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数理解析研究所講究録
第2074巻 2018年 63-67

A DUAL FORM OF THE SHARP NASH INEQUALITY AND
ITS WEIGHTED GENERALIZATION

Eric A. Carlen1 and Elliott H. Lieb2

1 Department of Mathematics, Hill Center,Rutgers University, 110 Frelinghuysen Road Piscataway NJ 08854‐8019 USA

2 Departments of Mathematics and Physics, Jadwin Hall, Princeton University
Washington Rd., Princeton, NJ 08544.

September 15, 2017

Abstract

The well known duality between the Sobolev inequality and the Hardy‐Littlewood‐
Sobolev inequality suggests that the Nash inequality should also have an interesting dualform. We provide one here. This dual inequality relates the L^{2} norm to the infimal
convolution of the L^{\infty} and H^{-1} norms. The computation of this infimal convolution
is a minimization problem, which we solve explicitly, thus providing a new proof of thesharp Nash inequality itself. This proof, via duality, also yields the sharp form of someweighted generalizations of the Nash inequality and the dual of these weighted variants.

1Introduction

The subject of this talk is an example of how Kato motivated others by asking goodquestions. The story starts with a letter from Kato to Eric Carlen and Michael Loss, inwhich he asks whether it is possible to compute the sharp constant in Nash’s inequality[5].
Eric and Michael solved that problem in 1993 [2] and showed, surprisingly, that every
optimal function has compact support. The unanswered question hanging in the air was
What is the dual of Nash’s inequality? We have a solution of this problem and the result
is even more surprising—as one might expect.

Let us review the situation by starting with Sobolev’s inequality.
0This is a summary of a talk given by Elliott Lieb on September 4, 2017 at the conference at the Universityof Tokyo in honor of the centenary of Tosio Kato. The full paper is at arXiv:1704.08720.

lWork partially supported by U.S. National Science Foundation grant DMS 1501007.
2Work partially supported by U.S. National Science Foundation grant PHY 1265118© 2017 by the authors. This paper may be reproduced, in its entirety, for non‐commercial purposes.

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From Sobolev to Nash

The Sobolev inequality in \mathbb{R}^{n} ,( n \geq

3only), (see [1, 7], DD) is\Vert\nabla f\Vert_{2}\geq

S_{n}\Vert f\Vert_{2n/(n-2)}.
This is an inequality between two (convex) integrals and has an unambiguous dual, which is
the Hardy‐Littlewood inequality (HLS) (see [4]) and which is valid for all n and 0< $\lambda$<n,\displaystyle \int\int g(x)|x-y|^{- $\lambda$}g(y)dxdy\leq C_{n}( $\lambda$)\Vert g\Vert_{2n/(2n- $\lambda$)}^{2}.
The special case $\lambda$=n-2 is the dual of Sobolev, but we see that HLS covers many morecases. We learn here that it is sometimes useful to study duals because they can lead us tonew mathematics. When n=3 , Sobolev tells us about kinetic energy, while its dual, HLS,
is the story of the Coulomb potential and ‘potential theory’, which has quite a different
flavor.

Nash’s inequality involves three integrals and is valid for all n.

C_{n}\Vert\nabla f\Vert_{2}^{n/(n+2)} \Vert f\Vert_{1}^{2/(n+2)} \geq \Vert f\Vert_{2}.
Carlen and Loss [2] found the sharp C_{n} and the optimizers, which always have compact
support.

For n \geq 3 , Nash’s inequality can be derived from Sobolev’s inequality (but with abad constant) by using Hölder’s incquality. Thus, Nash is weaker than Sobolev— but it
is extremely useful for problems in which the L^{1} ‐norm is either conserved or monotone
decreasing.

Kato was interested in the two‐dimensional Navier Stokes equation in the vorticity
formulation, which is just such a problem. Nash had applications to fluid dynamics inmind when he wrote his famous 1958 parabolic regularity paper in which his inequality firstappeared. Many applications have been found in probability theory.

3

Our dual Nash inequality

L_{n}\displaystyle \Vert g\Vert^{\frac{2n+4}{2n+4} \geq \inf_{h}\{\frac{1}{2}\Vert(- $\Delta$)^{-1/2}(g-h)\Vert_{2}^{2}+\Vert h\Vert_{\infty} \}What this says is, given a function g\in L^{2}(\mathbb{R}^{n}) , try to minimizc its Coulomb cncrgy
by subtracting another function h . The price to be paid, however, is the L^{\infty} ‐norm of h.
There are three topics to be discussed:(1.) Where docs this funny inequality come from and what is its connection to Nash?
(2.) Does there exist a minimizing h for this new problem and what does it look like?
(3.) Does there exist an optimizing g (and h) that gives the smallest value of L_{n} ?
How is this g related to the optimizer for Nash?

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4Generalities about dual inequalities

Suppose we have two convex functionals, A(f) , B(f) and A(f)-B(f)\geq 0, \forall f , as in
the Sobolev inequality. We can then take the Legendre transforms:

A^{*}(g) :=\displaystyle \sup_{f}\{\int fg-A(f)\}, B^{*}(g):=\sup_{f}\{\int fg-B(f)\}.
Let F be an (approximate) maximizer for B^{*}(g) , whence we have the dual inequality:

B^{*}(g)-A^{*}(g)\displaystyle \geq\int Fg-B(F)-\int Fg+A(F)\geq 0.
Thus, the dual of A\geq B is B^{*}\geq A^{*} . Since A, B are convex, the ‘dual of the dual’ isthe original inequality A\geq B. )In the case of Nash, there are 3 functionals and the right sidc is not convcx. Hclp!
We must combine 2 of thcm into one convex functional, and this will lead us to the strange

construction called infimal convolution. (see [6].)5Second law of thermodynamics and infimal convolution

Lct systems A and \mathrm{B} have energy dependent entropy functions S_{A}(E) and S_{B}(E) . These
functions are concave, of course. The systems are brought into equilibrium with to‐tal energy U . According to the second law they distribute the energy so that thetotal entropy ts maximized. Thus

S_{AB}(U)=\displaystyle \sup_{E}\{S_{A}(U-E)+S_{B}(E)\}.
The amazing thing is this: Despite the supremumE, the resulting S_{AB} is a concave

function — as requircd by thc second law. (For convex functions everything is reversed.)
The general theorem, (1 line proof!) of which this ‘convolution’ is a special case, isthis:

If F(X, Y) is a jointly concave function of X, Y then \displaystyle \sup_{Y}F(X, Y) is concave!
Let us apply this to thc product \Vert\nabla f\Vert_{2}^{n/(n+2)}\Vert f\Vert_{1}^{2/(n+2)} of functions of f , that appear
on the ‘large side’ of Nash. This product is NOT a convex functional. To deal with thisproblem we shall first reformulate Nash.
To convert the product into one convex function using infimal convolution, we mustfirst convert them into a sum of functions.

By using thc f‐scaling properties of the various norms, we can rewrite this inequalityasC_{n}^{(2n+4)/n}\Vert\nabla f\Vert_{2}^{2} + $\Phi$(f)\geq \Vert f\Vert_{2}^{(2n+4)/n},where$\Phi$(f)=\left\{begin{ar y}{l 0&\Vertf\Vert_{1}\leq1\ \infty&\Vertf\Vert_{1}> , \end{ar y}\right.

and whose Legendre transform is \Vert g\Vert_{\infty}.

The Legendre transform of of \Vert\nabla f\Vert_{2} is our beloved Coulomb potential \Vert(-\triangle)^{-1/2}g\Vert_{2}^{2}.

The fundamental theorem of convex analysis is: the Legendre transform of the sum oftwo convex functions is the infimal convolution of the two Legendre transforms.

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Conclusion: By taking the infimal convolution of these two convex functions, andscaling g , we get a dual of the Nash inequality (in which both sides are convex ing ):L_{n}\displaystyle \Vert g\Vert^{\frac{2n+4}{2n+4} \geq \inf_{h}\{\frac{1}{2}\Vert(-\triangle)^{-1/2}(g-h)\Vert_{2}^{2}+\Vert h\Vert_{\infty} \}Unfortunately, because of the ‘infh’, this is useless unless we can find h6

Facts about hThis is the fun part! We cannot compute h (except in one case), but we can say, more orless, what h looks like.

As a preliminary step we can try to minimize \Vert(-\triangle)^{-1/2}(g-h)\Vert_{2}^{2} under the condition
that \Vert h\Vert_{\infty} \leq c . Call this K(c) and, as a second, easy step, minimizc K(c)+c . So let us
discuss only the first step, with c fixed and |h(x)| \leq c, \forall x.

It is not hard to prove (everyone here can surely do it) that a unique minimizing hexists for K(c) . Let us then move on to the Euler‐Lagrange equation for h , which is$\psi$(x)\left{bgin{ary}l \geq0,\mathr{i}\mathr{f}(x)=c\ 0,\mathr{i}\mathr{f}-c<h(x)c\ leq0,\mathr{i}\mathr{f}(x)=-c,\end{ary}\ight.

with$\psi$=(-\triangle)^{-1}(g-h) .

0
An important fact about Laplacians (in the scnse of distributions) is that \triangle f
almost cvcrywhere on the set \{x: f(x)=0\} . Since \triangle $\psi$=h-g , we conclude that almost=everywhere
eitherh(x)=\pm c or else h(x)=g(x)and|g(x)| <cThis kind of argument goes back to the 2016 ‘no‐flat‐spots for strictly subharmonic

functions’ theorem of Frank & Lieb [3].
In case g\geq 0 one can also show that h\geq 0.

Another thing that one can easily prove is that \displaystyle \int h=\int g for any c>0 . (Otherwisethe Coulomb energy would be infinite.)
Unfortunately, we cannot find a formula for h except in one special, but important
case: The case in which g is a symmetric decreasing, non‐negative radial function. Trivial
proof!

h(x)=\{7]c\mathrm{i}\mathrm{f}|x| \leq R,g(x)\mathrm{i}\mathrm{f}|x| >R.

and volume of\displaystyle \{x: |x| <R\}=\frac{1}{c}\int g.

The sharp constantThc sharp constant C_{n} in Nash and L_{n} in dual Nash (\mathrm{d}\mathrm{N}) are trivially related, just as
are the sharp constants for Sobolev and HLS. Assume you have not read the Carlen‐Losspaper for C_{n} , and let us compute L_{n} directly. This will gives us an alternative proof of C_{n}.

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Let G be the maximizing g in \mathrm{d}\mathrm{N} . By Faber‐Krahn (i.e., rearrangement inequality forthc Laplacian) the optimizcrs for \mathrm{N} arc symmctric decreasing. By the 1:1 corrcspondence
between optimizers for \mathrm{N} and \mathrm{d}\mathrm{N} , we see that G also wants to be svmmetric decreasing.
In this case, we know the optimum H , as we just saw at the end of the previous slide.
Let us compute the Euler‐Lagrange equation for G. L_{n}G=(-\triangle)^{-1}(G-H)(Note: The variation w.r. \mathrm{t}. H=H_{G} vanishes sincc H is a minimizer for G ).

With $\phi$=G-H we havein a ball of radius R , and $\phi$ satisfies Dirichlet,
and also Neumann boundary conditions on the ball. This eigenvalue problem is exactly
what Carlen‐Loss found for Nash, and which they solved explicitly.

8The weighted version]

To conclude this story, let me briefly explain the word ‘weighted’ in thc titlc.
The sharp weighted Nash inequality for p>0 generalizes \mathrm{C}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{e}\mathrm{n}/\mathrm{L}\mathrm{o}\mathrm{s}\mathrm{s}/\mathrm{N}\mathrm{a}\mathrm{s}\mathrm{h} :\Vert f\Vert_{2}^{2+n/(4+2p)} \leq C_{n,p}\Vert\nabla f\Vert_{2}^{2} \Vert|x|^{p}f\Vert_{1}^{n/(4+2p)}.

Legendre transforming, as before, the equivalent dual weighted Nash inequality is:

L_{n,p}| g\displaystyle \Vert^{\frac{2n+4}{2n+4}} \geq \inf_{h}\{\frac{1}{2}\Vert(-\triangle)^{-1/2}(g-h)\Vert_{2}^{2}+\Vert|x|^{-p}h\Vert_{\infty} \}In contrast to the unweighted case, neither sharp constant was known.
When pis an even integer, however, the method we just described can be applied and yields a
new result: The Lsharpvalues
sharp values of \mathrm{C}_{nn} andnWeights different from |x|^{p} are possible, but only in this case can we easily find aformula for the sharp constants.

References[1] T. AUBIN, Problemes isopérimetriques et espaces de Sobolev. Journal of differential
geometry 11, No. 4 (1976), pp. 573‐598.
[2] E. A. CARLEN AND M. Loss Sharp constant in Nash’s inequality, Internat. Math.
Res. Notices, 1993, (1993) , pp. 213‐215[3] R. L. FRANK AND E. H. LIEB, A ‘liquid‐solid’ phase transition in a simple modelfor swarming, based on the {}^{t}no flat‐spots’ theorem for subharmonic functions, arXiv
preprint arXiv:1607.07971.

[4] E. H. LIEB, Sharp constants in the Hardy‐Littlewood‐Sobolev and related inequalities,Ann. Math. (2), 118 (1983), pp. 349‐374.

[5] J. F. NASH Continuity of solutions of parabolic and elliptic equations, Amer. J. Math.,
80, (1958), PP. 931‐954.
[6] ROCKAFELLAR, Convex Analysis, Princeton Univ. Press, Princeton, 1970

[7] G. TALENTI, Best constants in Sobolev Inequality. Ann. Mat. Pura Appl. 110 (1976),
pp. 353‐372.