{{\left| {f\left( {{y_1},{y_2}} \right)} \right|}^{{p_1}}}} {\text{d}}{y_1}} \right]}^{\frac{1}{{{p_1}}}}}{\text{d}}{y_2}} .} \end{array} $

Theorem B tells us that, when $ f\in {{L}^{{{p}_{1}}, {{p}_{2}}}}\left( {{\mathbb{R}}^{{{n}_{1}}}}\times {{\mathbb{R}}^{{{n}_{2}}}} \right)$, there holds

$ \begin{array}{*{20}{c}} {\left\| \; \right\|\frac{1}{{{{\left| {B\left( {0,\left| {{x_2}} \right|} \right)} \right|}^{1 - \frac{{{\beta _2}}}{{{n_2}}}}}}}\int_{\left| {{y_2}} \right| < \left| {{x_2}} \right|} {{{\left[ {\int_{{\mathbb{R}^{{n_1}}}} {{{\left| {f\left( {{y_1},{y_2}} \right)} \right|}^{{p_1}}}} {\text{d}}{y_1}} \right]}^{\frac{1}{{{p_1}}}}}} } \\ {{\text{d}}{y_2}\left\| {_{{L^{{p_1}}}\left( {{\mathbb{R}^{{n_1}}}} \right)}} \right.\left\| {_{{L^{{q_2},\infty }}\left( {{\mathbb{R}^{{n_2}}}} \right)}} \right. \leqslant 1 \cdot {{\left\| f \right\|}_{{L^{{p_1},{p_2}}}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}}.} \end{array} $

Based on the above estimates, we immediately obtain the inequality (11).

In order to show that C₁ is the best, we construct a sharp function

$ F\left( {{x_1},{x_2}} \right) = {f_1}\left( {{x_1}} \right){f_2}\left( {{x_2}} \right), $

where

$ {f_1}\left( {{x_1}} \right) = \frac{1}{{{{\left( {1 + {{\left| {{x_1}} \right|}^{{q_1}{\beta _1}}}} \right)}^{1 + \frac{{{n_1}}}{{{q_1}{\beta _1}}}}}}}, $

and

$ {f_2}\left( {{x_2}} \right) = {\chi _1}\left( {\left| {{x_2}} \right|} \right). $

As stated in Theorem B, it means

$ \frac{{{{\left\| {{\mathbb{H}_{{\beta _{\text{1}}}}}{f_1}} \right\|}_{{L^{{p_1}}}\left( {{\mathbb{R}^{{n_1}}}} \right)}}}}{{{{\left\| {{f_1}} \right\|}_{_{{L^{{p_1}}}\left( {{\mathbb{R}^{{n_1}}}} \right)}}}}} = {C_1}. $

(13)

From the proof of Theorem A, we have

$ \frac{{{{\left\| {{\mathbb{H}_{{\beta _{\text{2}}}}}{f_2}} \right\|}_{w{L^{{p_2}}}\left( {{\mathbb{R}^{{n_2}}}} \right)}}}}{{{{\left\| {{f_2}} \right\|}_{_{{L^{{p_2}}}\left( {{\mathbb{R}^{{n_2}}}} \right)}}}}} = 1. $

(14)

Combining (13) with (14) we obtain

$ \begin{array}{*{20}{c}} {{{\left\| {{\mathbb{H}_{{\beta _1},{\beta _2}}}} \right\|}_{{L^{{p_1},{p_2}}}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right) \to {L^{{Q_I}}}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}}} \\ { = \mathop {\sup }\limits_{{{\left\| F \right\|}_{{L^{{p_1},{p_2}}}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}} \ne 0} \frac{{{{\left\| {{\mathbb{H}_{{\beta _1},{\beta _2}}}F} \right\|}_{{L^{{Q_I}}}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}}}}{{{{\left\| F \right\|}_{{L^{{p_1},{p_2}}}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}}}}} \\ { \geqslant \frac{{{{\left\| {{\mathbb{H}_{{\beta _1},{\beta _2}}}{f_1}{f_2}} \right\|}_{{L^{{Q_I}}}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}}}}{{{{\left\| {{f_1}{f_2}} \right\|}_{{L^{{p_1},{p_2}}}\left( {{\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}}} \right)}}}}} \\ { = \frac{{{{\left\| {{\mathbb{H}_{{\beta _1}}}{f_1}} \right\|}_{{L^{{p_1}}}\left( {{\mathbb{R}^{{n_1}}}} \right)}}}}{{{{\left\| {{f_1}} \right\|}_{{L^{{p_1}}}\left( {{\mathbb{R}^{{n_1}}}} \right)}}}} \cdot \frac{{{{\left\| {{\mathbb{H}_{{\beta _2}}}{f_2}} \right\|}_{{L^{{p_2}}}\left( {{\mathbb{R}^{{n_2}}}} \right)}}}}{{{{\left\| {{f_2}} \right\|}_{{L^{{p_2}}}\left( {{\mathbb{R}^{{n_2}}}} \right)}}}} = {C_1}.} \end{array} $

Thus we finish the proof.

We would like to remark that similar results have been deduced by authors of Ref.[11] for Hardy operator on higher-dimensional product spaces.