The best constant in the embedding of $W^{N,1}({\mathbb R}^N)$ into $L^\infty({\mathbb R}^N)$

TL;DR

This paper determines the optimal constant for embedding the Sobolev space $W^{N,1}({ m I ext{R}}^N)$ into $L^ ext{I ext{f}}({ m I ext{R}}^N)$, generalizing previous results to all dimensions using elliptic operator theory.

Contribution

It extends the known embedding constant results from dimensions one and two to all dimensions by identifying a fundamental solution of a specific elliptic operator.

Findings

Computed the best embedding constant for all dimensions.

Identified $ ext{log}|x|$ as a fundamental solution of an elliptic operator.

Extended previous low-dimensional results to general $N$.

Abstract

We compute the best constant in the embedding of $W^{N,1}({\mathbb R}^N)$ into $L^\infty({\mathbb R}^N)$, extending a result of Humbert and Nazaret in dimensions one and two to any $N$. The main tool is the identification of $\log |x|$ as a fundamental solution of a certain elliptic operator of order $2N$.

Equations

$$\Vert u{\Vert }_{\infty }\le C{\int }_{{\mathbb{R}}^N}|{\nabla }^{N}u|dx,\forall u\in C_c^{\infty }({\mathbb{R}}^N),$$

$${\nabla }^{N}u={\left\{\frac{{\partial }^{N}u}{\partial x_{i_1}\cdots \partial x_{i_N}}\right\}}_{i_1,\dots ,i_N\in I_N},$$

$$|{\nabla }^{N}u|={\left\{\sum _{i_1,\dots ,i_N\in I_N}{\left(\frac{{\partial }^{N}u}{\partial x_{i_1}\cdots \partial x_{i_N}}\right)}^2\right\}}^{1/2}.$$

$$\Vert u{\Vert }_{\infty }\le A{\int }_M|{\nabla }_g^{N}u{|}_{g}d{v}_g+B\Vert u{\Vert }_{W^{N-1,1}(M)},\forall u\in C^{\infty }(M),$$

$$|{\nabla }^N\log |x||=\frac{\sqrt{{\ell }_N}}{|x{|}^N},$$

$$\Vert u{\Vert }_{\infty }\le {\left(\sqrt{l_N}{\omega }_{N-1}\right)}^{-1}{\int }_{{\mathbb{R}}^N}|{\nabla }^{N}u|dx,\forall u\in W^{N,1}({\mathbb{R}}^N),$$

$$\Vert u{\Vert }_{\infty }\le {\left(\sqrt{l_N}{\omega }_{N-1}\right)}^{-1}{\int }_{\Omega }|{\nabla }^{N}u|dx,\forall u\in W_0^{N,1}(\Omega ),$$

$$\mathcal{L}=(-1{)}^N\sum _{i_1,\dots ,i_N\in I_N}\frac{{\partial }^N}{\partial x_{i_1}\cdots \partial x_{i_N}}\left(|x{|}^N\frac{{\partial }^N}{\partial x_{i_1}\cdots \partial x_{i_N}}\right),\text{ (see Proposition refprop:prop below)}.$$

$${\nabla }^{m}u$$

$$|{\nabla }^{m}u|$$

$$|{\nabla }^m{u}_A(x)|=|({\nabla }^{m}u)(Ax)|,\forall x\in B_R(0)\setminus \{0\}.$$

$${\int }_{{\mathbb{R}}^N}|{\nabla }^m{u}_A|={\int }_{{\mathbb{R}}^N}|{\nabla }^{m}u|.$$

$$\frac{\partial}{\partial x_{i}}\big{(}u(Ax)\big{)}=\sum_{j=1}^{N}a_{j,i}\frac{\partial u}{\partial y_{j}}(y)$$

$$\frac{{\partial }^m{u}_A}{\partial x_{i_1}\cdots \partial x_{i_m}}(x)=\sum _{j_1,\dots ,j_m\in I_N}{a}_{j_1,i_1}{a}_{j_2,i_2}\cdots a_{j_m,i_m}\frac{{\partial }^{m}u}{\partial y_{j_1}\cdots \partial y_{j_m}}(y).$$

$$\begin{gathered} |{\nabla }^m{u}_A(x){|}^2=\sum _{i_1,\dots ,i_m\in I_N}{\left(\frac{{\partial }^m{u}_A}{\partial x_{i_1}\cdots \partial x_{i_m}}(x)\right)}^2= \\ \sum _{\begin{array}{c}{i}_1,\dots ,i_m\in I_N\\ j_1,\dots ,j_m\in I_N\\ k_1,\dots ,k_m\in I_N\end{array}}{a}_{j_1,i_1}{a}_{j_2,i_2}\cdots a_{j_m,i_m}{a}_{k_1,i_1}{a}_{k_2,i_2}\cdots a_{k_m,i_m}\left(\frac{{\partial }^{m}u}{\partial y_{j_1}\cdots \partial y_{j_m}}(y)\right)\left(\frac{{\partial }^{m}u}{\partial y_{k_1}\cdots \partial y_{k_m}}(y)\right) \\ =\sum _{\begin{array}{c}{j}_1,\dots ,j_m\in \{1,\dots ,N\}\\ k_1,\dots ,k_m\in \{1,\dots ,N\}\end{array}}\left\{\prod _{s=1}^m\left(\sum _{i_s=1}^N{a}_{j_s,i_s}{a}_{k_s,i_s}\right)\right\}\left(\frac{{\partial }^{m}u}{\partial y_{j_1}\cdots \partial y_{j_m}}(y)\right)\left(\frac{{\partial }^{m}u}{\partial y_{k_1}\cdots \partial y_{k_m}}(y)\right). \end{gathered}$$

$$\sum _{i_s=1}^N{a}_{j_s,i_s}{a}_{k_s,i_s}={\delta }_{j_s,k_s},s=1,\dots ,m.$$

$$|{\nabla }^m{u}_A(x){|}^2=\sum _{j_1,\dots ,j_m\in I_N}{\left(\frac{{\partial }^{m}u}{\partial y_{j_1}\cdots \partial y_{j_m}}(y)\right)}^2=|({\nabla }^{m}u)(y){|}^2,$$

$$|{\nabla }^m\log |x||=\frac{\sqrt{{\ell }_N^m}}{|x{|}^m},x\in {\mathbb{R}}^N\setminus \{0\}.$$

$$|{\nabla }^{m}u(x)|=|{\nabla }^{m}u(Ax)|,\forall x\in {\mathbb{R}}^N\setminus \{0\},\forall A\in O(N).$$

$${\ell }_N^m=m!\sum _{l=0}^{\lfloor m/2\rfloor }(m-2l)!l!{\left(\frac{N-3}{2}+l\right)}_l{\left(\sum _{n=\lceil m/2\rceil }^{m-l}{2}^{2n-m+l}\frac{(-1{)}^n}{2n}\left(\genfrac{}{}{0ex}{}{n}{m-n}\right)\left(\genfrac{}{}{0ex}{}{m-n}{l}\right)\right)}^2,$$

$$(\nu {)}_k=\{\begin{array}{ll}\prod _{j=0}^{k-l}(\nu -j)& \text{ for }\nu \in \mathbb{R},k\in \mathbb{N},\\ 1& \text{ for }\nu \in \mathbb{R},k=0.\end{array}$$

$$K_1=\frac{1}{2},K_2=\frac{1}{2\pi \sqrt{2}},K_3=\frac{1}{4\pi \sqrt{28}}.$$

$$(-1{)}^N\sum _{i_1,\dots ,i_N\in I_N}\frac{{\partial }^N}{\partial x_{i_1}\cdots \partial x_{i_N}}\left(|x{|}^N\frac{{\partial }^N\log |x|}{\partial x_{i_1}\cdots \partial x_{i_N}}\right)=-{\ell }_N{\omega }_{N-1}{\delta }_0.$$

$$F(x):=(-1{)}^N\sum _{i_1,\dots ,i_N\in I_N}\frac{{\partial }^N}{\partial x_{i_1}\cdots \partial x_{i_N}}\left(|x{|}^N\frac{{\partial }^N\log |x|}{\partial x_{i_1}\cdots \partial x_{i_N}}\right)$$

$$\sum _{i_1,\dots ,i_N\in I_N}\frac{{\partial }^N}{\partial x_{i_1}\cdots \partial x_{i_N}}\left(|x{|}^N\frac{{\partial }^N{u}_A}{\partial x_{i_1}\cdots \partial x_{i_N}}\right)(x)=\sum _{i_1,\dots ,i_N\in I_N}\frac{{\partial }^N}{\partial y_{i_1}\cdots \partial y_{i_N}}\left(|y{|}^N\frac{{\partial }^{N}u}{\partial y_{i_1}\cdots \partial y_{i_N}}\right)(y),$$

$$F(x)=c|x{|}^{-N},$$

$$\phi (t)=\{\begin{array}{ll}0& \text{ for }t\in [0,1/2],\\ 1& \text{ for }t\ge 1,\end{array}$$

$${\int }_{{\mathbb{R}}^N}|{\nabla }^N{v}_{\delta }|\le C,\text{ uniformly in }\delta .$$

$$|x|^{N}\big{|}\frac{1}{\sqrt{\ell_{N}}}\nabla^{N}\log|x|\big{|}=1,$$

$${\int }_{{\mathbb{R}}^N}|{\nabla }^N{v}_{\delta }|\ge {\int }_{\{\delta /2<|x|<2\}}|{\nabla }^N{v}_{\delta }|\ge \left|\frac{1}{\sqrt{{\ell }_N}}{\int }_{\{\delta /2<|x|<2\}}\left({\nabla }^N{v}_{\delta }\right)\cdot \left(|x{|}^N{\nabla }^N\log |x|\right)\right|.$$

$$\begin{gathered} \frac{1}{\sqrt{{\ell }_N}}{\int }_{\{\delta /2<|x|<2\}}\left({\nabla }^N{v}_{\delta }\right)\cdot \left(|x{|}^N{\nabla }^N\log |x|\right)=\frac{1}{\sqrt{{\ell }_N}}{\int }_{\{\delta /2<|x|<2\}}F{v}_{\delta } \\ =\frac{c}{\sqrt{{\ell }_N}}{\int }_{\{\delta /2<|x|<2\}}\frac{v_{\delta }}{|x{|}^N}=\frac{c}{\sqrt{{\ell }_N}}{\int }_{\{\delta <|x|<1\}}\frac{1}{|x{|}^N}+O(1)=\frac{c}{\sqrt{{\ell }_N}}{\omega }_{N-1}\log (1/\delta )+O(1), \end{gathered}$$

Full text

1 Introduction

It is well known that the space $W^{N,1}({\mathbb{R}}^{N})$ is continuously embedded in $L^{\infty}({\mathbb{R}}^{N})$. Actually, it is embedded in $C_{0}({\mathbb{R}}^{N})$ – the subspace of $C({\mathbb{R}}^{N})$ consisting of functions satisfying $\lim_{|x|\to\infty}u(x)=0$. This follows from the density of $C^{\infty}_{c}({\mathbb{R}}^{N})$ in $W^{N,1}({\mathbb{R}}^{N})$ and the inequality

[TABLE]

see e.g. Brezis [1, Remark 13, Chapter 9]. Above we used the notation

[TABLE]

where $I_{N}:=\{1,\ldots,N\}$ (so that $\nabla^{N}u$ is a tensor of size $N^{N}$) and

[TABLE]

It is natural to look for the optimal constant $C$ in (1.1) (i.e., the smallest constant for which the inequality holds). Of course it is equivalent to consider the inequality for either $u\in C^{\infty}_{c}({\mathbb{R}}^{N})$ or $u\in W^{N,1}({\mathbb{R}}^{N})$. Following Humbert and Nazaret [4] we denote the optimal constant in (1.1) by $K_{N}$. The constant $K_{N}$ played an important role in their study of best constants in the inequality

[TABLE]

for a smooth compact Riemannian $N$–manifold $(M,g)$. In [4] they computed the value of $K_{N}$ for $N=1,2$, and left open the question of computing its value for $N\geq 3$. We answer this question in the current paper. Our argument is valid for every $N\geq 1$.

Remark 1**.**

The particular choice of the norm as in (1.3) is important, since as we shall see below, it is invariant with respect to the orthogonal group $O(N)$.

In order to prescribe the value of the best constant we recall (see Morii, Sato and Sawano [2]) the following property of the function $\log|x|$:

[TABLE]

for some positive constant $l_{N}$ (see also Corollary 2.2 below). Its explicit (and complicated!) value was calculated in [2] (see Remark 3 below). Our main result is the following ($\omega_{m}$ denotes the surface area of the unit sphere ${\mathbb{S}}^{m}$):

Theorem 1.1**.**

(i) The value of $K_{N}$ is $\left(\sqrt{l_{N}}\,\omega_{N-1}\right)^{-1}$. That is,

[TABLE]

*and one cannot replace $\left(\sqrt{l_{N}}\,\omega_{N-1}\right)^{-1}$ by a smaller constant.

(ii) Moreover, for $N\geq 2$ there is no function in $W^{N,1}({\mathbb{R}}^{N})$ (except the zero function) for which equality holds in (1.6).*

Remark 2**.**

An easy consequence of Theorem 1.1 is that the same result holds in the space $W^{N,1}_{0}(\Omega)$ (the closure of $C^{\infty}_{c}(\Omega)$ in $W^{N,1}(\Omega)$), for every $\Omega\subset{\mathbb{R}}^{N}$. More precisely,

[TABLE]

the constant $K_{N}=\left(\sqrt{l_{N}}\,\omega_{N-1}\right)^{-1}$ is optimal, and the inequality is strict, unless $u$ is the zero function. This follows from the invariance of the $W^{N,1}$-seminorm with respect to scalings.

The main ingredient of the proof of Theorem 1.1 is another property of the function $\log|x|$, namely, it is a fundamental solution for the elliptic operator

[TABLE]

Acknowledgments.

The author was supported by the Israel Science Foundation (Grant No. 999/13). Part of this work was done while he was visiting the University Claude Bernard Lyon 1. He thanks the Mathematics Department for its hospitality.

2 Preliminaries

We will consider also the generalization of (1.2)–(1.3) for any $m\geq 1$. So for each $u\in C^{m}(\Omega)$, $m\geq 1$, where $\Omega$ is a domain in ${\mathbb{R}}^{N}$, we set

[TABLE]

We will use the same notation for functions in $W^{m,1}(\Omega)$.

A simple, yet important feature of our analysis is the invariance of the norm in (2.2) with respect to the orthogonal group $O(N)$. This is the content of the next Lemma.

Lemma 2.1**.**

For $u\in C^{m}(B_{R}(0)\setminus\{0\})$, $m\geq 1$, and $A\in O(N)$ set $u_{A}(x)=u(Ax)$. Then

[TABLE]

The identity (2.3) holds a.e. on ${\mathbb{R}}^{N}$ for $u\in W^{m,1}({\mathbb{R}}^{N})$. In particular, for any $u\in W^{m,1}({\mathbb{R}}^{N})$ and $A\in O(N)$, $u_{A}(x):=u(Ax)$ satisfies

[TABLE]

Proof.

Put $y=Ax$. From the basic formula

[TABLE]

we deduce that for every $m$-tuple $(i_{1},\ldots,i_{m})\in{(I_{N})}^{m}$ we have

[TABLE]

Taking the square of (2.5) and summing over all $m$-tuples yields

[TABLE]

Next we notice that since $A$ is orthogonal we have

[TABLE]

Using it in (2.6) gives

[TABLE]

and (2.3) follows.

The statement about $u\in W^{m,1}({\mathbb{R}}^{N})$ and then the equality (2.4) follow from (2.3) and the density of $C^{\infty}_{c}({\mathbb{R}}^{N})$ in $W^{m,1}({\mathbb{R}}^{N})$. ∎

Part (ii) of the next Corollary was proved in [2]; we present a more elementary proof. It is based on part (i) which in turn follows immediately from Lemma 2.1.

Corollary 2.2**.**

*(i) If $u\in C^{m}(B_{R}(0)\setminus\{0\})$ or $u\in W^{m,1}(B_{R}(0))$, $m\geq 1$, is radial then the function $x\mapsto|\nabla^{m}u(x)|$ is also radial.

(ii) For every $m\geq 1$ there exists a positive constant $\ell^{m}_{N}$ such that*

[TABLE]

Proof.

(i) This is a direct consequence of Lemma 2.1 which gives in our case

[TABLE]

(ii) By (i), $|\nabla^{m}\log|x||$ is a radial function. Since each derivative $\frac{\partial^{m}\log|x|}{\partial x_{i_{1}}\cdots\partial x_{i_{m}}}$ is clearly homogenous of order $-m$, the same is true for $|\nabla^{m}\log|x||$. Thus, the function $|\nabla^{m}\log|x||$ is necessarily of the form (2.7). ∎

For the case $m=N$ we use the shorthand $\ell_{N}=\ell^{N}_{N}$ (see (1.5)).

Remark 3**.**

The authors of [2] computed an explicit expression for $\ell_{N}^{m}$:

[TABLE]

with the notation

[TABLE]

The first values of $\ell_{N}$ are: $\ell_{1}=1,\ell_{2}=2,\ell_{3}=28$, which by Theorem 1.1 imply

[TABLE]

3 Proof of Theorem 1.1

The proof of Theorem 1.1 is divided into two parts. In the first part we compute the value of $K_{N}$ and in the second we prove that equality cannot hold in (1.6), unless $u\equiv 0$.

3.1 The value of $K_{N}$

The main ingredient of the proof of Theorem 1.1 is the identification of $\log|x|$ as a fundamental solution of a certain operator of order $2N$:

Proposition 3.1**.**

For all $N\geq 2$ we have, in the sense of distributions:

[TABLE]

Proof.

The function

[TABLE]

clearly belongs to $C^{\infty}({\mathbb{R}}^{N}\setminus\{0\})$. We claim that $F$ is a radial function. Indeed, A similar computation to the one used in the proof of Lemma 2.1 shows that for a smooth $u$ on either ${\mathbb{R}}^{N}$ or ${\mathbb{R}}^{N}\setminus\{0\}$ and $A\in O(N)$, the function $u_{A}(x)=u(Ax)$ satisfies

[TABLE]

with $y=Ax$. Since $\log|x|$ is radial, this implies that $F(Ax)=F(x)$, whence $F$ is radial.

From (3.2) it is clear that $F$ is homogenous of degree $-N$. Therefore it must be of the form

[TABLE]

for some constant $c\in{\mathbb{R}}$. We claim that $c=0$.

Assume by contradiction that $c\neq 0$. Fix a function $\varphi\in C^{\infty}[0,\infty)$ taking values in $[0,1]$ and satisfying

[TABLE]

and then, for any $\delta\in(0,1)$ let $\varphi_{\delta}(t)=\varphi(t/\delta)$. Fix also a function $\zeta\in C^{\infty}_{c}({\mathbb{R}}^{N})$, taking values in $[0,1]$ and satisfying $\zeta=1$ on $B_{1}(0)$ and $\zeta=0$ on ${\mathbb{R}}^{N}\setminus B_{2}(0)$. Finally, define $v_{\delta}(x)=\zeta(x)\varphi_{\delta}(|x|)$ which belongs to $C^{\infty}_{c}({\mathbb{R}}^{N})$. Since $v_{\delta}\equiv 1$ on $\{\delta<|x|<1\}$ and $|\nabla^{N}\varphi_{\delta}(|x|)|\leq C/\delta^{N}$ for $|x|\leq\delta$ we have

[TABLE]

On the other hand, by (2.7),

[TABLE]

whence

[TABLE]

Applying integration by parts to the integral on the R.H.S. of (3.7) and using (3.3) gives

[TABLE]

where $O(1)$ denotes a bounded quantity, uniformly in $\delta$. Combining (3.7)–(3.8) with (3.5) leads to a contradiction for $\delta$ small enough, whence $c=0$ as claimed.

From the above we deduce that the distribution

[TABLE]

satisfies $\operatorname{supp}(\mathcal{F})\subset\{0\}$. By a celebrated theorem of L. Schwartz [5] it follows that

[TABLE]

for some multi-indices $\alpha_{1},\ldots,\alpha_{L}$. But by (2.7) each term on the R.H.S. of (3.9) can be written as

[TABLE]

with $g$ satisfying $|g(x)|\leq C/|x|^{N-1}$, so that $g\in L^{1}_{\text{loc}}({\mathbb{R}}^{N})$. It follows that $\mathcal{F}$ in (3.9) is a sum of first derivatives of functions in $L^{1}_{\text{loc}}$, whence for some $\mu\in{\mathbb{R}}$,

[TABLE]

It remains to determine the value of $\mu$ in (3.11). For that matter we will use a family of test functions $\{u_{\varepsilon}\}$, for small $\varepsilon>0$. Let $\varphi_{\varepsilon}(t)=\varphi(t/\varepsilon)$ with $\varphi$ given by (3.4) and define on $[0,\infty)$, $f_{\varepsilon}(t)=(-\log\varepsilon)-\int_{\varepsilon}^{t}\frac{\varphi_{\varepsilon}(s)}{s}\,ds$. Finally, let $u_{\varepsilon}(x)=\zeta(x)f_{\varepsilon}(|x|)$ on ${\mathbb{R}}^{N}$, where $\zeta$ is the same function used in the proof of Proposition 3.1 (recall $\zeta=1$ on $B_{1}(0)$ while $\zeta=0$ outside $B_{2}(0)$). It is easy to verify that $u_{\varepsilon}\in C^{\infty}_{c}({\mathbb{R}}^{N})$ (in fact, $\text{supp}(u_{\varepsilon})\subset B_{2}(0)$) and it satisfies:

[TABLE]

where $O(1)$ stands for a quantity which is bounded uniformly in $\varepsilon$.

Since $u_{\varepsilon}\in C^{\infty}_{c}({\mathbb{R}}^{N})$, we get by the definition of $\mathcal{F}$ (see (3.9)) and (3.11) that

[TABLE]

By (3.13)–(3.15) we get for the R.H.S. of (3.16),

[TABLE]

where we also used (3.6). Using (3.16)–(3.17) in conjunction with (3.12) yields $\mu=-\ell_{N}\omega_{N-1}$, as claimed. ∎

Proof of part (i) of Theorem 1.1.

Clearly it is enough to consider $u\in C^{\infty}_{c}({\mathbb{R}}^{N})$ and without loss of generality we may assume $u(0)=\|u\|_{\infty}$. By (3.1),

[TABLE]

Using (3.6) to bound the R.H.S. of (3.18) from above (as in (3.7)) yields

[TABLE]

whence

[TABLE]

To prove that equality holds in (3.19) it suffices to consider $u_{\varepsilon}$ constructed in the course of the proof of Proposition 3.1. Indeed, the arguments used there yield

[TABLE]

which in conjunction with (3.12) gives

[TABLE]

This clearly implies equality in (3.19). ∎

3.2 Nonexistence of an optimizer in (1.6)

Proof of Theorem 1.1(ii).

Looking for contradiction, assume that for some $N\geq 2$ there exists $u\in W^{N,1}({\mathbb{R}}^{N})$, $u\neq 0$, for which equality holds in (1.6). We may assume without loss of generality that $u(0)=\|u\|_{\infty}$.

We first show that such $u$ can be assumed radial. Indeed, notice that for every $A_{1},A_{2}\in O(N)$, the function $v(x):=(u(A_{1}x)+u(A_{2}x))/2$ satisfies $v(0)=\|v\|_{\infty}=\|u\|_{\infty}$ and

[TABLE]

where in the last equality we used (2.4). It follows that $v$ too realizes equality in (1.6). We can apply the same principle also for continuous averaging. Indeed, the function

[TABLE]

where the integration is with respect to the (normalized) Haar measure on $SO(N)$ (see [3]), belongs to $W^{N,1}({\mathbb{R}}^{N})$ and satisfies $\widetilde{u}(0)=\|\widetilde{u}\|_{\infty}=\|u\|_{\infty}$ and

[TABLE]

Hence, equality must hold in the last inequality and $\widetilde{u}$ is a (nontrivial) radial function for which equality holds in (1.6).

Let $\{u_{\varepsilon}\}_{\varepsilon>0}\subset C^{\infty}_{c}({\mathbb{R}}^{N})$ satisfy $u_{\varepsilon}\to\widetilde{u}$ in $W^{N,1}({\mathbb{R}}^{N})$, whence also in the uniform norm on ${\mathbb{R}}^{N}$. Since (3.18) holds for $u=u_{\varepsilon}$, passing to the limit yields

[TABLE]

Applying the Cauchy-Schwarz inequality to the integrand on the R.H.S. of (3.22) in conjunction with (1.5) and our assumption that equality holds in (1.6) for $\widetilde{u}$ yields

[TABLE]

It follows from (3.23) that $\nabla^{N}\widetilde{u}(x)\parallel\nabla^{N}\log|x|$, a.e. in ${\mathbb{R}}^{N}$. Since both $|\nabla^{N}\log|x||$ and $|\nabla^{N}\widetilde{u}(x)|$ are radial (see Corollary 2.2), it follows that there exists a function $a(t)$ such that

[TABLE]

For a smooth *radial * function $u$ on ${\mathbb{R}}^{N}$ we compute, introducing the variable $s=r^{2}/2=|x|^{2}/2$,

[TABLE]

Here and in the sequel, with a slight abuse of notation, we will consider a radial function $u(x)$ also as a function of $s$. A simple induction shows that for any multi-index $\alpha=(\alpha_{1},\ldots,\alpha_{N})$ with $|\alpha|=\sum_{j=1}^{N}\alpha_{j}=m$ we have

[TABLE]

where each $P^{(\alpha)}_{m-2i}$ is either an homogenous polynomial of degree $m-2i$ with positive integer coefficients, or the zero polynomial. It follows from (3.25) that the tensor $\nabla^{m}u$ can be written as

[TABLE]

where each ${\mathcal{P}}_{m-2i}$ is a tensor whose nonzero elements are taken from the set

[TABLE]

We claim that none of the tensors $\{{\mathcal{P}}_{m-2i}\}_{i=0}^{\left\lfloor{m/2}\right\rfloor}$ is the zero tensor. Indeed, this follows from the simple observation that for each fixed $j\in I_{N}$ we have

[TABLE]

with positive integer coefficients $\{b_{i}\}$. Now we can rewrite (3.26) as

[TABLE]

with $(\tilde{x}_{1},\ldots,\tilde{x}_{N})=(1/|x|)(x_{1},\ldots,x_{N})\in{\mathbb{S}}^{N-1}$. Of course, the above formulas continue to hold when we replace the smooth $u$ by a function belonging to $W^{m,1}$.

Going back to (3.24), using (3.27) for $u=\widetilde{u}$ and $u=\log|x|$, we conclude that

[TABLE]

Since for $i_{1}\neq i_{2}$ the monomials in the components of ${\mathcal{P}}_{N-2i_{1}}$ and ${\mathcal{P}}_{N-2i_{2}}$ have different degrees, it follows from (3.28) that

[TABLE]

Using (3.29) for $i=i_{0}:=\left\lfloor{N/2}\right\rfloor$ and $i=i_{0}-1$ yields that $\widetilde{v}:=\frac{d^{N-i_{0}}\widetilde{u}}{ds^{N-i_{0}}}$ satisfies

[TABLE]

Integrating (3.30) gives $\widetilde{v}=\frac{c}{s^{N-i_{0}}}$ for some constant $c$, whence

[TABLE]

where $Q_{N-i_{0}-1}$ is a polynomial of degree less or equal to $N-i_{0}-1$ and $\tilde{c}$ is another constant. For $\widetilde{u}$ as in (3.31), the requirements $\widetilde{u},|\nabla^{N}\widetilde{u}|\in L^{1}({\mathbb{R}}^{N})$ clearly impose $\widetilde{u}=0$. Contradiction. ∎

Remark 4**.**

It was shown in [4] that for $N=1$ the function $u(x)=e^{-|x|}$ satisfies $u(0)=\|u\|_{\infty}=(1/2)\int_{\mathbb{R}}|u^{\prime}|$, that is, equality holds in (1.6). In fact, this is true for any $u\in W^{1,1}({\mathbb{R}})$ satisfying $\operatorname{sgn}u^{\prime}(x)=-\operatorname{sgn}x$, a.e. on ${\mathbb{R}}$.