Sharp bounds for Hardy type operators on higher-dimensional product spaces

In this paper, we investigate a class of fractional Hardy type operators $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ defined on higher-dimensional product spaces $\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}}$. We use novel methods to obtain two main results. One is that we obtain the operator $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ is bounded from $L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}},|x|^{\gamma})$ to $L^{q}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}},|x|^{\alpha})$ and the bounds of the operator $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ is sharp worked out. The other is that when $\alpha=\gamma=(0,\cdots,0)$, the norm of the operator $\mathscr{H}_{\beta_{1},\cdots,\beta_{m}}$ is obtained.

Obviously, the operator H β 1 ,β 2 ,··· ,βm is natural generalization of the classical Hardy operators, such as the operator H [3,4], the fractional Hardy operator H β [7], the Hardy operator H on the n-dimensional product space [11] and the Hardy type operator H m on m-dimensional product spaces [6].
If f is a non-negative measurable function on G = (0, ∞), the classical Hardy operator defined as The following Theorem A due to Hardy [3,4] is well known.
Theorem A If f is a non-negative measurable function on G, let 1 < p < ∞ and α < p − 1, then the following two inequalities and hold, and both constants p p−1 and p p−1−α are sharp. Recall that, for a nonnegative measurable function f on R n , the n-dimensional fractional Hardy operator H β with spherical mean is defined by |B(0, |x|)| 1− β n |y|<|x| f (y)dy, (1.2) where x ∈ R n \{0} and 0 ≤ β < n.
In 2015, Lu and Zhao considered the operator defined by (1.2) and obtained the following Theorem B.
Theorem B If f is a nonnegative measurable function on R n , let 0 < β < n, 1 < p < q < ∞ and 1 q = 1 p − β n , then the inequality For the weighted case, in 1985, Sawyer [10] considered the weighted Hardy inequalities with general weight functions u and v only on two-dimensional product space. However, it is much hard to apply the method in [10] to the case dimensional greater than two. In [11], Wang, Lu and Yan studied the Hardy operator H with power weight on the m-dimensional case, and obtained the following result.
Theorem 1.1 Suppose that 1 < p < q < ∞, 0 < β i < n i and 1 (1.6) Moreover, For two differences power weight, we have following result. Moreover, It is worth mentioning that that proof in [11] is not suitable to the operator H β 1 ,β 2 ,··· ,βm . Although the idea in the paper is motivated by the reference [9], there are some essential differences. The difficulty is how to deal with the product space case. In this paper we will use the novel method to become as result in [7]. The reconstruct some auxiliary functions to achieve the sharp bounds, which is quite different from [9].
Throughout the note, we use the following notation. The definition of the usual beta function is defined by where z and w are complex numbers with the positive real parts. The set B(0, |x|) denotes a open ball with center at the original point and radius |x|, and |B(0, |x|)| denotes the volume of the ball B(0, |x|). For one m-dimensional vector α = (α 1 , α 2 , · · · , α m ) and For a real number p, 1 < p < ∞, p ′ is the conjugate number of p, that is, 1/p + 1/p ′ = 1.

Preliminaries
To reduce the dimension of function space, we need the following lemma which was obtained by some ideas and methods used in [6]. and P roof . We merely the proof with the case m = 2 for the sake of clarity in writing, and the same is true for the general case m > 2.
For two differences power weight of fractional Hardy operator H β with β = 0 (write as H), we have following lemma, which can be found in the paper [9]. Lemma 2.3 Suppose that 1 < p < q < ∞, n ∈ N, x ∈ R n , γ < n(p − 1) and α+n q = γ+n p . If f ∈ L p (R n , |x| γ ), then we have

8)
where C * pq is sharp and equal to In fact, by using Persson and Samko [9] result: therefore, we find that With the help of previous consequences, we shall prove our main statements.

Proof of main results
First we use the Lu and Zhao [7] result to derive a new constant, which is sharp in (1.6) for each p ∈ (1, q). P roof of T heorem 1.1 We merely the proof with the case m = 2 for the sake of clarity in writing, and the same is true for the general case m > 2.
According to Lu and Zhao [7] estimate for the high dimensional fractional Hardy operator in the case 1 < p < q < ∞ we find that where A 1 is a constant in the inequality of (1.3) with n = n 1 . By applying the generality Minkowski's inequality with the power q p , we obtain that where A 2 is a constant in the inequality of (1.3) with n = n 2 . Therefore, it implies that Next, we need to proved the converse inequality. It follows from Lemma 2.1 that the norm of the operator H β 1 ,β 2 from L p (R n 1 ×R n 2 ) to L q (R n 1 ×R n 2 ) is equal to the norm that H β 1 ,β 2 restricts to radial functions. Consequently, without loss of generality, it suffices to carry out the proof the converse inequality by assuming that f is a nonnegative, radial, smooth function with compact support on R n 1 × R n 2 .
Using the polar coordinate transformation, we can rewrite (3.9) as that the left side of (3.10) is It is easy to verify that Therefore, This completes the proof of Theorem 1.1. ✷ P roof of T heorem 1.2 We merely the proof with the case m = 2 for the sake of clarity in writing, and the same is true for the general case m > 2.
Without loss of generality, it follows from Lemma 2.1 we can assuming that f is a nonnegative, radial, smooth function with compact support on R n 1 × R n 2 .