1 Introduction

Let \(\mathbb {R}^n_+\) stand for the following half-space \(\begin{aligned}\mathbb {R}^n_+:=\{(x',x_n)\;|\;x'=(x_1,\ldots ,x_{n-1})\in \mathbb {R}^{n-1}, x_n>0\},\;n\in\mathbb{N}.\end{aligned}\) The Hardy inequality in \(\mathbb {R}^n_+\) asserts that i f \(p>1\) then

$$\begin{aligned} \int _{\mathbb {R}^n_+}|\nabla u|^p\mathrm {d}x\ge \left( \frac{p-1}{p}\right) ^p\int _{\mathbb {R}^n_+}\frac{|u|^p}{x_n^p}\mathrm {d}x\quad {\text {for \, all }}\,\, u\in C_c^\infty (\mathbb {R}^n_+), \end{aligned}$$
(1)

with the best possible constant. In particular, an integration by parts shows that

$$\begin{aligned} \int _{\mathbb {R}^n_+}\frac{|u|^{p-1}|u|_{x_n}}{x_n^{p-1}}\mathrm {d}x= \frac{1}{p}\int _{\mathbb {R}^n_+}\frac{(|u|^p)_{x_n}}{x_n^{p-1}}\mathrm {d}x= \frac{p-1}{p}\int _{\mathbb {R}^n_+}\frac{|u|^p}{x_n^p}\mathrm {d}x. \end{aligned}$$

An application of Hölder’s inequality with conjugate exponents p and \(p/(p-1)\) on the left term gives the stronger form of (1) with \(|u_{x_n}|\) in place of \(|\nabla u|\).Footnote 1

The Hardy–Sobolev–Maz’ya inequality: For \(p=2\), the critical Sobolev norm can be added on the right hand side of (1). More precisely, Maz’ya in his treatise [10] proved that for \(n\ge 3\) there exists a positive constant C such that

$$\begin{aligned} \left( \int _{\mathbb {R}^n_+}|\nabla u|^2\mathrm {d}x - \frac{1}{4}\int _{\mathbb {R}^n_+}\frac{u^2}{x_n^2}\mathrm {d}x\right) ^{1/2} \ge C\left( \int _{\mathbb {R}^n_+}|u|^{2^*}\mathrm {d}x\right) ^{1/2^*} \quad {\text {for \, all }}\,\, u\in C_c^\infty (\mathbb {R}^n_+), \end{aligned}$$
(2)

where \(2^*:=2n/(n-2)\). In [2] the optimal constant C in three dimensions is found to be the same with the best constant in the Sobolev inequality, while in [13] it has been shown that this fails in higher dimensions.

The Hardy–Sobolev inequality: The p-version of (2) for \(2\le p<n\) is

$$\begin{aligned} \bigg (\int _{\mathbb {R}^n_+}|\nabla u|^p\mathrm {d}x - \bigg (\frac{p-1}{p}\bigg )^{p}\int _{\mathbb {R}^n_+}\frac{u^p}{x_n^p}\mathrm {d}x\bigg )^{1/p} \ge C\bigg (\int _{\mathbb {R}^n_+}|u|^{p^*}\mathrm {d}x\bigg )^{1/p^*} \quad {\text {for \, all }} \, \, u\in C_c^\infty (\mathbb {R}^n_+), \end{aligned}$$
(3)

where \(p^\star =np/(n-p)\). This has been established in [4] (see §3 of this note) and later on with a different method in [6]. However, both approaches seem to fail giving (3) for \(1<p<2\).

Question 1: Is (3) true for \(1<p<2\)?

Having in mind the Sobolev embedding theorem, it is natural to ask for the corresponding inequalities when \(p\ge n\).

The Hardy–Morrey inequality: In [5] (see §4 of this note) the complete answer in the case \(p>n\) was given. More precisely, if \(p>n\ge 2\) there exists a positive constant C such that

$$\begin{aligned} \sup _{\begin{array}{c} x,y\in \mathbb {R}^n_+\\ x\ne y \end{array}}\frac{|u(x)-u(y)|}{|x-y|^{1-n/p}} \le C \bigg (\int _{\mathbb {R}^n_+}|\nabla u|^p\mathrm {d}x-\bigg (\frac{p-1}{p}\bigg )^p\int _{\mathbb {R}^n_+}\frac{|u|^p}{x_n^p}\mathrm {d}x\bigg )^{1/p} \quad {\text {for \, all }} \,\, u\in C_c^\infty (\mathbb {R}^n_+). \end{aligned}$$
(4)

Moreover, (4) fails for \(n=1\) (see [5, §7] for a sharp substitute in this case).

The Hardy–Moser–Trudinger inequality: In the case \(p=n=2\), the following sharp result has been established in [9]: There exists a positive constant C such that

$$\begin{aligned}&\int _{\mathbb {R}^2_+}\frac{e^{4\pi u^2}-1-4\pi u^2}{x_2^2}\mathrm {d}x \le C,\\&\quad {\text {for \, all }} \, u\in C_c^\infty (\mathbb {R}^2_+) \, {\text { satisfying }} \int _{\mathbb {R}^2_+}|\nabla u|^2\mathrm {d}x-\frac{1}{4}\int _{\mathbb {R}^2_+}\frac{u^2}{x_2^2}\mathrm {d}x\le 1. \end{aligned}$$

The proof uses the Riemann mapping theorem and it is natural to ask for a dimensional free proof and the following generalization

Question 2: Let \(n\in \mathbb {N}{\setminus }\{1\}\). Does there exists a positive constant \(C>0\) such that

$$\begin{aligned}&\int _{\mathbb {R}^n_+}\frac{\exp \bigg \{(n\omega _{n}^{1/n}|u|)^{n/(n-1)}\bigg \}-\sum _{j=0}^{n-1}\frac{(n\omega _{n}^{1/n}|u|)^{jn/(n-1)}}{j!}}{x_n^n}\mathrm {d}x \le C, \\&\quad {\text {for \, all }} \,\, u\in C_c^\infty (\mathbb {R}^n_+) {\text { satisfying }} \int _{\mathbb {R}^n_+}|\nabla u|^n\mathrm {d}x-\bigg (\frac{n-1}{n}\bigg )^n\int _{\mathbb {R}^n_+}\frac{|u|^n}{x_n^n}\mathrm {d}x\le 1? \end{aligned}$$

Here we have denoted by \(\omega _{n}\) the volume of the unit ball in \(\mathbb {R}^n\). Some subcritical results have been obtained in [5]: a Hardy–John–Nirenberg inequality and also Theorem B for \(p=n\) there, which in this note is the outcome of (6) applied to (7) and taking \(p=n\).

2 Two lower estimates on the Hardy difference

We recall here two estimates that we are going to use in the proofs of (3) and (4).

2.1 A lower estimate from the ground state transform

In [1] the authors obtained various auxiliary lower bounds for the Hardy difference:

$$\begin{aligned} I_p[u;\mathbb {R}^n_+] :=\int _{\mathbb {R}^n_+}|\nabla u|^p\mathrm {d}x-\left( \frac{p-1}{p}\right) ^p\int _{\mathbb {R}^n_+}\frac{|u|^p}{x_n^p}\mathrm {d}x, \quad u\in C_c^\infty (\mathbb {R}^n_+). \end{aligned}$$

In particular, the ground state transform

$$\begin{aligned} u=x_n^{1-1/p}v, \end{aligned}$$
(5)

implies

$$\begin{aligned} I_p[u;\mathbb {R}^n_+] = \int _{\mathbb {R}^n_+}\bigg \{\bigg |\frac{p-1}{p}x_n^{-1/p}v\mathbf {e}_n+x_n^{1-1/p}\nabla v\bigg |^p - \bigg |\frac{p-1}{p}x_n^{-1/p}v\mathbf {e}_n\bigg |^p\Big \}\mathrm {d}x. \end{aligned}$$

This together with the vectorial inequality (see [8])

$$\begin{aligned} |a+b|^p-|a|^p \ge (2^{p-1}-1)^{-1}|b|^p+p|a|^{p-2}a\cdot b, \quad {\text {for \, all }} \,\, a,b\in \mathbb {R}^n{\text { and }}\, p\ge 2, \end{aligned}$$

gives the following lower estimate on \(I_p[u;\mathbb {R}^n_+]\) (see [1, Lemma 3.3])

$$\begin{aligned} I_p[u;\mathbb {R}^n_+]&\ge c_p \int _{\mathbb {R}^n_+}x_n^{p-1}|\nabla v|^p\mathrm {d}x+\bigg (\frac{p-1}{p}\bigg )^{p-1}\int _{\mathbb {R}^n_+}\nabla |v|^p\cdot \mathbf {e}_n\mathrm {d}x \nonumber \\&= c_p\int _{\mathbb {R}^n_+}x_n^{p-1}|\nabla v|^p\mathrm {d}x, \end{aligned}$$
(6)

where \(c_p=(2^{p-1}-1)^{-1}\) and \(p\ge 2\).

2.2 A lower estimate from an inequality by Cabré and Ros-Oton

In [5] the following sharp lower estimate of the functional that appears on the right hand side of (6) is given: Let b, p, q satisfy

$$\begin{aligned} -1<b\le 0,1\le p<\frac{n}{b+1},\quad {\text {and}} \,\, q:=\frac{np}{n-p(b+1)}. \end{aligned}$$

There exists a positive constant C such that

$$\begin{aligned} \int _{\mathbb {R}^n_+}x_n^{p-1}|\nabla v|^p\mathrm {d}x \ge C\bigg (\int _{\mathbb {R}^n_+}\bigg (x_n^{b+1/p'}|v|\bigg )^{q}\mathrm {d}x\bigg )^{p/q} \quad {\text {for \, all }} \, \, v\in C_c^\infty (\mathbb {R}^n_+). \end{aligned}$$
(7)

This is to be compared with the case where the monomial weight in [3, Theorem 1.3], degenerates to the distance from the boundary of the half-space. In particular, by the choice \(A_i=0\) for all \(i=1,\ldots ,n-1\) and \(A_n=p-1\) in [3], one deduces the following weighted Sobolev inequality

$$\begin{aligned} \int _{\mathbb {R}^n_+}x_n^{p-1}|\nabla v|^p\mathrm {d}x \ge C\Big (\int _{\mathbb {R}^n_+}x_n^{p-1}|v|^{p(p+n-1)/(n-1)}\mathrm {d}x\Big )^{(n-1)/(p+n-1)} \quad {\text {for \, all }} \,\, v\in C_c^\infty (\mathbb {R}^n), \end{aligned}$$
(8)

which for \(v\in C_c^\infty (\mathbb {R}^n_+)\) is a special case of (7), as one can easily check by taking \(b=-(p-1)/(p+n-1)\). Let us mention that the best constant C in the above inequality is obtained in [3], and that for \(p=2\) this inequality (with its sharp constant) was known before by a result of Maz’ya and Shaposhnikova (see [11, §6]).

3 Proof of the Hardy–Sobolev inequality

Let \(u\in C_c^\infty (\mathbb {R}^n_+)\). Following [4], we start from the Gagliardo-Nirenberg inequality

$$\begin{aligned} n\omega _n^{1/n}\Big (\int _{\mathbb {R}^n}|f|^{n/(n-1)}\mathrm {d}x\Big )^{1-1/n} \le \int _{\mathbb {R}^n}|\nabla f|\mathrm {d}x\quad {\text {for \, all }} \, f\in W^{1,1}(\mathbb {R}^n), \end{aligned}$$

and setting \(f=|u|^{p^\star (1-1/n)}\) we get

$$\begin{aligned} n\omega _n^{1/n}\frac{n-p}{p(n-1)}\Big (\int _{\mathbb {R}^n_+}|u|^{p^\star }\mathrm {d}x\Big )^{1-1/n} \le \int _{\mathbb {R}^n_+}|u|^{p^\star (1-1/p)}|\nabla u|\mathrm {d}x. \end{aligned}$$
(9)

To estimate the term of the right hand side of (9), we set \(u=x_n^{1-1/p}v\) to obtain

$$\begin{aligned} \int _{\mathbb {R}^n_+}|u|^{p^*(1-1/p)}|\nabla u|\mathrm {d}x&= \int _{\mathbb {R}^n_+}|u|^{p^*(1-1/p)}\Big |x_n^{1-1/p}\nabla v+\frac{p-1}{p}x_n^{-1/p}v\mathbf {e}_n\Big |\mathrm {d}x\nonumber \\&\le \underbrace{\int _{\mathbb {R}^n_+}|u|^{p^*(1-1/p)}x_n^{1-1/p}|\nabla v|\mathrm {d}x}_{=:A} \nonumber \\&\quad + \, \frac{p-1}{p}\underbrace{\int _{\mathbb {R}^n_+}x_n^{p^*(1-1/p)^2-1/p}|v|^{p^*(1-1/p)+1}\mathrm {d}x.}_{=:B} \end{aligned}$$
(10)

To ease the computation, we set

$$\begin{aligned} \beta := p^*(1-1/p)^2+1-1/p, \end{aligned}$$

so that

$$\begin{aligned} B = \int _{\mathbb {R}^n_+}x_n^{\beta -1}|v|^{\beta p/(p-1)}\mathrm {d}x. \end{aligned}$$

To estimate B we integrate by parts as follows

$$\begin{aligned} B= & {} \frac{1}{\beta }\int _{\mathbb {R}^n_+}\nabla x_n^\beta \cdot \mathbf {e}_n\;|v|^{\beta p/(p-1)}\mathrm {d}x, \\= & {} - \frac{p}{p-1}\int _{\mathbb {R}^n_+}x_n^\beta |v|^{\beta p/(p-1)-1}\nabla |v|\cdot \mathbf {e}_n\;\mathrm {d}x\\= & {} - \frac{p}{p-1}\int _{\mathbb {R}^n_+}|u|^{p^*(1-1/p)}x_n^{1-1/p}\nabla |v|\cdot \mathbf {e}_n\;\mathrm {d}x\\\le & {} \frac{p}{p-1}A. \end{aligned}$$

Inserting this into (10), we obtain

$$\begin{aligned} \int _{\mathbb {R}^n_+}|u|^{p^*(1-1/p)}|\nabla u|\mathrm {d}x\le 2A. \end{aligned}$$
(11)

Now we estimate A using Hölder’s inequality as follows

$$\begin{aligned} A&= \int _{\mathbb {R}^n_+}\Big \{|u|^{p^*(1-1/p)}\Big \}\Big \{x_n^{1-1/p}|\nabla v|\Big \}\mathrm {d}x\\&\le \Vert u\Vert _{L^{p^*}(\mathbb {R}^n_+)}^{p^*(1-1/p)}\Big (\int _{\mathbb {R}^n_+}x_n^{p-1}|\nabla v|^p\mathrm {d}x\Big )^{1/p} \\&\le c_p^{-1/p}\Vert u\Vert _{L^{p^*}(\mathbb {R}^n_+)}^{p^*(1-1/p)}(I_p[u;\mathbb {R}^n_+])^{1/p}, \end{aligned}$$

where we have used (6). Inserting the above estimate of A in (11), we get

$$\begin{aligned} \int _{\mathbb {R}^n_+}|u|^{p^*(1-1/p)}|\nabla u|\mathrm {d}x\le 2c_p^{-1/p}\Vert u\Vert _{L^{p^*}(\mathbb {R}^n_+)}^{p^*(1-1/p)}(I_p[u;\mathbb {R}^n_+])^{1/p}. \end{aligned}$$

Coupling this with (9) we deduce (3).

4 Proof of the Hardy–Morrey inequality

We first recall Morrey’s “Dirichlet growth” theorem (see [12, Theorem 3.5.2] or [7, Theorem 7.19]).

Theorem 4.1

Let \(\Omega\) be a domain in \(\mathbb {R}^n,~n\ge 1.\) Let \(u\in C_c^\infty (\Omega )\) and suppose that for some \(M>0\) and \(\alpha \in (0,1]\) the following estimate is true for all \(B_r\subset \mathbb {R}^n\)

$$\begin{aligned} \int _{B_r}|\nabla u|\mathrm {d}x \le Mr^{n-1+\alpha }. \end{aligned}$$
(12)

Then there exists \(c(n,\alpha )>0\) such that for all \(B_r\subset \mathbb {R}^n\)

$$\begin{aligned} \sup _{x,y\in B_r}|u(x)-u(y)| \le cMr^\alpha , \end{aligned}$$

or, equivalently (since u is compactly supported)

$$\begin{aligned} \sup _{\begin{array}{c} x,y\in \Omega \\ x\ne y \end{array}}\frac{|u(x)-u(y)|}{|x-y|^\alpha } \le cM. \end{aligned}$$

In view of the above theorem, for (4) to be true, it is enough to establish the following estimate

$$\begin{aligned} \int _{B_r}|\nabla u|\mathrm {d}x \le c\big (I[u;\mathbb {R}^n_+]\big )^{1/p}r^{n(1-1/p)}, \end{aligned}$$
(13)

for all \(r>0\) and for some positive constant c that depends only on n. To this end, let \(B_r\subset \mathbb {R}^n\) such that \(B_r\cap \mathbb {R}^n_+\ne \emptyset .\) Setting \(u=x_n^{1-1/p}v\) we have

$$\begin{aligned} \int _{B_r}|\nabla u|\mathrm {d}x \le \underbrace{\int _{B_r}x_n^{1-1/p}|\nabla v|\mathrm {d}x}_{=:K_r} + \frac{p-1}{p}\underbrace{\int _{B_r}x_n^{-1/p}|v|\mathrm {d}x}_{=:L_r}. \end{aligned}$$

Using first Hölder’s inequality and then (6) we get

$$\begin{aligned} K_r&\le \bigg (\int _{B_r}x_n^{p-1}|\nabla v|^p\mathrm {d}x\bigg )^{1/p}(\omega _{n}r^n)^{1-1/p} \nonumber \\&\le C(n,p)\big (I_p[u;\mathbb {R}^n_+]\big )^{1/p}r^{n(1-1/p)}. \end{aligned}$$
(14)

We will next estimate \(L_r\). Setting \(q:=p(p+n-1)/(n-1)\) we apply Holder’s inequality as follows

$$\begin{aligned} L_r&= \int _{B_r}\{x_n^{(p-1)/q}|v|\}\{x_n^{-1/p-(p-1)/q}\}\mathrm {d}x \nonumber \\&\le \bigg (\int _{B_r}x_n^{p-1}|v|^q\mathrm {d}x\bigg )^{1/q}\bigg (\int _{B_r\cap \mathbb {R}^n_+}x_n^{-\theta }\mathrm {d}x\bigg )^{1-1/q}; \theta :=\bigg (\frac{1}{p}+\frac{p-1}{q}\bigg )\frac{q}{q-1}. \end{aligned}$$
(15)

To estimate the right factor in (15), let \(Q_{2r}\) be the cube with the same center as \(B_r\) and edges of length 2r that are parallel to the coordinate axes. Then

$$\begin{aligned} \int _{B_r\cap \mathbb {R}^n_+}x_n^{-\theta }\mathrm {d}x&\le \int _{Q_{2r}\cap \mathbb {R}^n_+}x_n^{-\theta }\mathrm {d}x\\&= (2r)^{n-1}\int _{\max \{0,y_n-r\}}^{y_n+r}x_n^{-\theta }\mathrm {d}x_n \\&= \frac{1}{1-\theta }(2r)^{n-1}\big ((y_n+r)^{1-\theta }-\max \{0,y_n-r\}^{1-\theta }\big ), \end{aligned}$$

where \(y_n\) is the n-th coordinate of the center of \(B_r.\) If \(y_n\le r\) then

$$\begin{aligned} \int _{B_r\cap \mathbb {R}^n_+}x_n^{-\theta }\mathrm {d}x \le \frac{1}{1-\theta }(2r)^{n-\theta }. \end{aligned}$$
(16)

If \(y_n>r,\) then since \(\theta \in (0,1)\) there holds \(\alpha ^{1-\theta }-\beta ^{1-\theta }\le (\alpha -\beta )^{1-\theta },\) for all \(\alpha \ge \beta \ge 0,\) and thus (16) again holds true.

The left factor in (15) increases if we integrate in the whole \(\mathbb {R}^n_+\) and we may use first (8) and then (6) to estimate it by the Hardy difference. Altogether, we arrive at

$$\begin{aligned} L_r \le C(n,p) \big (I_p[u;\Omega ]\big )^{1/p}r^{(n-\theta )(q-1)/q}. \end{aligned}$$

This is the desired estimate (13), since

$$\begin{aligned} (n-\theta )\frac{q-1}{q}=n\frac{p-1}{p}. \end{aligned}$$

The proof is complete. \(\square\)