Mixed radial-angular bounds for Hardy-type operators on Heisenberg group

1.   Introduction

The classic Hardy operator and its dual operator are defined by

for the locally integrable function $f$ on $\mathbb{R}$ and $x\neq0.$ The classic Hardy operator was introduced by Hardy and he showed the following Hardy inequalities

where $1 < p < \infty$, the constants $\frac{p}{p-1}$, $p$ are best possible.

Faris^[6] first extended Hardy-type operator to higher dimension, Christ and Grafakos^[2] gave an equivalent version of $n$-dimensional Hardy operator $\mathcal{H}$ for nonnegative function $f$ on $\mathbb{R}^n$,

where $\Omega_n = \frac{\pi^{\frac{n}{2}}}{\Gamma(1+\frac{n}{2})}$ is the volume of the unit ball in $\mathbb{R}^n$. By a direct computation, the dual operator of $\mathcal{H}$ can be defined by setting, for nonnegative function $f$ on $\mathbb{R}^n$,

Christ and Grafakos^[2] proved that the norms of $\mathcal{H}$ and $\mathcal{H}^*$ on $L^p(\mathbb{R}^n)(1 < p < \infty)$ are also $\frac{p}{p-1}$ and $p$, which is the same as that in the 1-dimensional case and is also independent of the dimension. The sharp weak estimate for $\mathcal{H}$ was obtained by Zhao et al.^[19]. For $1\leq p\leq\infty$,

where 1 is best constant.

In recent years, the research on Hardy operator related issues is receiving increasing attention, Hardy et al. provided us with the early development and application of Hardy's inequalities. In ^[8,9,15], Fu et al. have engaged in many related discuss, which provide convenience for our research.

In this paper, we will investigate the sharp bound for Hardy-type operators in the setting of the Heisenberg group, which plays important role in several branches of mathematics. Now, allow us to introduce some basic knowledge about the Heisenberg group which will be used in the following. The Heisenberg group $\mathbb{H}^n$ is a non-commutative nilpotent Lie group, with the underlying manifold $\mathbb{R}^{2n}\times\mathbb{R}$ with the group law

and

where $x = (x_1, \cdots, x_{2n}, x_{2n+1})$, $y = (y_1, \cdots, y_{2n}, y_{2n+1})$. The Haar measure on $\mathbb{H}^n$ coincides with the usual Lebesgue measure on $\mathbb{R}^{2n+1}$. We denote the measure of any measurable set $E \subset \mathbb{H}^n$ by $|E|$. Then

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

The Heisenberg distance derived from the norm

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

This distance $d$ is left-invariant in the sense that $d(p, q)$ remains unchanged when $p$ and $q$ are both left-translated by some fixed vector on $\mathbb{H}^n$. Furthermore, $d$ satisfies the triangular inequality ^[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

respectively. And we have

where

is the volume of the unit ball $B(0, 1)$ on $\mathbb{H}^n$, and the area of the unit sphere $\mathbb{S}^{Q-1}$ is $\omega_Q = Q\Omega_Q$ (see^[4]). More about Heisenberg group can refer to ^[7,11,16].

The $n$-dimensional Hardy operator and its dual operator on Heisenberg group is defined by Wu and Fu ^[18]

where $x\in\mathbb{H}^n\backslash\{0\}$, $f$ be a locally integrable function on $\mathbb{H}^n$. They proved that $\mathcal{H}_h$ and $\mathcal{H}_h^*$ is bounded from $L^p(\mathbb{H}^n)$ to $L^p(\mathbb{H}^n)$, $1 < p\leq\infty$. Moreover,

This is the same as the result on $\mathbb{R}^n$.

In ^[10,13], León-Saavedra and González studied the behavior of Ces$\grave{a}$ro operator, Chu et al. in ^[3] defined the $n$-dimensional weighted Hardy operator on Heisenberg group $\mathcal{H}_{hw}$ and $n$-dimensional weighted Ces$\grave{a}$ro operator on Heisenberg group $\mathcal{H}^*_{hw}$. Let us recall their definition.

Definition 1. Let $w:[0, 1]\rightarrow [0, \infty)$ be a measurable function. For a measurable function $f$ on $\mathbb{H}^n$, the $n$-dimensional weighted Hardy operator on Heisenberg group $\mathcal{H}_{hw}$ is defined by

For a measurable complex-valued function $f$ on $\mathbb{H}^n$, nonnegative function $w:[0, 1]\rightarrow (0, \infty)$,

and

the $n$-dimensional weighted Cesàro operator is defined by

which satisfies

where $f\in L^p(\mathbb{H}^n)$, $g\in L^q(\mathbb{H}^n)$, $1 < p < \infty$, $q = p/(p-1)$, $\mathcal{H}_{hw}$ is bounded on $L^p(\mathbb{H}^n)$ and $\mathcal{H}^*_{hw}$ is bounded on $L^q(\mathbb{H}^n)$.

Remark 1. In ^[3], Chu et al. proved the equality

was established when $w(t) = Qt^{Q-1}$ and $f$ is radial function.

Recently, many operators in harmonic analysis have been proved to be bounded on mixed radial-angular spaces, for instance, Duoandikoetxea and Oruetxebarria ^[5] built the extrapolation theorems on mixed radial-angular spaces to study the boundedness of a large class of operators which are weighted bounded. In ^[17], Wei and Yan studied the sharp bounds for $n$-dimensional Hardy operator and its dual in mixed radial-angular spaces on Euclidean space. Inspired by them, we will investigate the sharp bounds for $n$-dimensional Hardy operator and its dual operator in mixed radial-angular spaces on Heisenberg groups.

Now, we give the definition of mixed radial-angular spaces on Heisenberg group.

Definition 2. For any $n\geq2$, $1\leq p$, $\bar{p}\leq\infty$, the mixed radial-angular space $L^p_{|x|_h} L^{\bar{p}}_\theta(\mathbb{H}^n)$ consists of all functions $f$ in $\mathbb{H}^n$ for which

where $\mathbb{S}^{Q-1}$ denotes the unit sphere in $\mathbb{H}^n$.

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

2.   Mixed radial-angular bounds for $\mathcal{H}_h$ and $\mathcal{H}^*_h$

Theorem 1. Let $n\geq2, 1 < p, \bar{p}_1, \bar{p}_2 < \infty$. Then $\mathcal{H}_h$ is bounded from $L_{|x|_h}^p L_\theta^{\bar{p}_1}(\mathbb{H}^n)$ to $L_{|x|_h}^p L_\theta^{\bar{p}_2}(\mathbb{H}^n)$. Moreover,

Theorem 2. Let $n\geq2, 1 < p, \bar{p}_1, \bar{p}_2 < \infty$. Then $\mathcal{H}_h^*$ is bounded from $L_{|x|_h}^p L_\theta^{\bar{p}_1}(\mathbb{H}^n)$ to $L_{|x|_h}^p L_\theta^{\bar{p}_2}(\mathbb{H}^n)$. Moreover,

Proof of Theorem 1. Set

then $g$ is a radial function. Moreover, we have

where $g(r)$ can be defined as $g(r) = g(x)$ for any $x\in\mathbb{H}^n$ with $|x|_h = r$ since $g$ is a radial function. By using Hölder inequality, we have

Next, we use another form of Hardy operator

By change of variables, we can get

Thus, we have obtained

which implies the operator $\mathcal{H}$ and its restriction to radial function have same norm from $L_{|x|_h}^p L_\theta^{\bar{p}_1}$ to $L_{|x|_h}^p L_\theta^{\bar{p}_2}$. Without loss of generality, we can assume that $f$ is a radial function in the rest of proof. Consequently, we have

where $\mathcal{H}_h(f)(r)$ can be defined as $\mathcal{H}_h(f)(r) = \mathcal{H}_h(f)(x)$ for any $|x|_h = r$. Using Minkowski's inequality, we can get

Therefore, we have

On the other hand, for $0 < \epsilon < 1$, take

Then we can obtain

and

So, we have

Thus, we have obtained

Since $\epsilon^\epsilon\rightarrow1$ as $\epsilon\rightarrow0$, by letting $\epsilon\rightarrow0$, we have

Combining (2.2) and (2.3), we can get

This completes the proof of Theorem 1. □

Proof of Theorem 2. The proof of Theorem 2 is similar to prove of Theorem 1, we omit the details. □

3.   Mixed radial-angular bounds for $\mathcal{H}_{hw}$ and $\mathcal{H}^*_{hw}$

Theorem 3. Let $w:[0, 1]\rightarrow (0, \infty)$ be a function, $n\geq2, 1 < p, \bar{p}_1, \bar{p}_2 < \infty$. Then the $n$-dimensional weighted Hardy operator on Heisenberg group $\mathcal{H}_{hw}$ is bounded from $L_{|x|_h}^p L_\theta^{\bar{p}_1}(\mathbb{H}^n)$ to $L_{|x|_h}^p L_\theta^{\bar{p}_2}(\mathbb{H}^n)$ if and only if

Moreover,

Theorem 4. Let $w:[0, 1]\rightarrow (0, \infty)$ be a function, $n\geq2, 1 < p, \bar{p}_1, \bar{p}_2 < \infty$. Then the $n$-dimensional weighted Ces$\grave{a}$ro operator on Heisenberg group $\mathcal{H}^*_{hw}$ is bounded from $L_{|x|_h}^p L_\theta^{\bar{p}_1}(\mathbb{H}^n)$ to $L_{|x|_h}^p L_\theta^{\bar{p}_2}(\mathbb{H}^n)$ if and only if

Moreover,

The proof methods for Theorems 3 and 4 are the same, and similar to the proof method for Theorem 1. But as a special case, here we will give the proof of Theorem 4.

Proof of Theorem 4. Inspired by proof of Theorem 1, we have

where $\mathcal{H}^*_{hw}(f)(r)$ can be defined as $\mathcal{H}^*_{hw}(f)(r) = \mathcal{H}^*_{hw}(f)(x)$ for any $|x|_h = r$. Using Minkowski's inequality, we can get that

Therefore, we have

On the other, taking

and for $f\in L^{p}_{|x|_h}L^{\bar{p}_2}_\theta(\mathbb{H}^n)$, we obtain

For any $\epsilon > 0$, taking

then we have

and

where $\mathcal{H}^*_{hw}(f_\epsilon)(x)$ satisfied $\mathcal{H}^*_{hw}(f_\epsilon)(x) = \mathcal{H}^*_{hw}(f_\epsilon)(r)$ for any $|x|_h = r$.

So we have

By change of variable $|x|_h = \delta_{1/\epsilon}|y|_h$, we have

This implies that

Let $\epsilon\rightarrow0$, we have

Thus, we have finished the proof of Theorem 4.

It should be noted that operators $\mathcal{H}_{hw}$ and $\mathcal{H}_{hw}^*$ are very special cases of a general Hausdorff operator over locally compact groups, introduced in ^[14]. □

4.   Conclusions

In this article, we investigated the sharp bound for Hardy-type operators in the setting of the Heisenberg group, which plays important role in several branches of mathematics. Firstly, we studied $n$-dimensional Hardy operator and its dual in mixed radial-angular spaces on Heisenberg group and obtain their sharp bounds by using the rotation method. Furthermore, the sharp bounds of $n$-dimensional weighted Hardy operator and weighted Cesàro operator are also obtained.

Use of AI tools declaration

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

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 12271232) and Shandong Jianzhu University Foundation (Grant No. X20075Z0101).

Conflict of interest

The authors declare that they have no conflict of interest and competing interests. All procedures were in accordance with the ethical standards of the institutional research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. All authors contributed equally to this work. The manuscript is approved by all authors for publication. Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.