Reverse Stein–Weiss Inequalities on the Upper Half Space and the Existence of Their Extremals

1 Introduction

The classical Stein–Weiss inequality, established by Stein and Weiss [41] in the 1950s, states that

where p, q′, α, β and λ satisfy the following conditions:

The smallest Cn,α,β,p,q′ for (1.1) to hold is often referred as the best constant of the Stein–Weiss inequality. Throughout this paper, we always let p′, q′ denote the conjugate of p, q, respectively. We note that an alternative proof of establishing the Stein–Weiss inequalities has been recently found in [25] by using conditions on weights to guarantee the weighted boundedness of fractional integrals given in [37]. Lieb [29] used the method based on the symmetrization argument and the Riesz rearrangement to establish the existence of extremals for inequality (1.1) in the case p<q and α,β≥0. Furthermore, in the case of p=q, the extremals cannot be expected to exist (see Lieb [29] and also Herbst [28] for the case λ=n-1, p=q=2, α=0, β=1). In the case of p=q, Beckner [5, 4] obtained the sharp constant of the Stein–Weiss inequalities (1.1) by establishing a Stein–Weiss lemma with a general kernel. The precise estimate of the sharp constant of the Stein–Weiss inequalities for the case of p≠q was also established in [4]. For more results about proving precise estimates for Stein–Weiss functionals in conjunction with the study of Pitt’s type inequalities and their multilinear versions, we refer the reader to the works of Beckner [2, 3, 1, 6, 7]. We note that the existence of extremal functions for the Stein–Weiss inequalities in the case p<q, under the assumption α+β≥0, has been established by Chen, Lu and Tao [13], which extends Lieb’s result under the stronger assumption that α≥0 and β≥0, using the concentration-compactness principle of Lions [31, 32].

In the special case of α=β=0, the Stein–Weiss inequality (1.1) becomes the Hardy–Littlewood–Sobolev (HLS) inequality, which can be stated as follows:

with 1<q′, p<∞, 0<λ<n and 1q′+1p+λn=2. The above inequality was established by Hardy and Littlewood [27] for n=1 and Sobolev [38] for n≥2 (see also [39, 40]). Lieb and Loss [30] applied the layer cake representation formula to give an explicit upper bound for the sharp constant Cn,p,q′. More precisely, he showed that the best constant Cn,p,q′ satisfies the following estimate:

In the diagonal case q′=p=2⁢n2⁢n-λ, Lieb [29] classified the positive solutions of the Euler–Lagrange equations related to the Hardy–Littlewood–Sobolev inequality and obtained the best constant

Furthermore, Lieb also established the existence of extremals of (1.2) for general index p and q′ through the rearrangement inequality. Recently, Frank and Lieb [21, 22, 23], and Carlen, Carrillo and Loss [8] developed a rearrangement-free argument to obtain the sharp Hardy–Littlewood–Sobolev inequality and extremal functions (1.2). In particular, in [23], Frank and Lieb established a remarkable sharp Hardy–Littlewood–Sobolev inequality on the Heisenberg group which extends the celebrated work of Jerison and Lee on the sharp Sobolev inequality on the Heisenberg group. The Stein–Weiss inequality on the Heisenberg group was also established in [25] by using a condition on weighted inequalities for integral operators in [37]. Han [24] proved the existence of extremals of the Hardy–Littlewood–Sobolev inequality on the Heisenberg group for all the case of q′ and p through the concentration-compactness principle of Lions [31, 32]. More recently, the existence of the extremal functions of the Stein–Weiss inequalities proved in [25] have also been established by Chen, Lu and Tao [13]. The radial symmetry of positive solutions for the Hardy–Sobolev equation and system were studied in [10, 16, 15, 34]. For more results about the Stein–Weiss inequality, see [3, 5] and the references therein.

We mention that we recently have established the following Hardy–Littlewood–Sobolev and Stein–Weiss inequalities with the fractional Poisson kernel in [12, 14]:

where

f ∈ L p ⁢ ( ∂ ⁡ ℝ + n ) , g∈Lq′⁢(ℝ+n), and p,q′∈(1,∞) satisfy

We also proved that there exist extremals for the Stein–Weiss inequality (1.3), and these extremals must be radially decreasing about the origin. We further provided the regularity and asymptotic estimates of positive solutions to the integral systems which are the Euler–Lagrange equations of the extremals to the Stein–Weiss inequality (1.3) with the fractional Poisson kernel. These results are inspired by the work of Hang, Wang and Yan [26], where the Hardy–Littlewood–Sobolev type inequality was first established when γ=2 and α=β=0. The proof of the Stein–Weiss inequality (1.3) with the fractional Poisson kernel relies on our work on the Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel [14].

We now return to the discussion on the reverse Stein–Weiss inequalities. Chen, Liu, Lu and Tao [11] established the following reverse Stein–Weiss inequality which states that for n≥1, p,q′∈(0,1), λ>0, 0≤α<-nq and 0≤β<-np′ satisfying

there exists a constant Cn,α,β,p,q′>0 such that for any nonnegative functions f∈Lq′⁢(ℝn), g∈Lp⁢(ℝn), there holds

Chen, Liu, Lu and Tao [11] used Fourier analysis techniques and the fractional integral operators acting on radial functions to establish the above inequality (1.4). They also obtained the extremal functions for all the indices by using symmetry and a rearrangement argument. The work of [11] was motivated by the sharp reverse Hardy–Littlewood–Sobolev inequality by Beckner [7, 9], and by Dou and Zhu [18]. The result of Beckner in [7] was used by Carneiro [9] to establish the sharp inequality for the Strichartz norm. In the case q′=p=2⁢n2⁢n+λ, Beckner [7], and Dou and Zhu [18] established the sharp constants and extremal functions for the reverse Hardy–Littlewood–Sobolev inequality. Subsequently, Ngô and Nguyen [35] verified that extremal functions still exist for general p and q′ through adapting similar arguments to Lieb in [29]. We note that the range of the exponents in the reverse Stein–Weiss inequality (1.4) are quite different from those in the Stein–Weiss inequality (1.1). They are mainly reflected in the difference that the power of the kernel |x-y| is negative in (1.1) and positive in (1.4), and p,q′>1 in (1.1) and p,q′<1 in (1.4).

Recently, Dou [17] established the following double weighted Hardy–Littlewood–Sobolev inequality on the upper half space ℝ+n, which can be seen as an extension of the Stein–Weiss inequality (1.1):

with p, q′, α, β and λ satisfying the following conditions:

In the case α, β>0, he applied the rearrangement inequality to obtain the existence of extremals for inequality (1.5) when p<q. In the special case of α=β=0, the Stein–Weiss inequality (1.5) on the upper half space ℝ+n becomes the following Hardy–Littlewood–Sobolev inequality on the upper half space:

where 1<q′, p<∞, 0<λ<n-1 and 1q′+n-1n⁢1p+λ+1n=2. In the case of q′=2⁢n2⁢n-λ and p=2⁢n-22⁢n-2-λ, Dou and Zhu [19] applied the moving spheres in integral form to classify the positive solutions of the Euler–Lagrange equations of the extremals for the Hardy–Littlewood–Sobolev inequality on ℝ+n and obtained the sharp constant. Ngô and Nguyen [36] studied the sharp reverse Hardy–Littlewood–Sobolev inequality on ℝ+n. More recently, the reverse Hardy–Littlewood–Sobolev and Stein–Weiss inequalities on the Heisenberg group have been established in [33].

Motivated by the aforementioned works [4, 11, 17, 19, 18, 35, 36], we will show in this paper that there exist a reverse version of the Stein–Weiss inequality on the upper half space ℝ+n. Furthermore, we prove that such an inequality has an extremal function for general indices. To answer these questions, we consider the following reverse Stein–Weiss inequality on the upper half space ℝ+n. Our first main result is the following.

Next, we will study whether the sharp constant Cn,α,β,p′,q′ for the reverse Stein–Weiss inequality on the upper half space can be attained. Namely, we will consider whether there exist extremal functions for inequality (1.6). To this end, we can consider the following minimizing problem:

where the double weighted operator Vλ⁢(g)⁢(x) is given by

It is easy to verify that the extremals for (1.6) are those solving the minimizing problem (1.7). Then we can prove that the constant Cn,α,β,p,q′ could actually be achieved.

Once we have established the existence of extremals of inequality (1.6), we can then consider the corresponding Euler–Lagrange system. First of all, one can minimize the functional

under the constraint ∥f∥Lq′⁢(ℝ+n)=∥g∥Lp⁢(∂⁡ℝ+n)=1. By the Euler–Lagrange multiplier theorem, we can derive the following integral system:

Let u=c1⁢fq′-1, v=c2⁢gp-1, 1q′-1=-p1 and 1p-1=-p2, and pick two suitable constants c1 and c2. Then system (1.8) is simplified as

where 1p1-1+1p2-1⁢n-1n=α+β+λn.

Next, we give some asymptotic estimates of positive solutions to system (1.9) around the origin and near infinity.

Finally, we are interested in studying the following weighted integral system:

With the help of the Pohozaev identity, we establish some necessary conditions for the existence of positive solutions to the weighted integral system (1.10).

An immediate corollary is that we can obtain the following Liouville theorem to the above integral system (1.10).

In the special case ν1=ν2=0, system (1.10) becomes

Clearly, system (1.11) corresponds to the Euler–Lagrange system of the extremal functions of the reverse Hardy–Littlewood–Sobolev inequality on the upper half space ℝ+n:

for any nonnegative functions f∈Lq′⁢(ℝn), g∈Lp⁢(ℝn) and p,q′∈(0,1), λ>0 such that n-1n⁢1p+1q′-λ-1n=2, with 1q′-1=-p1 and 1p-1=-p2. Therefore, we can deduce the following sufficient and necessary condition from Theorem 1.2 and Theorem 1.4.

As an application of Theorem 1.1, we also establish a spherical form of the reverse Stein–Weiss inequality (1.6) on the upper half space in the case of q′=2⁢n2⁢n+λ+α+β and p=2⁢n-22⁢n-2+λ+α+β.

In view of the Stein–Weiss inequality (1.5) and stereographic projection, we can also obtain the following Stein–Weiss inequality on the upper sphere 𝕊+n.

This paper is organized as follows. In Section 2, we employ the reverse weighted Hardy inequality to establish the reverse Stein–Weiss inequality on the upper half space and prove Theorem 1.1. In Section 3, we obtain the existence results for extremals of the inequality (1.6) with the help of the rearrangement inequality and prove Theorem 1.2. In Section 4, we investigate some asymptotic estimates of positive solutions to the corresponding Euler–Lagrange system around the origin and near infinity and establish Theorem 1.3. In Section 5, we use the Pohozaev identity in integral form to obtain necessary conditions for the existence of any positive solutions of system (1.10) and prove Theorem 1.4. The reverse Stein–Weiss inequality and the Stein–Weiss inequality on the upper sphere 𝕊+n are considered in Sections 6 and 7 through the stereographic projection. Theorems 1.7 and 1.9 are proved in these two sections. We mention that the reverse Stein–Weiss inequality on the sphere was already established in [11].

2 The Proof of Theorem 1.1

Throughout this section, we shall establish the reverse Stein–Weiss inequality on the upper half space. We need the following reverse weighted Hardy inequality on the upper half space. Our proof of the reverse weighted Hardy inequality on the upper half space is similar to that of the weighted Hardy inequality on whole space proved by Drábek, Heing and Kufner [20]. However, it needs a more careful analysis on the upper half space. In order to state the reverse Stein–Weiss inequality on the upper half space, we give the following notations. Define

For x=0, we write BR=BR⁢(0), BRn-1=BRn-1⁢(0), BR+=BR+⁢(0).