The Weighted $\boldsymbol{L}^{\boldsymbol{p}}$ estimates for the fractional Hardy operator and a class of integral operators on the Heisenberg group

1.   Introduction

It is well-known that averaging operators play an important role in harmonic analysis. It is often desirable to obtain sharp norm estimates for them in different function spaces. Our starting point is the Hardy operator and its duality form:

where $x > 0$. Hardy ^[11] established the well-known Hardy integral inequalities

and

where $p^{\prime} = p/(p-1)$. He proved that the constant $p/(p-1)$ is sharp. The corresponding higher-dimensional Hardy operator was introduced by Faris ^[1] in his study of quantum mechanics. Christ and Grafakos ^[4] gave the following equivalent definition of $n$-dimensional Hardy operators:

where $x\in \mathbb{R} ^n\setminus \{\theta \}$, and $\theta$ is the origin in $\mathbb{R}^n$. They showed that

Here $\omega _n$ is the superficial area of the unit ball in $\mathbb{R}^n$. The Lebesgue spaces with power weights are another kind of function space to consider the sharp estimates of the Hardy operator. The method in ^[4] is invalid in this case. Fu et al. ^[7], by the method of rotation, established the following estimate:

For more information about the Hardy operator, we refer to the reader to ^[14].

Meanwhile, the fractional Hardy operator is also very interesting since it is a useful tool to study the embedding properties of function spaces. Mizuta et al. ^[15] showed that the optimal bound of the fractional Hardy operator implies the sharp embedding properties of function spaces. There is much literature on function spaces. Let $f$ be a locally integrable function on $\mathbb{R} ^n$. Then the $n$-dimensional fractional Hardy operator and its duality form are

where $0 < \alpha < n$, $x\in \mathbb{R} ^n\backslash\left\{ 0\right\}$. If $\alpha$ = 0, the fractional Hardy operator is the classic Hardy operator. There is much literature on the boundness of these operators ^[9,13,21,22]. Among them, Lu et al. ^[21] obtained the following estimates. Suppose

Then

where

If $p = 1$, then

The optimal $L^p\rightarrow L^q$ estimate was later obtained in ^[22]:

where

and $B\left(\cdot, \cdot \right)$ is the beta function defined by

where $z$ and $\omega$ are complex numbers with positive real parts. Comparing with the complicated bounds in the power weighted spaces, the sharp weak bounds for $H_{\alpha}$ and $H_{\alpha}^{*}$ seem easier to understand. Gao and Zhao ^[9] set up

It is then a nature problem to obtain the operator norms of $H_\alpha$ and its dual operator $H_\alpha^*$ in corresponding power weighted spaces. For the latter, Gao et al. ^[8] set up

and

Then Yu et al. ^[20] set up

and

Inspired by them, we will study the sharp weak bound for the fractional Hardy operator on the Heisenberg group, which plays an important role in several branches of mathematics. Now, allow us to introduce some basic knowledge about the Heisenberg group which will be used later.

The Heisenberg group is a very typical non-commutative group, and research on up-modulation and analytic problems is an extension of Euclidean space upharmonic and analytical problems, which is an important part of non-commutative harmonic analysis ^[6,18]. Harmonic analysis on the Heisenberg group has been drawing more and more attention, see^[5,23].

Let us introduce some basic knowledge about the Heisenberg group. The Heisenberg group $\mathbb{H}^n$ is a non-commutative nilpotent Lie group, with the underlying manifold $\mathbb{R}^{2n}\times\mathbb{R}$ and the group law.

Let

and then

The Heisenberg group $\mathbb{H}^n$ is a homogeneous group with dilations

The Haar measure on $\mathbb{H}^n$ coincides with the usual Lebesgue measure on $\mathbb{R}^{2n+1}$. We denote any measurable set $E\subset \mathbb{H}^n$ by $\left|E\right|$, and then

where $Q = 2n+2$ is called the homogeneous dimension of $\mathbb{H}^n$.

The Heisenberg distance derived from the norm

where $x = (x_1, x_2, \dots, x_{2n}, x_{2n+1})$, is given by

This distance $d$ is left-invariant, meaning that $d(p, q)$ remains constant when both $p$ and $q$ are left shifted by some fixed vector on $\mathbb{H}^n$. Furthermore, $d$ satisfies the trigonometric inequality defined by ^[12]:

For $r > 0$ and $x\in \mathbb{H}^n$, the ball and sphere with center $x$ and radius $r$ on $\mathbb{H}^n$ are given by

and

Then we obtain

where

represents the volume of the unit sphere $B(0, 1)$ on $\mathbb{H}^n$, and $\omega _Q = Q\varOmega _Q$. The reader is referred to ^[16,18] for more details.

Except for the fractional Hardy operator, we also study the sharp bounds for some $m$-linear $n$-dimensional integral operators on the Heisenberg group. In 2017, Batbold and Sawano ^[3] studied one-dimensional $m$-linear Hilbert-type operators that include the Hardy-Littlewood-Pólya operator on weighted Morrey spaces, and they obtained the sharp bounds. He et al. ^[10] extended the results in ^[3] and obtained the sharp bound for the generalized Hardy-Littlewood-Pólya operator on weighted central and noncentral homogenous Morrey spaces. He set up

where

In 2011, Wu and Fu ^[19] got the sharp estimate of the $m$-linear $p$-adic Hardy operator on Lebesgue spaces with power weights. Zhang et al. ^[25] obtained the sharp estimate for the $m$-linear $n$-dimensional Hausdorff operator on the weighted Morrey space.

Inspired by the above, on the Heisenberg group, we first study the sharp weak estimate for the $n$-dimensional fractional Hardy operator from $L^p$ to $L^{q, \infty}$. Second, we study a more general operator which includes Hardy, Hardy-Littlewood-Pólya, and Hilbert operators as a special case and consider their operator norm on weighted Lebesgue space. Finally, we also find the sharp bound for the Hausdorff operator on Lebesgue space, which generalizes the previous results.

To get the main conclusion, it is necessary to introduce some fundamental knowledge and definitions. In the setting of the Heisenberg group, these operators and spaces are the fractional Hardy operator, $m$-linear $n$-dimensional Hardy operator, $m$-linear $n$-dimensional Hardy-Littlewood-Pólya operator, $m$-linear $n$-dimensional Hilbert operator, $m$-linear $n$-dimensional Hausdorff operator, and weighted $L^p$ and $L^{q, \infty}$.

Definition 1.1. Let $f$ be nonnegative locally integrable functions on $\mathbb{H}^n$ and $Q = 2n+1$, $0 < \alpha < Q$. The $n$-dimensional fractional Hardy operator is defined by

where $x\in \mathbb{H}^n\backslash\left\{ 0\right\}$.

Definition 1.2. Let $m$ be a positive integer and $f_1, \dots, f_m$ be nonnegative locally integrable functions on $\mathbb{H}^n$. The $m$-linear $n$-dimensional Hardy operator is defined by

where $x\in \mathbb{H}^n\backslash\left\{ 0\right\}$.

Definition 1.3. Let $m$ be a positive integer and $f_1, \dots, f_m$ be nonnegative locally integrable functions on $\mathbb{H}^n$. The $m$-linear $n$-dimensional Hardy-Littlewood-Pólya operator is defined by

where $x\in \mathbb{H}^n\backslash\left\{ 0\right\}$.

Definition 1.4. Let $m$ be a positive integer and $f_1, \dots, f_m$ be nonnegative locally integrable functions on $\mathbb{H}^n$. The $m$-linear $n$-dimensional Hilbert operator is defined by

where $x\in \mathbb{H}^n\backslash\left\{ 0\right\}$.

Definition 1.5. Let $m$ be a positive integer, $f_1, \dots, f_m$ be nonnegative locally integrable functions on $\mathbb{H}^n$, and $\Phi$ be a nonnegative function on the Heisenberg group. The $m$-linear $n$-dimensional Hausdorff operator is defined by

where $x\in \mathbb{H}^n\backslash\left\{ 0\right\}$.

Definition 1.6. Let $1\leqslant p < \infty$. The Lebesgue space on the Heisenberg $L^p\left(\mathbb{H} ^n\right)$ is defined by

where

Definition 1.7. Let $\omega : \mathbb{H} ^n\rightarrow \left(0, \infty \right)$ be a positive measurable function, $1\leqslant p < \infty$. The weighted Lebesgue space on the Heisenberg $L^p\left(\mathbb{H} ^n, \omega \right)$ is defined by

where

Definition 1.8. Let $\omega : \mathbb{H} ^n\rightarrow \left(0, \infty \right)$ be a positive measurable function, $1\leqslant p < \infty$. The weighted weak Lebesgue space on the Heisenberg $L^p\left(\mathbb{H} ^n, \omega \right)$ is defined by

where

Next, we will provide the main results of this article.

2.   Sharp estimate for the fractional Hardy operator

In this section, we will study the weighted $L^p$ estimate for the fractional Hardy operator on the Heisenberg group. For the $n$-dimensional fractional Hardy operator, our results have a restricted condition: $\beta = 0$ when $p = 1$ and $\beta > 0$ when $p > 1$. Removing this restrictive condition requires a more complicated argument, and it will be presented in a future paper.

Theorem 2.1. Let $1 < p < \infty$, $1\leqslant q < \infty$, $\beta < Q\left(p-1 \right)$, $Q+\gamma > 0$, and $0\leqslant\alpha < \frac{\beta}{p-1}$.

If

then

Theorem 2.2. Let $Q+\beta > 0$ and $0 < \alpha < Q$. Then

Proof of Theorem 2.1. Noticing $Q-\frac{\beta}{p-1} > Q-\frac{Q\left(p-1 \right)}{p-1} = 0$, by $\mathrm{H}\ddot{\mathrm{o}}\mathrm{lder}$'s inequality, we have

where

Noticing $|\mathcal{H} _{\alpha}f\left(x \right) |\leqslant C_{p, Q, \beta, f}\left| x \right|_{h}^{-\frac{Q}{p}-\frac{\beta}{p}+\alpha}$, we have $\left\{ x:|\mathcal{H} _{\alpha}f\left(x \right) | > \lambda \right\} \subset \{ x:C_{p, Q, \beta, f}\left| x \right|_{h}^{-\frac{Q}{p}-\frac{\beta}{p}+\alpha} > \lambda \}$.

Since

we have

Thus

On the other hand, let

Noticing $Q+\beta (1-\frac{p}{p-1}) = Q-\frac{\beta}{p-1} > 0$, we have

So we have proved that $f_0\in L^p(\mathbb{H} ^n, \left| x \right|_{h}^{\beta})$. Then we calculate $\mathcal{H} _{\alpha}\left(f_0 \right) \left(x \right)$.

Denote $C_{p, Q, \beta} = \frac{\omega _Q\left(p-1 \right)}{pQ-Q-\beta}$ and

When $0 < \lambda < C_{p, Q, \beta}$, noticing $\alpha < \frac{\beta}{p-1}\, \,$ and $\beta < Q\left(p-1 \right)$, we have $\alpha < Q$ and

When $\lambda \geqslant C_{p, Q, \beta}$, noticing $\alpha < \frac{\beta}{p-1}\, \,$ and $\beta < Q\left(p-1 \right)$, we have $\alpha < Q$ and

Based on the above analysis, we have

Now we first calculate $M_1$. Since

and

we have

Then we calculate $M_2$, noticing $\left\| f_0 \right\| _{L^p(\mathbb{H} ^n, \left| x \right|_{h}^{\beta})} = \left(\frac{\omega _Q\left(p-1 \right)}{pQ-Q-\beta} \right) ^{\frac{1}{p}}$, $\gamma > -Q$, and

and we have

Its easy to see that $M_1 = M_2$, and then

This finishes the proof of Theorem 2.1.

Proof of Theorem 2.2. It is easy to see that

Notice $\left| \mathcal{H} _{\alpha}f\left(x \right) \right|\leqslant |x|_{h}^{\alpha -Q}\left\| f \right\| _{L^1\left(\mathbb{H} ^n \right)}$, and we have $\left\{ x:\left| \mathcal{H} _{\alpha}f\left(x \right) \right| > \lambda \right\} \subset \{ x:|x|_{h}^{\alpha -Q}\left\| f \right\| _{L^1\left(\mathbb{H} ^n \right)} > \lambda \}$. Since $Q-\alpha > 0$ and $Q+\gamma > 0$, we have

Thus

On the other hand, let $f_0\left(x \right) = \chi _{\{ x:\left| x \right|_h\leqslant 1 \}}\left(x \right)$. Then we have

and so $f_0\in L^1\left(\mathbb{H} ^n \right)$ and

Denote $C_Q = \frac{\omega _Q}{Q}$ and

When $\lambda \geqslant C_Q$, noticing $0 < \alpha < Q$, we have $\left\{ x:|\mathcal{H} _{\alpha}\left(f_0 \right) \left(x \right) | > \lambda \right\} = \varnothing$.

When $0 < \lambda < C_Q$, noticing $0 < \alpha < Q$, we have

We have

When $\lambda \geqslant C_Q$, then $\left\{ x:|\mathcal{H} _{\alpha}f_0\left(x \right) | > \lambda \right\} = \varnothing$, and we have $M_4 = 0$. Then, we only need to calculate $M_3$. In addition, noticing

we have

Then

This finishes the proof of Theorem 2.2. Notices that Theorem 2.2 no longer holds when $\alpha = 0$.

3.   Sharp weighted $\boldsymbol{L}^{\boldsymbol{p}}$ estimate for the integral operator with a kernel

In this section, we will study the $m$-linear $n$-dimensional integral operator with a kernel on the Heisenberg group. Let $K:\mathbb{H} ^n\times \cdots \times \mathbb{H} ^n\rightarrow \left(0, \infty \right)$ be a measurable radial kernel such that $K\left(y_1, ..., y_m \right) = K(\left| y_1 \right|_h, ..., \left| y_m \right|_h)$, satisfying

where $\alpha_j$, $q_j$, $q$, and $Q$ are the pre-defined indicator and some fixed indices, $j = 1, 2, ..., m$. The $m$-linear $n$-dimensional integral operator with a kernel is defined by

where $x\in \mathbb{R} ^n\backslash \left\{ 0 \right\}$ and $f_j$ is a measurable radial function on $\mathbb{H} ^n$ with $j = 1, 2, ..., m$. Note that $\mathcal{H}^h$ is in fact an integral operator having a homogeneous radial $K$ of degree $-mn$.

In this paper, we will give the weighted $L^p$ estimate for the $m$-linear $n$-dimensional integral operator with a kernel on the Heisenberg group.

Theorem 3.1. Let $m\in \mathbb{N}$, $1 < q < \infty$, $\frac{1}{q} = \frac{1}{q_1}+\cdots +\frac{1}{q_m}$, $\alpha = \alpha _1+\cdots +\alpha _m$, $1 < q_j < \infty$ with $j = 1, ..., m$, and $f_j$ be a radial function in $L^{q_{\boldsymbol{j}}}(\mathbb{H} ^n, \left| x \right|_{h}^{\frac{q_{\boldsymbol{j}}\alpha _{\boldsymbol{j}}}{q}})$. Assume that the kernel $K$ is the constant defined by (3.1).

Then

Proof. Consider that

Obviously, $g_j$ satisfies $g_j\left(x \right) = g_j\left(\left| x \right|_h \right)$, and $\mathcal{H} ^h\left(f_1, ..., f_m \right) \left(x \right)$ is equal to

In the fourth to fifth lines, we let $z_j = \delta _{| y_j |_h}\xi _j$. From the fifth to sixth lines, we perform an integral permutation. In the sixth to seventh lines, we set $y_j = \delta _{| z_j |_h}t_j$. On the other hand, by applying Hölder's inequality, we conclude that

From the second to third lines, we apply Hölder's inequality. In the fourth to fifth lines, we let $z_j = \delta _{\left| x \right|_h}\xi _j$. From the fifth to sixth lines, we perform an integral permutation. In the sixth to seventh lines, we set $x = \delta _{| z_j |_h}t_j$. Therefore we have

which implies that the operator $\mathcal{H} ^h$ and its restriction to the function $g$ satisfying $g_j\left(x \right) = g_j(\left| x \right|_h)$ have the same operator norm in $L^q(\mathbb{H} ^n, \left| x \right|_{h}^{\alpha})$. So without loss of generality, we assume that $f_j\in L^{q_{\boldsymbol{j}}}(\mathbb{H} ^n, \left| x \right|_{h}^{\frac{q_{\boldsymbol{j}}\alpha _{\boldsymbol{j}}}{q}})$ with $j = 1, 2, ..., m$ satisfies that $f_j\left(x \right) = f_j(\left| x \right|_h)$ in the rest of the proof. Let $q^{\prime}$ is the conjugate number of $q$ and $g\in L^{q^{\prime}}(\mathbb{H} ^n, \left| x \right|_{h}^{\alpha})$. Using duality and Holder's inequality, and making a change of variables, we obtain the following sequence of inequalities:

This proves the first part of our theorem.

For the second part, we will show that if the kernel $K$ is nonnegative, then the operator norm $\| \mathcal{H}^h \|$ of $\mathcal{H}^h$ is equal to $C^h$. For a positive number $\varepsilon$ and $i = 1, 2, ..., m$, we define the sequences of functions $g_{\varepsilon}$ and $f_{j, \varepsilon}$ by

By a simple computation, we have

Therefore, we have

where $L_i$ is defined as

Here $\overset{\land}{dr_i}$ means that we do not integrate with respect to the variable $r_i$. The last equality follows from integration by parts and the observation that, if we let

then

where the upper index $(i)$ means that $x$ replaces the variable $r_i$ in the $i$-th position. By means of the previous step, we have

Let $E_i$ denote the domain of integral $L_i$ above (3.4), that is,

Taking into account that $\frac{1}{q} = \frac{1}{q_1}+\cdots +\frac{1}{q_m}$, we can bound the integrand of $\frac{\varepsilon L_i}{\omega _Q}$ on $E_i$ as follows:

For the integrand of the first term in (3.4) on $\left[0, 1 \right] ^m$, we also have that

Since the condition of kernel $K$ (3.1) is equivence to

using assumption (3.5), we can use the Lebesgue dominated convergence theorem, which implies that

and

Furthermore, we have

and for $i, j = 1, ..., m$, any of the intersection sets $\left[0, 1 \right] ^m\cap E_i$, $E_i\cap E_j$, $i\ne j$, has Lebesgue measure zero in $\mathbb{H} ^n$. Consequently, (3.4), (3.6), and (3.7) imply that

This finishes the proof of Theorem 3.1.

4.   Application: Sharp weighted $\boldsymbol{L}^{\boldsymbol{p}}$ estimates for a class of integral operators

By taking a particular kernel $K$ in operator $T$ defined by (3.1), we can obtain sharp weighted $L^p$ estimates for Hardy, Hardy-Littlewood-Pólya, and Hilbert operators on the Heisenberg group. Our results in this section are as follows.

Corollary 4.1. Assume that the real paramenters $q$, $q_j$, $\alpha$, and $\alpha_j$ with $j = 1, 2, ..., m$ are the same as in Theorem 3.1, and $f_j$ is a radial function in $L^{q_{\boldsymbol{j}}}(\mathbb{H} ^n, \left| x \right|_{h}^{\frac{q_{\boldsymbol{j}}\alpha _{\boldsymbol{j}}}{q}})$. Assume also that $-\frac{\alpha _j}{q}-\frac{Q}{q_j}+Q > 0$. Then

Corollary 4.2. Assume that the real paramenters $q$, $q_j$, $\alpha$, and $\alpha_j$ with $j = 1, 2, ..., m$ are the same as in Theorem 3.1, $f_j$ is a radial function in $L^{q_{\boldsymbol{j}}}(\mathbb{H} ^n, \left| x \right|_{h}^{\frac{q_{\boldsymbol{j}}\alpha _{\boldsymbol{j}}}{q}})$. Assume also that $-\frac{\alpha _j}{q}-\frac{Q}{q_j}+Q > 0$ and $-\frac{\alpha}{q}-\frac{Q}{q} < 0$. Then

Corollary 4.3. Assume that the real paramenters $q$, $q_j$, $\alpha$, and $\alpha_j$ with $j = 1, 2, ..., m$ are the same as in Theorem 3.1, and $f_j$ is a radial function in $L^{q_{\boldsymbol{j}}}(\mathbb{H} ^n, \left| x \right|_{h}^{\frac{q_{\boldsymbol{j}}\alpha _{\boldsymbol{j}}}{q}})$. Assume also that $-\frac{\alpha _j}{q}-\frac{Q}{q_j}+Q > 0$ and $-\frac{\alpha}{q}-\frac{Q}{q} < 0$. Then

Proof of Corollary 4.1. Next, we will use the methods in ^[7,17]. If we take the kernel

in Theorems 3.1, by a change of variables, it is easy to verify that $\mathcal{H}^h = \mathcal{H}_1^h$, and then $\mathcal{H}_1^h$ can be denoted by

Then all things reduce to calculating

To calculate this integral, employing the polar coordinates $y_j = \rho _j\xi _j$, $j = 1, 2, ..., m$, and Fubini's theorem, we obtain

We use the $m$-dimensional spherical coordinates

where $r\geqslant 0$ is the radial coordinate and $\varphi _j$, $j = 1, 2, ..., m-1$, are angular coordinates, $\varphi _j\in \left[0, \pi \right]$, $j = 1, 2, ..., m-2$, $\varphi _{m-1}\in \left[0, 2\pi \right)$, and the known fact that the associated Jacobian is

in (4.5). Since $-\frac{\alpha _j}{q}-\frac{Q}{q_j}+Q > 0$, we have

We also use the fact $\varphi _j\in \left(0, \pi /2 \right)$, $j = 1, 2, ..., m-1$. From the seventh to eighth lines, we let $t_j = \sin \varphi _j$ for $j = 1, 2, ..., m-1$. From the eighth to ninth lines, we set $s_j = t_j^2$ for $j = 1, 2, ..., m-1$, as well as the definition of the beta function (see, e.g., ^[24]).

By using the following well-known relation between Euler's beta and gamma functions:

(see, for example, ^[24]), after some simple calculations, we see that the following relations hold:

Then we have

This finishes the proof of Corollary 4.1.

Proof of Corollary 4.2. Next, we refer to the methods in ^[3]. If we take the kernel

in Theorem 2.1, by a change of variables, we have $\mathcal{H}^p = \mathcal{H}_{2}^{h}$, and then $\mathcal{H}_{2}^{h}$ can be denoted by

Then, we reduce to calculating

To calculate this integral, we divide the integral into $m$ parts. Let

It is clear that

and $E_i\cap E_j = \varnothing \left(i\ne j \right)$. Let

and then we have

Now let us calculate $J_j$, $j = 1, 2, ..., m$. Since $-\frac{\alpha _j}{q}-\frac{Q}{q_j}+Q > 0$, using the polar coordinate transformation, we have

For $j = 1$, since $-\frac{\alpha _j}{q}-\frac{Q}{q_j}+Q > 0$ and $-\frac{\alpha}{q}-\frac{Q}{q} < 0$, we have

Similar for $i = 2, ..., m-1$, we have

When $i = m$, similar to the previous step, we show that

Then, it yields that

This finishes the proof of Corollary 4.2.

Proof of Corollary 4.3. If we take the kernel

in Theorem 2.1, by a change of variables, we have $\mathcal{H}^p = \mathcal{H}_{3}^{h}$, and then $\mathcal{H}_{3}^{h}$ can be denoted by

Then, we reduce to calculating

Actually, this method stems from Benyi and Oh ^[2], who investigated the one-dimensional case. Following their method, it is easy to find the higher-dimensional case, as well. For completeness, we give the details. Employing the polar coordinates and making a change of variables, we have

Let $\rho _{j}^{Q} = t_j$, and we have

Let us denote the integral on the right by

By making the change of variables $t_m = \left(1+t_1+\cdots +t_{m-1} \right) t$ and performing integration with respect to $dt$, we have the following identity:

Observe that, if we make the change of variables $t+1 = 1/s$, we obtain

for $a, b > 0$. According to known conditions, obviously, we have $(Q-\frac{\alpha _j}{q}-\frac{Q}{q_j}) /Q > 0$. Now, we will show that $m-(Q-\frac{\alpha _m}{q}-\frac{Q}{q_m}) /Q-\cdots -(Q-\frac{\alpha _k}{q}-\frac{Q}{q_k}) /Q > 0$ with $k = m, m-1, ..., 1$. Since $-\frac{\alpha _j}{q}-\frac{Q}{q_j}+Q > 0$ and $-\frac{\alpha}{q}-\frac{Q}{q} < 0$, we have

Therefore, if we recall the relationship between the beta and gamma functions, we obtain

and

By a simple induction argument, we obtain from this recurrence that

Then

This finishes the proof of Corollary 4.3.

5.   Further results: Sharp weighted $\boldsymbol{L}^{\boldsymbol{p}}$ estimate for the Hausdorff operator

In this section, we will use the previous results to give the weighted $L^p$ estimates for the $m$-linear $n$-dimensional Hausdorff operator on the Heisenberg group.

Corollary 5.1. Assume that the real paramenters $q$, $q_j$, $\alpha$, and $\alpha_j$ with $j = 1, 2, ..., m$ are the same as in Theorem 3.1. A nonnegative function $\Phi$ on $\mathbb{H} ^n$ satisfies

Then

Proof. By a change of variables, the $m$-linear $n$-dimensional Hausdorff operator become

We can obtain

This is similar to the proof of Theorem 3.1, so we omit the details. This finishes the proof of Corollary 5.1.

6.   Conclusions

First, in the setting of the Heisenberg group, the $n$-dimensional fractional Hardy operator has a sharp weak estimate from $L^p$ to $L^{q, \infty}$. The weak estimate bound is given by

Additionally, for the $L^1$ case, we have

Second, we derive the sharp bounds for the $m$-linear $n$-dimensional integral operator with a kernel on weighted Lebesgue spaces:

Finally, as an application, the sharp bounds for Hardy, Hardy-Littlewood-Pólya, and Hilbert operators on weighted Lebesgue spaces are obtained. Moreover, we also find the estimate for the Hausdorff operator on weighted $L^p$ spaces:

Author contributions

Tianyang He: Conceptualization, methodology; Zhiwen Liu: Writing-original draft; Ting Yu: Writing-review and editing, validation, methodology. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China with code 12471461 and the Fundamental Research Funds for the Central Universities. The funders had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Conflict of interest

The authors declare that they have no conflict of interest.