Stronger Versions of Stein–Weiss Inequalities

Abstract

In this paper, stronger versions of Stein–Weiss inequalities and reverse Stein–Weiss inequalities are established.

1. Introduction

As is well known, Hardy–Littlewood–Sobolev inequality [1] states that, for $p,r>1$, $0<\alpha <n$, and ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{r}}-{\displaystyle \frac{\alpha }{n}}=1$, the following inequality holds for any functions $f\in L^p\left({\mathbb{R}}^n\right)$ and $h\in L^r\left({\mathbb{R}}^n\right)$:

$$\begin{array}{c}\hfill C(n,\alpha ,p){\left|\right|f\left|\right|}_p{\left|\right|h\left|\right|}_r\ge \left|{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)h\left(y\right)}{{|x-y|}^{n-\alpha }}}dxdy\right|,\end{array}$$

(1)

where $C(n,\alpha ,p)$ is a sharp constant independent of f and h and satisfies

$$\begin{array}{c}\hfill C(n,\alpha ,p)\le {\displaystyle \frac{n}{\alpha }}{\omega }_n^{1-{\displaystyle \frac{\alpha }{n}}}{\displaystyle \frac{1}{pr}}\left({\left({\displaystyle \frac{1-\frac{\alpha }{n}}{1-1/p}}\right)}^{1-{\displaystyle \frac{\alpha }{n}}}+{\left({\displaystyle \frac{1-\frac{\alpha }{n}}{1-1/r}}\right)}^{1-{\displaystyle \frac{\alpha }{n}}}\right),\end{array}$$

(2)

where ${\omega }_n$ represents the volume of the unit ball in ${\mathbb{R}}^n$. In the special case where $p=r=2n/(n+\alpha )$, the constant $C(n,\alpha ,p)$ simplifies to

$$\begin{array}{c}\hfill C(n,\alpha ,p)=C(n,\alpha )={\pi }^{(n-\alpha )/2}{\displaystyle \frac{\mathsf{\Gamma }(\alpha /2)}{\mathsf{\Gamma }\left(\right(n+\alpha )/2)}}{\left\{{\displaystyle \frac{\mathsf{\Gamma }(n/2)}{\mathsf{\Gamma }\left(n\right)}}\right\}}^{-\alpha /n},\end{array}$$

where $\mathsf{\Gamma }$ is the gamma function. Equality in (1) holds if and only if $h=a_{0}f$ and $f\left(x\right)=a(b^2+|x-x_0{{|}^2)}^{-{\displaystyle \frac{n+\alpha }{2}}}$ for some $a_0$, $a\in \mathbb{R}$, $0\ne b\in \mathbb{R}$ and $x_0\in {\mathbb{R}}^n$.

In recent years, affine Hardy–Littlewood–Sobolev inequalities have attracted considerable attention. since they provide stronger estimates than the classical ones. J. Haddad and M. Ludwig (Theorem 1 [2]) established sharp-affine Hardy–Littlewood–Sobolev inequalities for $0<\alpha <n$ and non-negative $f\in L^{2n/(n+\alpha )}\left({\mathbb{R}}^n\right)$:

$$\begin{array}{ccc}\hfill {\gamma }_{n,\alpha }{∥f∥}_{{\displaystyle \frac{2n}{n+\alpha }}}^2& \ge & n{\omega }_n^{{\displaystyle \frac{n-\alpha }{n}}}{\left({\displaystyle \frac{1}{n}}{\int }_{{\mathbb{S}}^{n-1}}{\left({\int }_0^{\infty }{t}^{\alpha -1}{\int }_{{\mathbb{R}}^n}f\left(x\right)f\left(x+t\xi \right)dxdt\right)}^{{\displaystyle \frac{n}{\alpha }}}d\xi \right)}^{{\displaystyle \frac{\alpha }{n}}}\hfill \\ & \ge & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)f\left(y\right)}{{\left|x-y\right|}^{n-\alpha }}}dxdy.\hfill \end{array}$$

(3)

Equality holds in the first inequality if and only if the function f has the form $f\left(x\right)=a{\left(1+\left|\varphi {\left(x-x_0\right)}^2\right|\right)}^{-{\displaystyle \frac{n+\alpha }{2}}}$ for $x\in {\mathbb{R}}^n$, where $a\ge 0$, $\varphi \in GL\left(n\right)$ and $x_0\in {\mathbb{R}}^n$. Here, $GL\left(n\right)$ denotes general linear transformations. For the second inequality, equality is attained precisely if f is radially symmetric.

Here, the integral over the unit sphere ${\mathbb{S}}^{n-1}$ is taken with respect to the $(n-1)$-dimensional Hausdorff measure. The sharp-affine Hardy–Littlewood–Sobolev inequalities (3) are significantly stronger than the classical Hardy–Littlewood–Sobolev inequality (1) for $f=h$.

As a natural extension, affine inequalities have been generalized to the two-function setting. Y. Lin, J. Zhou, and J. Lan (Theorem 1.1 [3]) established generalized affine Hardy–Littlewood–Sobolev inequality for $0<\alpha <n$, $n\ge 1$ and $1<p,r<\infty$, satisfying

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{r}}-{\displaystyle \frac{\alpha }{n}}=1.\end{array}$$

Then, there is a constant $C\left(n,\alpha ,p\right)>0$ such that for all non-negative functions $f\in L^p\left({\mathbb{R}}^n\right)$ and $h\in L^r\left({\mathbb{R}}^n\right)$,

$$\begin{array}{ccc}\hfill C\left(n,\alpha ,p\right){∥f∥}_p{∥h∥}_r& \ge & n{\omega }_n^{{\displaystyle \frac{n-\alpha }{n}}}{\left({\displaystyle \frac{1}{n}}{\int }_{{\mathbb{S}}^{n-1}}{\left({\int }_0^{\infty }{t}^{\alpha -1}{\int }_{{\mathbb{R}}^n}h\left(x\right)f\left(x+t\xi \right)dxdt\right)}^{{\displaystyle \frac{n}{\alpha }}}d\xi \right)}^{{\displaystyle \frac{\alpha }{n}}}\hfill \\ & \ge & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)h\left(y\right)}{{\left|x-y\right|}^{n-\alpha }}}dxdy.\hfill \end{array}$$

(4)

If $p=r={\displaystyle \frac{2n}{n+\alpha }}$, there is equality in the first inequality if and only if h coincides with a translation of $a_{0}f$ where $a_0\in \mathbb{R}$ and

$$\begin{array}{c}\hfill f\left(x\right)=a{\left(b^2+\left|\varphi {\left(x-x_0\right)}^2\right|\right)}^{-{\displaystyle \frac{n+\alpha }{2}}}\end{array}$$

for some $a\in \mathbb{R}$, $0\ne b\in \mathbb{R}$, $\varphi \in GL\left(n\right)$, and $x_0\in {\mathbb{R}}^n$. There is equality in the second inequality if f and h are radially symmetric.

The generalized affine Hardy–Littlewood–Sobolev inequality (4) is stronger than Hardy–Littlewood–Sobolev inequality (1). When $p=r={\displaystyle \frac{2n}{n+\alpha }}$ and $f=h$, we can get the affine Hardy–Littlewood–Sobolev inequalities (3).

The case $\alpha >n$ corresponding to reverse-type inequalities has also been widely investigated. J. Dou and M. Zhu [4] and W. Beckner [5] obtained sharp Hardy–Littlewood–Sobolev inequalities for $\alpha >n$ as follows:

$$\begin{array}{c}\hfill C\left(n,\alpha ,p\right){∥f∥}_p{∥h∥}_r\le {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)h\left(y\right)}{{\left|x-y\right|}^{n-\alpha }}}dxdy\end{array}$$

(5)

for non-negative functions $f\in L^p\left({\mathbb{R}}^n\right)$ and $h\in L^r\left({\mathbb{R}}^n\right)$ satisfying ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{r}}-{\displaystyle \frac{\alpha }{n}}=1$, where $C\left(n,\alpha ,p\right)$ is defined in (2). When $p=r={\displaystyle \frac{2n}{n+\alpha }}$, equality in the inequality holds if and only if $h=a_{0}f$ and

$$\begin{array}{c}\hfill f\left(x\right)=a{\left(b^2+\left|{\left(x-x_0\right)}^2\right|\right)}^{-{\displaystyle \frac{n+\alpha }{2}}}\end{array}$$

for some $a_0$, $a\in \mathbb{R}$, $0\ne b\in \mathbb{R}$, and $x_0\in {\mathbb{R}}^n$.

In parallel with the affine development for $0<\alpha <n$, affine structures also appear in the reverse setting. J. Haddad and M. Ludwig (Theorem 14 [2]) established corresponding sharp-affine Hardy–Littlewood–Sobolev inequalities for $\alpha >n$ and non-negative $f\in L^{2n/(n+\alpha )}\left({\mathbb{R}}^n\right)$:

$$\begin{array}{c}\hfill {\gamma }_{n,\alpha }{∥f∥}_{{\displaystyle \frac{2n}{n+\alpha }}}^2\le n{\omega }_n^{{\displaystyle \frac{n-\alpha }{n}}}{\left|S_{\alpha }f\right|}^{{\displaystyle \frac{\alpha }{n}}}\le {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)f\left(y\right)}{{\left|x-y\right|}^{n-\alpha }}}dxdy.\end{array}$$

(6)

Equality holds in the first inequality if and only if the function f has the form $f\left(x\right)=a{\left(1+\left|\varphi {\left(x-x_0\right)}^2\right|\right)}^{-{\displaystyle \frac{n-\alpha }{2}}}$ for $x\in {\mathbb{R}}^n$, where $a\ge 0$, $\varphi \in GL\left(n\right)$ and $x_0\in {\mathbb{R}}^n$. For the second inequality, equality is attained precisely if f is radially symmetric.

Very recently, the above results have been further generalized to the two-function reverse case. Y. Lin, J. Zhou, and J. Lan (Theorem 1.2 [3]) established generalized affine Hardy–Littlewood–Sobolev inequality for $\alpha >n$, $n\ge 1$, and $0<p,r<1,$ satisfying

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{r}}-{\displaystyle \frac{\alpha }{n}}=1,\end{array}$$

where there is a constant $C\left(n,\alpha ,p\right)>0$ such that for all non-negative $f\in L^p\left({\mathbb{R}}^n\right)$ and $h\in L^r\left({\mathbb{R}}^n\right)$,

$$\begin{array}{c}\hfill C\left(n,\alpha ,p\right){∥f∥}_p{∥h∥}_r\le n{\omega }_n^{{\displaystyle \frac{n-\alpha }{n}}}{\left|S_{\alpha }\left(f,h\right)\right|}^{{\displaystyle \frac{\alpha }{n}}}\le {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)h\left(y\right)}{{\left|x-y\right|}^{n-\alpha }}}dxdy.\end{array}$$

(7)

If $p=r={\displaystyle \frac{2n}{n+\alpha }}$, there is equality in the first inequality if and only if h coincides with a translation of $a_{0}f$ where $a_0\in \mathbb{R}$ and

$$\begin{array}{c}\hfill f\left(x\right)=a{\left(b^2+\left|\varphi {\left(x-x_0\right)}^2\right|\right)}^{-{\displaystyle \frac{n+\alpha }{2}}}\end{array}$$

for some $a\in \mathbb{R},0\ne b\in \mathbb{R}$, $\varphi \in GL\left(n\right)$, and $x_0\in {\mathbb{R}}^n$. There is equality in the second inequality if f and h are radially symmetric.

The generalized affine Hardy–Littlewood–Sobolev inequality (7) is stronger than sharp Hardy–Littlewood–Sobolev inequality (5). When $p=r={\displaystyle \frac{2n}{n+\alpha }}$ and $f=h$, we can obtain the sharp-affine Hardy–Littlewood–Sobolev inequalities (6).

From now on, we focus on the Stein–Weiss inequality, which we also refer to as doubly weighted HLS inequality. The Stein–Weiss inequality on ${\mathbb{R}}^n$, which was proven by E. M. Stein and G. Weiss [6], states that for any $n\ge 1$, $1<p,q<+\infty$, $\alpha +\beta \ge 0$, $\alpha <{\displaystyle \frac{n}{p^{\prime }}}$, $\beta <{\displaystyle \frac{n}{q^{\prime }}}$, $0<\lambda <n$, ${\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{q^{\prime }}}=1$, ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{p^{\prime }}}=1$,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}\ge 1,\mathrm{and}{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}+{\displaystyle \frac{\alpha +\beta +\lambda }{n}}=2,\end{array}$$

(8)

there is a constant $C_{n,\alpha ,\beta ,p,q}>0$ such that

$${\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{|x-y|}^{\lambda }{\left|y\right|}^{\beta }}}dxdy\le C_{n,\alpha ,\beta ,p,q}{\parallel f\parallel }_p{\parallel g\parallel }_q.$$

(9)

Since its publication, this inequality has exerted a profound influence across multiple areas of mathematics. Over the decades, extensive research has been devoted to its generalizations. For example, D. Salim, S. A. Hazmy, Y. Soeharyadi, and W. S. Budhi [7] established the Stein–Weiss inequality on Morrey spaces. More precisely, for $0\le i\le n$ and $1<p<\infty$, the local Morrey space $L^{p,i}\left(0\right)$ is the collection of f with

$${\parallel f\parallel }_{L^{p,i}\left(0\right)}=\underset{r>0}{sup}{r}^{-{\displaystyle \frac{i}{p}}}{\parallel f\parallel }_{L^p\left(B(0,r)\right)}<\infty ,$$

and the Morrey space $L^{p,i}$ is the set of f with

$${\parallel f\parallel }_{L^{p,i}}=\underset{z\in {\mathbb{R}}^n}{sup}{\parallel f(z+·)\parallel }_{L^{p,i}\left(0\right)}<\infty .$$

In 2012, X. Han, G. Lu, and J. Zhu [8] established the Stein–Weiss type inequality on Heisenberg group. The n-dimensional Heisenberg group is ${\mathbb{H}}^n={\mathbb{C}}^n\times \mathbb{R}$ with group structure given by

$$uv=(z,t)(z^{\prime },t^{\prime })=\left(z+z^{\prime },t+t^{\prime }+2Im(z·\overline{z})\right)$$

for any two points $u=(z,t)$, $v=(z^{\prime },t^{\prime })\in {\mathbb{H}}^n$, where $z,z^{\prime }\in {\mathbb{C}}^n$, $t,t^{\prime }\in \mathbb{R}$ and $z·\overline{z}={\sum }_{j=1}^n{z}_j\overline{z_j}$. Haar measure on ${\mathbb{H}}^n$ is the Lebesgue measure $du=dzdt$, in which $z=x+iy$ with $x,y\in {\mathbb{R}}^n$. With the settings on the Heisenberg group ${\mathbb{H}}^n$, the authors considered the analogous weighted HLS inequality, namely the Stein–Weiss inequality on ${\mathbb{H}}^n$. Recently, L. Chen, Z. Liu, G. Lu, and C. Tao [9] established Stein–Weiss inequality with the fractional Poisson kernel and proved that there exist extremals for the Stein–Weiss inequality. Specifically, the authors establish the following Stein–Weiss inequality with the fractional Poisson kernel:

$${\int }_{{\mathbb{R}}_{+}^n}{\int }_{\partial {\mathbb{R}}_{+}^n}{\left|\xi \right|}^{-\alpha }{f\left(\xi \right)P(x,\xi ,\gamma )g\left(x\right)\left|x\right|}^{-\beta }d\xi dx\le C_{n,\alpha ,\beta ,p,q^{\prime }}{\parallel g\parallel }_{L^{q^{\prime }}\left({\mathbb{R}}_{+}^n\right)}{\parallel f\parallel }_{L^p(\partial {\mathbb{R}}_{+}^n)},$$

where $P(x,\xi ,\gamma )$ is the fractional Poisson kernel, i.e.,

$$P(x,\xi ,\gamma )={\displaystyle \frac{x_n}{\left(\right|x^{\prime }-\xi {{|}^2+x_n^2)}^{\frac{n+2-\gamma }{2}}}},2\le \gamma <n,$$

$f\in L^p(\partial {\mathbb{R}}_{+}^n)$, $g\in L^{q^{\prime }}\left({\mathbb{R}}_{+}^n\right)$ and $p,q^{\prime }\in (1,\infty )$ satisfy

$${\displaystyle \frac{n-1}{n}}{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q^{\prime }}}+{\displaystyle \frac{\alpha +\beta +2-\gamma }{n}}=1.$$

These three examples illustrate distinct generalizations of the Stein–Weiss inequality: extending it to broader function spaces such as Morrey spaces, generalizing the underlying geometric structure to non-Euclidean settings like the Heisenberg group, and modifying the integral operator by using boundary-to-interior kernels such as the fractional Poisson kernel. Together, these three directions constitute important branches of the theory of Stein–Weiss inequalities, demonstrating the remarkable versatility and robustness of this inequality across diverse mathematical structures. For more information about Stein–Weiss inequalities and HLS inequalities, we can refer to [1,3,10,11,12,13,14].

A notable development in this line of research is the reverse Stein–Weiss inequality. The reverse Stein–Weiss inequality on ${\mathbb{R}}^n$ was obtained by L. Chen, Z. Liu, G. Lu, and C. Tao [15]: For any $n\ge 1,p,q\in (0,1),\lambda <0,{\displaystyle \frac{n}{p^{\prime }}}<\alpha \le 0$, and ${\displaystyle \frac{n}{q^{\prime }}}<\beta \le 0$ satisfying

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}+{\displaystyle \frac{\alpha +\beta +\lambda }{n}}=2,\end{array}$$

(10)

there is a constant $C_{n,\alpha ,\beta ,p,q}^{\prime }>0$ such that for any non-negative functions $f\in$$L^p\left({\mathbb{R}}^n\right)$ and $g\in L^q\left({\mathbb{R}}^n\right)$,

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{|x-y|}^{\lambda }{\left|y\right|}^{\beta }}}dxdy\ge C_{n,\alpha ,\beta ,p,q}^{\prime }{\parallel f\parallel }_p{\parallel g\parallel }_q.\end{array}$$

(11)

Inspired by the inequalities (3), (4), (6), (7), which incorporate intermediate integral terms, and motivated by the classical Stein–Weiss inequality (9) and its reverse counterpart (11), we aim to establish in this paper stronger versions of both the Stein–Weiss inequality and the reverse Stein–Weiss inequality by introducing analogous intermediate integral terms.

To carry out this work, for any $\alpha \in \mathbb{R}$, $\beta \in \mathbb{R}$, $\lambda \in (0,n),p,q\in (1,\infty )$, or $\lambda \in (-\infty ,0),p,q\in (0,1)$, we first define the star-shaped set $S_{\alpha ,\beta ,\lambda }(f,g)$ for any non-negative functions $f\in$$L^p\left({\mathbb{R}}^n\right)$ and $g\in L^q\left({\mathbb{R}}^n\right)$ by its radial function,

$${{\rho }_{S_{\alpha ,\beta ,\lambda }(f,g)}\left(\xi \right)}^{n-\lambda }={\int }_0^{\infty }{r}^{n-\lambda -1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f(y+r\xi )g\left(y\right)}{{|y+r\xi |}^{\alpha }{\left|y\right|}^{\beta }}}dydr,\xi \in {\mathbb{R}}^n\setminus \left\{0\right\}.$$

(12)

In this paper, we prove the following strengthened Stein–Weiss inequalities.

Theorem 1.

Let $n\ge 1$, $1<p,q<+\infty$, $\alpha \ge 0$, $\beta \ge 0$, $\alpha <{\displaystyle \frac{n}{p^{\prime }}}$, $\beta <{\displaystyle \frac{n}{q^{\prime }}}$, $0<\lambda <n$ and let $n,p,q,\alpha ,\beta ,\lambda$ satisfy (8). Then, there is a constant $C_{n,\alpha ,\beta ,p,q}>0$ such that

$$\begin{array}{ccc}& & C_{n,\alpha ,\beta ,p,q}{∥f∥}_p{∥g∥}_q\hfill \\ & \ge & n{\omega }_n^{{\displaystyle \frac{\lambda }{n}}}{\left({\displaystyle \frac{1}{n}}{\int }_{{\mathbb{S}}^{n-1}}{\left({\int }_0^{\infty }{r}^{n-\lambda -1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(y+ru\right)g\left(y\right)}{{|y+ru|}^{\alpha }{\left|y\right|}^{\beta }}}dydr\right)}^{{\displaystyle \frac{n}{n-\lambda }}}du\right)}^{{\displaystyle \frac{n-\lambda }{n}}}\hfill \\ & \ge & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{\left|x-y\right|}^{\lambda }{\left|y\right|}^{\beta }}}dxdy\hfill \end{array}$$

(13)

for all non-negative functions $f\in L^p\left({\mathbb{R}}^n\right)$ and $g\in L^q\left({\mathbb{R}}^n\right)$.

By (12), we can write the first inequality in Theorem 1 in the form

$$\begin{array}{c}\hfill C_{n,\alpha ,\beta ,p,q}{\parallel f\parallel }_p{\parallel g\parallel }_q\ge n{\omega }_n^{{\displaystyle \frac{\lambda }{n}}}{\left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|}^{{\displaystyle \frac{n-\lambda }{n}}}.\end{array}$$

(14)

Removing the middle term in (13), we obtain the classical Stein–Weiss inequality (9). Thus, inequalities (13) are stronger than Stein–Weiss inequality (9).

In addition, we also obtain the following strengthened reverse Stein–Weiss inequalities.

Theorem 2.

Let $n\ge 1,p,q\in (0,1),\lambda <0,{\displaystyle \frac{n}{p^{\prime }}}<\alpha \le 0$, ${\displaystyle \frac{n}{q^{\prime }}}<\beta \le 0$, and (10) holds for $n,p,q,\alpha ,\beta ,\lambda$. Then, there is a constant $C_{n,\alpha ,\beta ,p,q}^{\prime }>0$ such that

$$\begin{array}{c}\hfill C_{n,\alpha ,\beta ,p,q}^{\prime }{\parallel f\parallel }_p{\parallel g\parallel }_q\le n{\omega }_n^{{\displaystyle \frac{\lambda }{n}}}{\left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|}^{{\displaystyle \frac{n-\lambda }{n}}}\le {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{|x-y|}^{\lambda }{\left|y\right|}^{\beta }}}dxdy\end{array}$$

(15)

for all non-negative functions $f\in L^p\left({\mathbb{R}}^n\right)$ and $g\in L^q\left({\mathbb{R}}^n\right)$.

Remove the middle term in (15), we get reverse Stein–Weiss inequality (11). Thus, inequalities (15) are stronger than reverse Stein–Weiss inequality (11).

2. Preliminaries

2.1. Star Bodies and Dual Mixed Volumes

A set $\mathsf{\Omega }\subseteq {\mathbb{R}}^n$ is said to be convex if $\left(1-\lambda \right)x+\lambda y\in \mathsf{\Omega }$ for any x, $y\in \mathsf{\Omega }$ and $\lambda \in \left[0,1\right]$. A convex body in ${\mathbb{R}}^n$ is a closed, bounded convex set that has non-empty interior.

We call a closed set $K\subseteq {\mathbb{R}}^n$ star-shaped with respect to the origin if $\left[0,x\right]\subseteq K$ for every $x\in K$. For a star-shaped set K, the radial function ${\rho }_K:{\mathbb{R}}^n\setminus \left\{0\right\}\to [0,\infty ]$ is defined as

$$\begin{array}{c}\hfill {\rho }_K\left(x\right)=sup\{\lambda \ge 0:\lambda x\in K\}.\end{array}$$

Additionally, the gauge function ${∥·∥}_K:{\mathbb{R}}^n\to \left[0,\infty \right]$ associated with K is defined by

$$\begin{array}{c}\hfill {\left|\right|·\left|\right|}_K=inf\{\lambda >0:x\in \lambda K\}.\end{array}$$

It is easily seen that ${\rho }_K={\left|\right|·\left|\right|}_K^{-1}$ holds. We say that two star-shaped sets K and L are dilates if there exists a constant $c\ge 0$ such that ${\rho }_K=c{\rho }_L$ almost everywhere on ${\mathbb{S}}^{n-1}$.

We say that a set K is a star body if ${\rho }_K$ is strictly positive and continuous in ${\mathbb{R}}^n\setminus \left\{0\right\}$. Set ${\left|\right|·\left|\right|}_{B^n}=|·|$, where $B^n$ is n-dimensional unit ball. The volume of a set K is denoted by $\left|K\right|$. The volume of a star-shaped set $K\subseteq {\mathbb{R}}^n$ can be expressed in terms of its measurable radial function:

$$\left|K\right|={\displaystyle \frac{1}{n}}{\int }_{{\mathbb{S}}^{n-1}}{\rho }_K{\left(\xi \right)}^{n}d\xi .$$

(16)

Let $\lambda \in (0,\infty )$ and $\lambda \ne n$. The dual mixed volume for star-shaped sets $K,L\subseteq {\mathbb{R}}^n$ endowed with measurable radial functions is defined by

$$\begin{array}{c}\hfill {\tilde{V}}_{\lambda }\left(K,L\right)={\displaystyle \frac{1}{n}}{\int }_{{\mathbb{S}}^{n-1}}{\rho }_K{\left(\xi \right)}^{n-\lambda }{\rho }_L{\left(\xi \right)}^{\lambda }d\xi .\end{array}$$

(17)

If $K=L$ in (17), we obtain

$$\begin{array}{c}\hfill {\tilde{V}}_{\lambda }\left(K,K\right)=\left|K\right|.\end{array}$$

The notion of dual mixed volume for star bodies was first introduced by E. Lutwak [16]. The dual mixed volume inequality for star-shaped sets $K,L\subseteq {\mathbb{R}}^n$ of finite volume when $0<\lambda <n$ can be expressed by

$$\begin{array}{c}\hfill {\tilde{V}}_{\lambda }(K,L)\le {\left|K\right|}^{{\displaystyle \frac{n-\lambda }{n}}}{\left|L\right|}^{{\displaystyle \frac{\lambda }{n}}}.\end{array}$$

(18)

The dual mixed volume inequality for star-shaped sets $K,L\subseteq {\mathbb{R}}^n$ when $\lambda <0$ or $\lambda >n$ can be expressed by

$$\begin{array}{c}\hfill {\tilde{V}}_{\lambda }(K,L)\ge {\left|K\right|}^{{\displaystyle \frac{n-\lambda }{n}}}{\left|L\right|}^{{\displaystyle \frac{\lambda }{n}}}.\end{array}$$

(19)

Inequalities (18) and (19) can be proved by Hölder’s inequality. When K and L are dilates, we can obtain the equalities in (18) and (19) for finite ${\tilde{V}}_{\alpha }\left(K,L\right)$ by the equality case in Hölder’s inequality. For more information regarding convex bodies, star bodies, and dual mixed volumes, we can refer to [17,18,19,20].

2.2. Symmetrization

Let $E\subseteq {\mathbb{R}}^n$ be a Borel set with finite measure. Define indicator function $1_E$ by $1_E\left(x\right)=1$ when $x\in E$ and $1_E\left(x\right)=0$ when $x\notin E$ for a subset $E\subseteq {\mathbb{R}}^n$. Schwarz symmetral of E, denoted by $E^{\star }$, is a closed Euclidean ball centered at origin whose volume coincides with that of E. Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a non-negative measurable function such that for any $t>0$, the superlevel set $\left\{f\ge t\right\}$ has finite measure. For almost every $x\in {\mathbb{R}}^n$, the layer cake formula is defined by

$$\begin{array}{c}\hfill f\left(x\right)={\int }_0^{\infty }{1}_{\left\{f\ge t\right\}}\left(x\right)dt.\end{array}$$

(20)

For some $x\in {\mathbb{R}}^n$, the Schwarz symmetrization$f^{\star }$ of f is defined by

$$\begin{array}{c}\hfill f^{\star }\left(x\right)={\int }_0^{\infty }{1}_{{\left\{f\ge t\right\}}^{\star }}\left(x\right)dt,\end{array}$$

where ${\left\{f\ge t\right\}}^{\star }$ is a closed Euclidean ball centered at origin whose volume coincides with that of $\left\{f\ge t\right\}$. Consequently, $f^{\star }$ is characterized almost everywhere by being radially symmetric and possessing superlevel sets whose measures coincide with those of f. We also say that $f^{\star }$ is the symmetric decreasing rearrangement of f. $f^{\star }$ is said to be strictly symmetric decreasing, if $f^{\star }\left(x\right)>f^{\star }\left(y\right)$ when $\left|x\right|<\left|y\right|$.

The proof of our results rely on the Riesz rearrangement inequality (see, for example, (Theorem 3.7 [1])).

Theorem 3

(Riesz’s rearrangement inequality). Let $f,g$, and h be three non-negative functions on ${\mathbb{R}}^n$. Then,

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}f\left(x\right)g(x-y)h\left(y\right)dxdy\le {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{f}^{\star }\left(x\right)g^{\star }(x-y)h^{\star }\left(y\right)dxdy,\end{array}$$

with the understanding that ${\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{f}^{\star }\left(x\right)g^{\star }(x-y)h^{\star }\left(y\right)dxdy=\infty$ iff ${\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}f\left(x\right)$$g(x-y)h\left(y\right)dxdy=\infty$.

3. The Star-Shaped Set ${\mathit{S}}_{\mathit{\alpha },\mathit{\beta },\mathit{\lambda }}(\mathit{f},\mathit{h})$

Let $f,g:{\mathbb{R}}^n\to [0,\infty )$ be measurable functions, $K\subseteq {\mathbb{R}}^n$ be a star-shaped set with a measurable radial function and $\alpha \in \mathbb{R}$, $\beta \in \mathbb{R}$, $\lambda \in (0,n),p,q\in (1,\infty )$ or $\lambda \in (-\infty ,0),p,q\in (0,1)$. First, we study the following double integral

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dxdy.\end{array}$$

According to Fubini’s theorem, polar coordinates, and (12), it can be concluded that

$$\begin{array}{ccc}\hfill {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dxdy& =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f(y+z)g\left(y\right)}{{|y+z|}^{\alpha }{∥z∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dzdy\hfill \\ & =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{S}}^{n-1}}{\int }_0^{\infty }{\displaystyle \frac{f(y+ru)g\left(y\right)}{{|y+ru|}^{\alpha }{∥ru∥}_K^{\lambda }{\left|y\right|}^{\beta }}}{r}^{n-1}drdudy\hfill \\ & =& {\int }_{{\mathbb{S}}^{n-1}}{{\rho }_K\left(u\right)}^{\lambda }{\int }_0^{\infty }{r}^{n-\lambda -1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f(y+ru)g\left(y\right)}{{|y+ru|}^{\alpha }{\left|y\right|}^{\beta }}}dydrdu\hfill \\ & =& {\int }_{{\mathbb{S}}^{n-1}}{{\rho }_K\left(u\right)}^{\lambda }{{\rho }_{S_{\alpha ,\beta ,\lambda }(f,g)}\left(u\right)}^{n-\lambda }du.\hfill \end{array}$$

Accordingly, from (17), we see that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dxdy=n{\tilde{V}}_{\lambda }(S_{\alpha ,\beta ,\lambda }(f,g),K).\end{array}$$

(21)

4. Strengthened Stein–Weiss Inequalities

In this section, we prove strengthened Stein–Weiss inequalities.

Proof of Theorem 1. 

Suppose $f,g:{\mathbb{R}}^n\to [0,\infty )$ are non-negative, non-zero, measurable functions with superlevel sets of finite measure. Let us first prove that

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{1_{\left\{f\ge t\right\}}\left(x\right)}{{\left|x\right|}^{\alpha }}}\right)}^{\star }\le {\displaystyle \frac{1_{{\left\{f\ge t\right\}}^{\star }}\left(x\right)}{{\left|x\right|}^{\alpha }}}\mathrm{and}{\left({\displaystyle \frac{1_{\left\{g\ge s\right\}}\left(y\right)}{{\left|y\right|}^{\beta }}}\right)}^{\star }\le {\displaystyle \frac{1_{{\left\{g\ge s\right\}}^{\star }}\left(y\right)}{{\left|y\right|}^{\beta }}}.\end{array}$$

(22)

Let $h_1\left(x\right)=1_{\left\{f\ge t\right\}}\left(x\right)$ and $h_2\left(x\right)={\left|x\right|}^{-\alpha }$. By $h_1\left(x\right)=1_{\left\{f\ge t\right\}}\left(x\right)$, it follows that $h_1{h}_2\le h_2$, and thus

$$\begin{array}{c}\hfill {\left(h_1{h}_2\right)}^{\star }\le h_2^{\star }.\end{array}$$

(23)

Let $x\in \left\{{\left(h_1{h}_2\right)}^{\star }\ge t\right\}$, by (23) and $supp{\left(h_1{h}_2\right)}^{\star }={\left\{f\ge t\right\}}^{\star }$, we can see that $x\in \left\{h_2^{\star }\ge t\right\}$ and $x\in {\left\{f\ge t\right\}}^{\star }$ and thus $x\in \left\{h_1^{\star }{h}_2^{\star }\ge t\right\}$. Therefore,

$$\begin{array}{c}\hfill \left\{{\left(h_1{h}_2\right)}^{\star }\ge t\right\}\subseteq \left\{h_1^{\star }{h}_2^{\star }\ge t\right\}.\end{array}$$

(24)

By layer cake Formulas (20) and (24),

$$\begin{array}{c}\hfill {\left(h_1{h}_2\right)}^{\star }\left(x\right)={\int }_0^{\infty }{1}_{\left\{{\left(h_1{h}_2\right)}^{\star }\ge t\right\}}\left(x\right)dt\le {\int }_0^{\infty }{1}_{\left\{h_1^{\star }{h}_2^{\star }\ge t\right\}}\left(x\right)dt=h_1^{\star }\left(x\right)h_2^{\star }\left(x\right)\end{array}$$

(25)

and we thus obtain the first inequality of (22). We can prove the second inequality of (22) following the same method as the proof of the first inequality of (22).

Suppose $K\subseteq {\mathbb{R}}^n$ is a star-shaped set whose radial function is measurable and whose volume satisfies $\left|K\right|>0$. By layer cake Equation (20) to the functions f and g and Fubini’s theorem, we obtain

$$\begin{array}{ccc}& & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dxdy\hfill \\ & =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\int }_0^{\infty }{\int }_0^{\infty }{\displaystyle \frac{1_{\left\{f\ge t\right\}}\left(x\right)1_{\left\{g\ge s\right\}}\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dtdsdxdy\hfill \\ & =& {\int }_0^{\infty }{\int }_0^{\infty }{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1_{\left\{f\ge t\right\}}\left(x\right)1_{\left\{g\ge s\right\}}\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dxdydtds.\hfill \end{array}$$

(26)

Let $h\left(x\right)={\displaystyle \frac{1_{\left\{f\ge t\right\}}\left(x\right)}{{\left|x\right|}^{\alpha }}}$ and $w\left(y\right)={\displaystyle \frac{1_{\left\{g\ge s\right\}}\left(y\right)}{{\left|y\right|}^{\beta }}}$. By Theorem 3 and (22), it follows that

$$\begin{array}{ccc}& & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}h\left(x\right){∥x-y∥}_K^{-\lambda }w\left(y\right)dxdy\hfill \\ & \le & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{h}^{\star }\left(x\right){∥x-y∥}_{K^{\star }}^{-\lambda }{w}^{\star }\left(y\right)dxdy\hfill \\ & =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\left({\displaystyle \frac{1_{\left\{f\ge t\right\}}\left(x\right)}{{\left|x\right|}^{\alpha }}}\right)}^{\star }{∥x-y∥}_{K^{\star }}^{-\lambda }{\left({\displaystyle \frac{1_{\left\{g\ge s\right\}}\left(y\right)}{{\left|y\right|}^{\beta }}}\right)}^{\star }dxdy\hfill \\ & \le & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1_{{\left\{f\ge t\right\}}^{\star }}\left(x\right)}{{\left|x\right|}^{\alpha }}}{∥x-y∥}_{K^{\star }}^{-\lambda }{\displaystyle \frac{1_{{\left\{g\ge s\right\}}^{\star }}\left(y\right)}{{\left|y\right|}^{\beta }}}dxdy\hfill \\ & =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1_{\left\{f^{\star }\ge t\right\}}\left(x\right)}{{\left|x\right|}^{\alpha }}}{∥x-y∥}_{K^{\star }}^{-\lambda }{\displaystyle \frac{1_{\left\{g^{\star }\ge s\right\}}\left(y\right)}{{\left|y\right|}^{\beta }}}dxdy\hfill \end{array}$$

(27)

for $s,t>0$. By (27) and (26), we can obtain

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dxdy\le {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f^{\star }\left(x\right)g^{\star }\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_{K^{\star }}^{\lambda }{\left|y\right|}^{\beta }}}dxdy.\end{array}$$

(28)

Applying (21) and (28), we can assert that

$$\begin{array}{ccc}\hfill {\tilde{V}}_{\lambda }(K,S_{\alpha ,\beta ,\lambda }(f,g))& =& {\displaystyle \frac{1}{n}}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dxdy\hfill \\ & \le & {\displaystyle \frac{1}{n}}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f^{\star }\left(x\right)g^{\star }\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_{K^{\star }}^{\lambda }{\left|y\right|}^{\beta }}}dxdy\hfill \\ & =& {\tilde{V}}_{\lambda }(K^{\star },S_{\alpha ,\beta ,\lambda }(f^{\star },g^{\star })).\hfill \end{array}$$

Then,

$$\begin{array}{c}\hfill {\tilde{V}}_{\lambda }(K,S_{\alpha ,\beta ,\lambda }(f,g))\le {\tilde{V}}_{\lambda }(K^{\star },S_{\alpha ,\beta ,\lambda }(f^{\star },g^{\star })).\end{array}$$

(29)

Suppose $f\in L^p\left({\mathbb{R}}^n\right)$ and $g\in L^q\left({\mathbb{R}}^n\right)$. Let us suppose that $|S_{\alpha ,\beta ,\lambda }(f,g)|<\infty$. Applying (29) for $K=S_{\alpha ,\beta ,\lambda }(f,g)$ and by dual mixed volume inequality (18) for $0<\lambda <n$, we have

$$\begin{array}{ccc}\hfill |S_{\alpha ,\beta ,\lambda }(f,g)|& =& {\tilde{V}}_{\lambda }\left(S_{\alpha ,\beta ,\lambda }(f,g),S_{\alpha ,\beta ,\lambda }(f,g)\right)\hfill \\ & \le & {\tilde{V}}_{\lambda }({\left(S_{\alpha ,\beta ,\lambda }(f,g)\right)}^{\star },S_{\alpha ,\beta ,\lambda }(f^{\star },g^{\star }))\hfill \\ & =& {\left|{\left(S_{\alpha ,\beta ,\lambda }(f,g)\right)}^{\star }\right|}^{{\displaystyle \frac{\lambda }{n}}}{\left|S_{\alpha ,\beta ,\lambda }(f^{\star },g^{\star })\right|}^{{\displaystyle \frac{n-\lambda }{n}}}\hfill \\ & =& {\left|S_{\alpha ,\beta ,\lambda }(f,g)\right|}^{{\displaystyle \frac{\lambda }{n}}}{\left|S_{\alpha ,\beta ,\lambda }(f^{\star },g^{\star })\right|}^{{\displaystyle \frac{n-\lambda }{n}}}.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill \left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|\le \left|S_{\alpha ,\beta ,\lambda }\left(f^{\star },g^{\star }\right)\right|.\end{array}$$

(30)

Assume that $|S_{\alpha ,\beta ,\lambda }(f,g)|=\infty$. For $k\ge 1$, define

$$\begin{array}{c}\hfill f_k\left(x\right)=f\left(x\right)1_{k{B}^n}\left(x\right),g_k\left(x\right)=g\left(x\right)1_{k{B}^n}\left(x\right).\end{array}$$

Note that $f_k$ and $g_k$ are non-decreasing with respect to k and converge pointwise to f and g, respectively. By monotone convergence theorem, we have

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\int }_0^{\infty }{r}^{n-\lambda -1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f_k(x+ru)g_k\left(x\right)}{{|x+ru|}^{\alpha }{\left|x\right|}^{\beta }}}dxdr={\int }_0^{\infty }{r}^{n-\lambda -1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f(x+ru)g\left(x\right)}{{|x+ru|}^{\alpha }{\left|x\right|}^{\beta }}}dxdr\end{array}$$

and the convergence is monotone. Further application of monotone convergence theorem, we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\int }_{{\mathbb{S}}^{n-1}}{\left({\int }_0^{\infty }{r}^{n-\lambda -1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f_k(x+ru)g_k\left(x\right)}{{|x+ru|}^{\alpha }{\left|x\right|}^{\beta }}}dxdr\right)}^{{\displaystyle \frac{n}{n-\lambda }}}du\\ \hfill ={\int }_{{\mathbb{S}}^{n-1}}{\left({\int }_0^{\infty }{r}^{n-\lambda -1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f(x+ru)g\left(x\right)}{{|x+ru|}^{\alpha }{\left|x\right|}^{\beta }}}dxdr\right)}^{{\displaystyle \frac{n}{n-\lambda }}}du.\end{array}$$

Thus,

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}|S_{\alpha ,\beta ,\lambda }(f_k,g_k)|=|S_{\alpha ,\beta ,\lambda }(f,g)|=\infty .\end{array}$$

(31)

For $f\in L^p\left({\mathbb{R}}^n\right)$ and $g\in L^q\left({\mathbb{R}}^n\right)$, the functions $f_k$ and $g_k$ have compact support and satisfy $|S_{\alpha ,\beta ,\lambda }(f_k,g_k)|<\infty$ for all $k\ge 1$. It is easy to show that ${\left(f_k\right)}^{\star }\le f^{\star }$ and ${\left(g_k\right)}^{\star }\le g^{\star }$ almost everywhere. Thus, by (30), we can see that

$$\begin{array}{c}\hfill |S_{\alpha ,\beta ,\lambda }(f_k,g_k)|\le |S_{\alpha ,\beta ,\lambda }({\left(f_k\right)}^{\star },{\left(g_k\right)}^{\star })|\le |S_{\alpha ,\beta ,\lambda }(f^{\star },g^{\star })|\end{array}$$

(32)

for $k\ge 1$. By (31) and (32), we have $|S_{\alpha ,\beta ,\lambda }(f^{\star },g^{\star })|=\infty$.

Then

$$\begin{array}{c}\hfill \left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|\le \left|S_{\alpha ,\beta ,\lambda }\left(f^{\star },g^{\star }\right)\right|.\end{array}$$

(33)

Since ${\left|\right|f\left|\right|}_p={\left|\right|f^{\star }\left|\right|}_p$ and ${\left|\right|g\left|\right|}_q={\left|\right|g^{\star }\left|\right|}_q$, by the classical Stein–Weiss inequality (9), (21), (18), and (33), we obtain

$$\begin{array}{ccc}\hfill C_{n,p,q,\alpha ,\beta }{\left|\right|f\left|\right|}_p{\left|\right|g\left|\right|}_q& \ge & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f^{\star }\left(x\right)g^{\star }\left(y\right)}{{\left|x\right|}^{\alpha }{\left|x-y\right|}^{\lambda }{\left|y\right|}^{\beta }}}dxdy\hfill \\ & =& n{\tilde{V}}_{\lambda }\left(B^n,S_{\alpha ,\beta ,\lambda }(f^{\star },g^{\star })\right)\hfill \\ & =& n{\omega }_n^{{\displaystyle \frac{\lambda }{n}}}{|S_{\alpha ,\beta ,\lambda }(f^{\star },g^{\star })|}^{{\displaystyle \frac{n-\lambda }{n}}}\hfill \\ & \ge & n{\omega }_n^{{\displaystyle \frac{\lambda }{n}}}{|S_{\alpha ,\beta ,\lambda }(f,g)|}^{{\displaystyle \frac{n-\lambda }{n}}}.\hfill \end{array}$$

(34)

Next, let $K=B^n$ in (21) and by dual mixed volume inequality (18), we can see that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{|x-y|}^{\lambda }{\left|y\right|}^{\beta }}}dxdy=n{\tilde{V}}_{\lambda }(B^n,S_{\alpha ,\beta ,\lambda }(f,h))\le n{\omega }_n^{{\displaystyle \frac{\lambda }{n}}}{|S_{\alpha ,\beta ,\lambda }(f,g)|}^{{\displaystyle \frac{n-\lambda }{n}}}.\end{array}$$

(35)

By (34) and (35), we can obtain the desired conclusion. □

5. Strengthened Reverse Stein–Weiss Inequalities

We will prove strengthened reverse Stein–Weiss inequalities.

Proof of Theorem 2. 

Suppose $f,g:{\mathbb{R}}^n\to [0,\infty )$ are non-negative, non-zero, measurable functions with superlevel sets of finite measure. Let us first prove that

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{1_{\left\{f\ge u\right\}}\left(x\right)}{{\left|x\right|}^{\alpha }}}\right)}^{\star }\ge {\displaystyle \frac{1_{{\left\{f\ge u\right\}}^{\star }}\left(x\right)}{{\left|x\right|}^{\alpha }}}\mathrm{and}{\left({\displaystyle \frac{1_{\left\{g\ge t\right\}}\left(y\right)}{{\left|y\right|}^{\beta }}}\right)}^{\star }\ge {\displaystyle \frac{1_{{\left\{g\ge t\right\}}^{\star }}\left(y\right)}{{\left|y\right|}^{\beta }}}.\end{array}$$

(36)

Let $h_1\left(x\right)=1_{\left\{f\ge u\right\}}\left(x\right)$ and $h_2\left(x\right)={\left|x\right|}^{-\alpha }$. Since ${\left\{0<h_1{h}_2\le u\right\}}^{\star }\subseteq \left\{0<h_1^{\star }{h}_2\le u\right\}$,

$$\begin{array}{c}\hfill {\int }_0^{\infty }(1-1_{{\left\{0<h_1{h}_2\le u\right\}}^{\star }}\left(x\right))du\ge {\int }_0^{\infty }(1-1_{\left\{0<h_1^{\star }{h}_2\le u\right\}}\left(x\right))du.\end{array}$$

(37)

By layer cake formula (20) and (37),

$$\begin{array}{c}\hfill {\left(h_1{h}_2\right)}^{\star }\left(x\right)={\int }_0^{\infty }{1}_{{\left\{h_1{h}_2\ge u\right\}}^{\star }}\left(x\right)du\ge {\int }_0^{\infty }{1}_{\left\{h_1^{\star }{h}_2\ge u\right\}}\left(x\right)du=h_1^{\star }\left(x\right)h_2\left(x\right),\end{array}$$

(38)

we therefore obtain the first inequality of (36). We can prove the second inequality of (36) by following the same method as when proving the first inequality of (36).

Let

$$\begin{array}{ccc}\hfill {∥z∥}_K^{-\lambda }& =& {\int }_0^{\infty }{1}_{[s,\infty )}\left({∥z∥}_K^{-\lambda }\right)ds\hfill \\ & =& {\int }_0^{\infty }{1}_{[s^{-1/\lambda },\infty )}\left({∥z∥}_K\right)ds\hfill \\ & =& {\int }_0^{\infty }{k}_s\left(z\right)ds,\hfill \end{array}$$

(39)

where $k_s\left(z\right)=1_{s^{-1/\lambda }\left({\mathbb{R}}^n\setminus K\right)}\left(z\right)$. Applying layer cake Formula (20) to the functions f and g, as well as (39) and Fubini’s theorem, we obtain

$$\begin{array}{ccc}& & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dxdy\hfill \\ & =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{\displaystyle \frac{1_{\left\{f\ge u\right\}}\left(x\right)k_s\left(x-y\right)1_{\left\{g\ge t\right\}}\left(y\right)}{{\left|x\right|}^{\alpha }{\left|y\right|}^{\beta }}}dudsdtdxdy\hfill \\ & =& {\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1_{\left\{f\ge u\right\}}\left(x\right)k_s\left(x-y\right)1_{\left\{g\ge t\right\}}\left(y\right)}{{\left|x\right|}^{\alpha }{\left|y\right|}^{\beta }}}dxdydudsdt.\hfill \end{array}$$

(40)

Let $h\left(x\right)={\displaystyle \frac{1_{\left\{f\ge u\right\}}\left(x\right)}{{\left|x\right|}^{\alpha }}}$ and $w\left(y\right)={\displaystyle \frac{1_{\left\{g\ge t\right\}}\left(y\right)}{{\left|y\right|}^{\beta }}}$. By Theorem 3 and (36), we see that

$$\begin{array}{ccc}& & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}h\left(x\right)k_s\left(x-y\right)w\left(y\right)dxdy\hfill \\ & =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}h\left(x\right)\left(1-1_{s^{-1/\lambda }K}\left(x-y\right)\right)w\left(y\right)dxdy\hfill \\ & =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}h\left(x\right)w\left(y\right)dxdy-{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}h\left(x\right)1_{s^{-1/\lambda }K}\left(x-y\right)w\left(y\right)dxdy\hfill \\ & \ge & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{h}^{\star }\left(x\right)w^{\star }\left(y\right)dxdy-{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{h}^{\star }\left(x\right)1_{s^{-1/\lambda }{K}^{\star }}\left(x-y\right)w^{\star }\left(y\right)dxdy\hfill \\ & =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{h}^{\star }\left(x\right)\left(1-1_{s^{-1/\lambda }{K}^{\star }}\left(x-y\right)\right)w^{\star }\left(y\right)dxdy\hfill \\ & =& {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{h}^{\star }\left(x\right)1_{s^{-1/\lambda }\left({\mathbb{R}}^n\setminus K^{\star }\right)}\left(x-y\right)w^{\star }\left(y\right)dxdy\hfill \\ & \ge & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1_{\left\{f^{\star }\ge u\right\}}\left(x\right)}{{\left|x\right|}^{\alpha }}}{1}_{s^{-1/\lambda }\left({\mathbb{R}}^n\setminus K^{\star }\right)}\left(x-y\right){\displaystyle \frac{1_{\left\{g^{\star }\ge t\right\}}\left(y\right)}{{\left|y\right|}^{\beta }}}dxdy\hfill \end{array}$$

(41)

for some $u,s,t>0$. From (41) and (40), we can obtain

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dxdy\ge {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f^{\star }\left(x\right)g^{\star }\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_{K^{\star }}^{\lambda }{\left|y\right|}^{\beta }}}dxdy.\end{array}$$

(42)

Suppose $K\subseteq {\mathbb{R}}^n$ is a star-shaped set whose radial function is measurable. From (21) and (42), we obtain

$$\begin{array}{ccc}\hfill {\tilde{V}}_{\lambda }(S_{\alpha ,\beta ,\lambda }(f,g),K)& =& {\displaystyle \frac{1}{n}}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_K^{\lambda }{\left|y\right|}^{\beta }}}dxdy\hfill \\ & \ge & {\displaystyle \frac{1}{n}}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f^{\star }\left(x\right)g^{\star }\left(y\right)}{{\left|x\right|}^{\alpha }{∥x-y∥}_{K^{\star }}^{\lambda }{\left|y\right|}^{\beta }}}dxdy\hfill \\ & =& {\tilde{V}}_{\lambda }(S_{\alpha ,\beta ,\lambda }\left(f^{\star },g^{\star }\right),K^{\star }).\hfill \end{array}$$

Then,

$$\begin{array}{c}\hfill {\tilde{V}}_{\lambda }\left(S_{\alpha ,\beta ,\lambda }(f,g),K\right)\ge {\tilde{V}}_{\lambda }\left(S_{\alpha ,\beta ,\lambda }\left(f^{\star },g^{\star }\right),K^{\star }\right).\end{array}$$

(43)

On the one hand, suppose $\left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|<\infty$. Applying (43) with $K=S_{\alpha ,\beta ,\lambda }\left(f,g\right)$ and by dual mixed volume inequality (19) for $\lambda <0$, we have

$$\begin{array}{ccc}\hfill \left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|& =& {\tilde{V}}_{\lambda }\left(S_{\alpha ,\beta ,\lambda }\left(f,g\right),S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right)\hfill \\ & \ge & {\tilde{V}}_{\lambda }\left(S_{\alpha ,\beta ,\lambda }\left(f^{\star },g^{\star }\right),S_{\alpha ,\beta ,\lambda }{\left(f,g\right)}^{\star }\right)\hfill \\ & \ge & {\left|S_{\alpha ,\beta ,\lambda }\left(f^{\star },g^{\star }\right)\right|}^{{\displaystyle \frac{n-\lambda }{n}}}{\left|{\left(S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right)}^{\star }\right|}^{{\displaystyle \frac{\lambda }{n}}}\hfill \\ & =& {\left|S_{\alpha ,\beta ,\lambda }\left(f^{\star },g^{\star }\right)\right|}^{{\displaystyle \frac{n-\lambda }{n}}}{\left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|}^{{\displaystyle \frac{\lambda }{n}}}.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill \left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|\ge \left|S_{\alpha ,\beta ,\lambda }\left(f^{\star },g^{\star }\right)\right|.\end{array}$$

(44)

From ${∥f∥}_p={∥f^{\star }∥}_p$, ${∥g∥}_q={∥g^{\star }∥}_q$, reverse Stein–Weiss inequality (11), (21), equality case of (19), and (44), we obtain

$$\begin{array}{ccc}\hfill C_{n,\alpha ,\beta ,p,q}^{\prime }{\parallel f\parallel }_p{\parallel g\parallel }_q& \le & {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f^{\star }\left(x\right)g^{\star }\left(y\right)}{{\left|x\right|}^{\alpha }{|x-y|}^{\lambda }{\left|y\right|}^{\beta }}}dxdy\hfill \\ & =& n{\tilde{V}}_{\lambda }\left(S_{\alpha ,\beta ,\lambda }\left(f^{\star },g^{\star }\right),B^n\right)\hfill \\ & =& n{\left|S_{\alpha ,\beta ,\lambda }\left(f^{\star },g^{\star }\right)\right|}^{{\displaystyle \frac{n-\lambda }{n}}}{\omega }_n^{{\displaystyle \frac{\lambda }{n}}}\hfill \\ & \le & n{\left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|}^{{\displaystyle \frac{n-\lambda }{n}}}{\omega }_n^{{\displaystyle \frac{\lambda }{n}}}.\hfill \end{array}$$

(45)

Suppose

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{|x-y|}^{\lambda }{\left|y\right|}^{\beta }}}dxdy<\infty .\end{array}$$

Taking $K=B^n$ in (21) and applying dual mixed volume inequality (19), we obtain

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{|x-y|}^{\lambda }{\left|y\right|}^{\beta }}}dxdy=n{\tilde{V}}_{\lambda }\left(S_{\alpha ,\beta ,\lambda }\left(f,g\right),B^n\right)\ge n{\left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|}^{{\displaystyle \frac{n-\lambda }{n}}}{\omega }_n^{{\displaystyle \frac{\lambda }{n}}}.\end{array}$$

(46)

On the other hand, assume that $\left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|=\infty$. Inequality (45) holds trivially, and we will next verify that inequality (46) holds under this assumption.

For $k\ge 1$, define

$$\begin{array}{c}\hfill f_k\left(x\right)=f\left(x\right)1_{k{B}^n}\left(x\right),g_k\left(x\right)=g\left(x\right)1_{k{B}^n}\left(x\right).\end{array}$$

Note that $f_k,g_k$ are non-decreasing with respect to k and converge pointwise to f and g, respectively. By monotone convergence theorem, we have

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\int }_0^{\infty }{r}^{n-\lambda -1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f_k(x+ru)g_k\left(x\right)}{{|x+ru|}^{\alpha }{\left|x\right|}^{\beta }}}dxdr={\int }_0^{\infty }{r}^{n-\lambda -1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f(x+ru)g\left(x\right)}{{|x+ru|}^{\alpha }{\left|x\right|}^{\beta }}}dxdr\end{array}$$

and the convergence is monotone. Further application of monotone convergence theorem, we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\int }_{{\mathbb{S}}^{n-1}}{\left({\int }_0^{\infty }{r}^{n-\lambda -1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f_k(x+ru)g_k\left(x\right)}{{|x+ru|}^{\alpha }{\left|x\right|}^{\beta }}}dxdr\right)}^{{\displaystyle \frac{n}{n-\lambda }}}du\\ \hfill ={\int }_{{\mathbb{S}}^{n-1}}{\left({\int }_0^{\infty }{r}^{n-\lambda -1}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f(x+ru)g\left(x\right)}{{|x+ru|}^{\alpha }{\left|x\right|}^{\beta }}}dxdr\right)}^{{\displaystyle \frac{n}{n-\lambda }}}du.\end{array}$$

Thus,

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}|S_{\alpha ,\beta ,\lambda }(f_k,g_k)|=|S_{\alpha ,\beta ,\lambda }(f,g)|=\infty .\end{array}$$

(47)

For $f\in L^p\left({\mathbb{R}}^n\right)$ and $g\in L^q\left({\mathbb{R}}^n\right)$, the functions $f_k$ and $g_k$ have compact support and satisfy $|S_{\alpha ,\beta ,\lambda }(f_k,g_k)|<\infty$ for all $k\ge 1$. Thus, by (46) for $\left|S_{\alpha ,\beta ,\lambda }\left(f_k,g_k\right)\right|<\infty$, we can see that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f_k\left(x\right)g_k\left(y\right)}{{\left|x\right|}^{\alpha }{|x-y|}^{\lambda }{\left|y\right|}^{\beta }}}dxdy=n{\tilde{V}}_{\lambda }\left(S_{\alpha ,\beta ,\lambda }\left(f_k,g_k\right),B^n\right)\ge n{\left|S_{\alpha ,\beta ,\lambda }\left(f_k,g_k\right)\right|}^{{\displaystyle \frac{n-\lambda }{n}}}{\omega }_n^{{\displaystyle \frac{\lambda }{n}}}\end{array}$$

(48)

for $k\ge 1$. By (47) and (48), we have ${\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)g\left(y\right)}{{\left|x\right|}^{\alpha }{|x-y|}^{\lambda }{\left|y\right|}^{\beta }}}dxdy=\infty$. Then, (46) holds for $\left|S_{\alpha ,\beta ,\lambda }\left(f,g\right)\right|=\infty$.

By (45) and (46), we obtain the desired conclusion. □

6. Conclusions

In this paper, we establish strengthened versions of both the classical Stein–Weiss inequalities and reverse Stein–Weiss inequalities on ${\mathbb{R}}^n$ by introducing intermediate integral terms. The constructed star-shaped set $S_{\alpha ,\beta ,\lambda }(f,g)$, together with the dual mixed volume inequality, the monotone convergence theorem, and Riesz’s rearrangement inequality, plays a crucial role in the proof of these inequalities.