Sagun CHANILLOJean VAN SCHAFTINGENPo-Lam YUNG

(To Häım Brezis in admiration and friendship)

1 Introduction

The Sobolev embedding theorem states that if˙W1,p(Rn)is the homogeneous Sobolev space,obtained by completing the set of compactly supported smooth functions Cc∞(Rn)under the norm?∇u?Lp(Rn),thenembeds into Lp∗(Rn),whenever n≥2,1≤p

with(Such a theorem would have been trivial by Hodge decomposition,if˙W1,n(Rn)were to embed into L∞(Rn).)The existing proofs of the above theorem are all long and complicated.On the contrary,a weaker version of this theorem,where one replaces the spacecan be obtained from the following theorem of Van Schaftingen[9],when?≤n−2.

Theorem 1.1(see[9])Suppose that f is a smooth vector field on Rn,with

Then for any compactly supported smooth vector field φ on Rn,we have

whereis the pointwise Euclidean inner product of two vector fields in Rn.

See e.g.[4,6].We refer the interested reader to the survey in[10],for a more detailed account of this circle of ideas.

The original direct proof of Theorem 1.1 in[9]proceeds by decomposing

and by estimating first directly the innermost(n−1)-dimensional integral.This gives the impression that the strategy is quite rigid.The first goal of this note is to prove Theorem 1.1 by averaging a suitable estimate over all unit spheres in Rn.

In a second part of this paper,we adapt this idea of averaging over families of sets to prove an analogue of Theorem 1.1,in the setting where Rnis replaced by the real hyperbolic space Hn.

Theorem 1.2Suppose that f is a smooth vector field on Hn,with

where divgis the divergence with respect to the metric g on Hn.Then for any compactly supported smooth vector field φ on Hn,we have

whereand dVgare the pointwise inner product and the volume measure with respect to g respectively,∇gφ is the(1,1)tensor given by the Levi-Civita connection of φ with respect to g,and

We note that the above theorem is formulated entirely geometrically on Hn,without the need of specifying a choice of coordinate chart.As explained in Appendix A,Theorem 1.2 can be proved indirectly by patching together known estimates on Rnvia a partition of unity,and by applying Hardy’s inequality to get rid of lower order terms.

We shall prove Theorem 1.2 by averaging a suitable estimate over a family of hypersurfaces in Hn,where the family of hypersurfaces is obtained from the orbit of a “vertical hyperplane”under all isometries in Hn.The latter shares a similar flavour to the proof we will give below of Theorem 1.1.The innovation in the proof of the result for the hyperbolic space is in deducing Theorem 1.2 from Proposition 3.1,and in establishing Lemma 3.4(see Section 3 for details).

2 Another Proof of Theorem 1.1

Theorem 1.1 will follow from the following proposition.

Proposition 2.1Let f,φ be as in Theorem 1.1.Write Bnfor the unit ball{x∈Rn:|x|<1}in Rn,and Sn−1for the unit sphere(i.e.,the boundary of Bn).Also write dσ for the standard surface measure on Sn−1,and νfor the outward unit normal to the sphere Sn−1.Then

Here?φ?W1,n(Sn−1)=?φ?Ln(Sn−1)+?∇Sn−1φ?Ln(Sn−1),where ∇Sn−1φis the(1,1)tensor on Sn−1given by the covariant derivative of the vector field φ.

The proof of Proposition 2.1 in turn depends on the following two lemmas.The first one is a simple lemma about integration by parts.

Lemma 2.1Let f,ν be as in Proposition 2.1.Then for any compactly supported smooth function ψ on Rn,we have

The second one is a decomposition lemma for functions on the sphere Sn−1.

Lemma 2.2Let ϕbe a smooth function on Sn−1.For any λ >0,there exists a decomposition

and an extensionof ϕ2to RnBn,such thatis smooth and bounded on RnBn,with

Here?∇Sn−1ϕ?is the norm of the gradient of the function ϕon Sn−1.We postpone the proofs of Lemmas 2.1–2.2 to the end of this section.

Now we are ready for the proof of Proposition 2.1.

Proof of Proposition 2.1Let f,φ be as in the statement of Theorem 1.1.Apply Lemma 2.2 to ϕ =?φ,ν?,where λ >0 is to be chosen.Then since

there exists a decomposition

and an extensionof ϕ2to RnBn,such that∈C∞∩L∞(RnBn),

and

Now

In the first term,we estimate trivially

To estimate the second term,we let θ be a smooth cut-o fffunction with compact support on Rn,such that θ(x)=1 whenever|x|≤ 1.For ε∈ (0,1),let θε(x)= θ(εx).Then θε=1 on Sn−1,so we can rewrite II as

for any ε∈(0,1).We then integrate by parts using Lemma 2.1,with ψ:=?ϕ2θε,and obtain

(The cut-o fffunction θεis inserted so that ψhas compact support.)We now let ε → 0+.The second term then tends to 0,since it is just

where f∈L1,∇θ(ε·)∈L∞and∈L∞on RnBn.On the other hand,the first term tends to

by dominated convergence theorem,since f∈L1and∈L∞on RnBn.As a result,

from which we see that

Together,by choosingwe get

as desired.

We will now deduce Theorem 1.1 from Proposition 2.1.The idea is to average(2.1)over all unit spheres in Rn.

Proof of Theorem 1.1First,for each fixed x∈Rn,we have

where we are identifying ω ∈ Sn−1with the corresponding unit tangent vector to Rnbased at the point x.Hence to estimate?f(x),φ(x)?dx,it suffices to estimate

which is the same as

by a change of variable(x,ω)?→ (z+ ω,ω).Now when z=0,the inner integral can be estimated by Proposition 2.1;for a general z?0,one can still estimate the inner integral by Proposition 2.1,since the proposition is invariant under translations.Thus the above double integral is bounded,in absolute value,by

Since

and

we proved that under the assumption of Theorem 1.1,we have

This is almost the desired conclusion,except that we have an additional zeroth order term on φ on the right-hand side of the estimate.But that can be scaled away by homogeneity.In fact,if f and φ satisfies the assumption of Theorem 1.1,then so does the dilations

Applying(2.3)to fεand φεinstead,we get

i.e.,

So,letting ε→ 0+,we get the desired conclusion of Theorem 1.1.

Proof of Lemma 2.1Note that?f,ν?ψ =?ψf,ν?,and νis the inward unit normal to∂(RnBn).So by the divergence theorem on Rn,we have

But since divf=0,we have

and the desired equality follows.

Proof of Lemma 2.2Suppose that ϕand λare as in Lemma 2.2.We will construct first a decomposition ϕ = ϕ1+ ϕ2on Sn−1,so that both ϕ1and ϕ2are smooth on Sn−1,and

(Here ∇Sn−1ϕ2is the gradient on Sn−1.)Once this is established,the lemma will follow,by extending ϕ2so that it is homogeneous of degree 0;in other words,we will then define

It is then straight forward to verify that

since the radial derivative of?ϕ2is zero.

To construct the desired decomposition on Sn−1,we proceed as follows.

If λ ≥ 1,we set ϕ2=fflSnϕ,so that∇Sn−1ϕ2=0 on Sn−1,then

and the estimate for ϕ1follows from the classical Morrey–Sobolev estimate.

If 0< λ <1,we pick a non-negative radial cut-o fffunction η ∈(Rn),with η=1 in a neighborhood of 0,and define ηλ(x)= η(λ−1x)for x ∈ Rn.We then consider the function

When restricted to x∈ Sn−1,this function is a constant independent of the choice of x∈ Sn−1,by rotation invariance of the integral.We then write cλfor this constant,i.e.,

Note that by our choice of η,when 0< λ <1,

Now we define,for x∈ Sn−1,that

Then for x∈ Sn−1,we have

by definition of cλ.But for x,y ∈ Sn−1,we have,by Morrey’s embedding,that

It follows that

Lettingwe see that the right-hand side above is just

But this last integral can be estimated by

by the support and L∞bound of.Hence using also(2.5),we see that

as desired.

Next,suppose that x ∈ Sn−1,and v is a unit tangent vector to Sn−1at x.Then

But if we differentiate both sides of the definition(2.4)of cλwith respect to ∂v,we see that

Multiplying(2.7)byand subtracting that from(2.6),we get

Using Morrey’s embedding again,we see that

Lettingwe see that the right-hand side above is just

But this last integral can be estimated by

by the support and L∞bound of ηλ.Hence using also(2.5),we see that

as desired.

3 A Borderline Sobolev Embedding on the Real Hyperbolic Space Hn

We now turn to a corresponding result on the real hyperbolic space Hn.We will first give a direct proof in this current section,in the spirit of the earlier proof of Theorem 1.1 by using spherical averages.In the appendix,we give a less direct proof,using a variant of Theorem 1.1 on Rn.

First we need some notations.We will use the upper half space model for the hyperbolic space.In other words,we take Hnto be

and the metric on Hnto be

We will use the following orthonormal frame of vector fields:

at every point of Hn.Note that if jn,then

(Here∇=∇gis the Levi-Civita connection with respect to the hyperbolic metric g.)In fact,since{e1,···,en}is an orthonormal basis,for any k=1,···,n,we have

Also,we have

This is because if j?n,then

by(3.1),and

To prove Theorem 1.2,note that we only need to consider the case n≥2,since when n=1,

and(1.2)follows trivially.Hence from now on we assume n≥2.

We will deduce Theorem 1.2 from the following proposition.

Proposition 3.1Assume n≥2.Let f,φ be as in Theorem 1.2.Write S for the copy of(n−1)-dimensional hyperbolic space inside Hn,given by

and X for the half-space

so that S is the boundary of X.Also writefor the volume measure on S with respect to the hyperbolic metric on S,and ν=e1for the unit normal to S.Then

Hereand all integrals on S on the right-hand side are with respect to

The S will be called a vertical hyperplane in Hn.It is a totally geodesic submanifold of Hn.We will consider all hyperbolic hyperplanes in Hn,that is the image of S under all isometries of Hn.The set of all such hypersurfaces in Hnwill be denoted by S;it will consist of all Euclidean parallel translates of S in the x?-direction,and all Euclidean northern hemispheres whose centers lie on the plane{xn=0}.

The proof of Proposition 3.1 in turn depends on the following two lemmas.The first one is a simple lemma about integration by parts,which is the counterpart of Lemma 2.1.

Lemma 3.1Assume n≥ 2.Let f,S,X,ν be as in Proposition 3.1.Then for any compactly supported smooth function ψ on Hn,we have

The second one is a decomposition lemma for functions on S,which is the counterpart of Lemma 2.2.

Lemma 3.2Assume n ≥ 2.Let ϕ be a smooth function with compact support on S.For any λ>0,there exists a decomposition

and an extensionof ϕ2to Hn,such thatis smooth with compact support on Hn,and

with

We postpone the proofs of Lemmas 3.1–3.2 to the end of this section.

Now we are ready for the proof of Proposition 3.1.

Proof of Proposition 3.1Let f,φ be as in the statement of Theorem 1.2.Apply Lemma 3.2 to ϕ =?φ,ν?g,where λ >0 is to be chosen.Then since

(this follows sincefor all k),there exists a decomposition

and an extensionof ϕ2to Hn,such that

and

Now

In the first term,we estimate trivially

In the second term,we first integrate by parts using Lemma 3.1,with ψ=?ϕ2,and obtain

so

Together,by choosingwe get

as desired.

We now deduce Theorem 1.2 from Proposition 3.1.The idea is to average(3.3)over all images of S under isometries in Hn(i.e.,all hypersurfaces in the collection S).

Proof of Theorem 1.2First,for each fixed x=(x?,xn)∈Hn,we have the following analogue of the identity(2.2),used in the proof of Theorem 1.1:

Here we are identifying ω ∈ Sn−1with the corresponding tangent vector to Hnbased at the point x.(Note then xnωhas length 1 with respect to the metric g at x,so xnωbelongs to the unit sphere bundle at x.)Furthermore,since the above integrand is even in ω,we may replace the integral over Sn−1by the integral only over the northern hemisphere:={ω ∈Sn−1:ωn>0}.Hence to estimate?f(x),φ(x)?gdVg,it suffices to estimate

We will compute this integral by making a suitable change of variables.

To do so,given x ∈and ω ∈,let S(x,ω)be the hyperbolic hypersurface in S passing through x with normal vector ω at x.In other words,S(x,ω)would be a Euclidean hemisphere,with center on the plane{xn=0};we denote the center of this Euclidean hemisphere by(z,0),where z=z(x,ω).

For each fixed x ∈,the map ω ?→ z(x,ω)provides an invertible change of variables fromto Rn−1.Thus we are led to parametrize the integral in(3.4)by z instead of ω.In order to do that,we observe that the vectors x − (z,0)and ωare collinear.This implies that if z=z(x,ω),then

(Here|x−(z,0)|is the Euclidean norm of x−(z,0).)Write Φx(z)for the right-hand side of the above equation.We view Φxas a map Φx:Rn−1→⊂ Rn,and compute the Jacobian of the map.We have

(Here we think of x,z as column vectors,and DΦxas an(n−1)×n matrix.)Thus

By computing the determinant in a basis that contains x?−z,we get

By a change of variable ω = Φx(z),and using Fubini’s theorem,we see that(3.4)is equal to

Now we fix z ∈ Rn−1,and compute the inner integral over x by integrating over successive hemispheres of radius r centered at(z,0).More precisely,let S(z,r)be the Euclidean northern hemisphere with center(z,0)and of radius r>0.Then S(z,r)∈ S,and for any z∈ Rn−1,we have

where dσ(x)is the Euclidean surface measure on S(z,r).However,if x ∈ S(z,r),then xnΦx(z)is precisely the upward unit normal to S(z,r)at x.Also,ifis the induced surface measure on S(z,r)from the hyperbolic metric on Hn,then

indeed if we writethen at x ∈ S(z,r),we have

(Here i denotes the interior product of a vector with a differential form.)Hence the integral(3.5)is just equal to

By Proposition 3.1 and its invariance under isometries of the hyperbolic space Hn,we have

Hence by H¨older’s inequality,(3.6)is bounded by

But undoing our earlier changes of variable,we see that

Similarly,

Altogether,(3.6)(and hence(3.4))is bounded by

This is almost what we want,except that on the right-hand side we have an additional?φ?Ln(Hn).To fix this,one applies Lemma 3.3 below,with p=n,and the desired conclusion of Theorem 1.2 follows.

Lemma 3.3Assume n ≥ 2.For any compactly supported smooth vector field φ on Hn,and any 1≤p<∞,we have

ProofIn fact,for any functionand any exponent 1≤p<∞,we have,from Hardy’s inequality,that

This is because

by Hardy’s inequality.(3.7)then follows by integrating over all x?∈ Rn−1with respect to dx?.Now we apply(3.7)to Φ =?φ,ej?,1 ≤ j≤ n −1.In view of(3.1),we have

so we get

Similarly,we can apply(3.7)to Φ =?φ,en?,and use(3.2)in place of(3.1).Altogether,we see that

as desired.

We now turn to the proofs of Lemmas 3.1–3.2.

Proof of Lemma 3.1Note that?f,ν?gψ =?ψf,ν?g,and νis the inward unit normal to∂X.Alsoagrees with the induced surface measure on S from Hn.So by the divergence theorem on Hn,we have

But since divgf=0,we have

and the desired equality follows.

The proof of Lemma 3.2 will be easy,once we establish the following lemma.

Lemma 3.4Let ϕ be a smooth function with compact support on Hm,m ≥ 1.For any p>m and λ>0,there exists a decomposition

such that ϕ2is smooth with compact support on Hm,and

with

We postpone the proof of this lemma to the end of this section.

Proof of Lemma 3.2Suppose that ϕand λare as in Lemma 3.2.We identify S with Hm,where m=n−1.(This is possible because the restriction of the metric of Hnto S induces a metric on S that is isometric to that of Hm.)Using Lemma 3.4,with p=n,we obtain a decomposition ϕ = ϕ1+ ϕ2on S,such that

with

We then extend ϕ2to Hnby setting for(x1,x??,xn)∈ R×Rn−2×R+,

One immediately checks thatis smooth with compact support on Hn,with

In view of(3.8),we obtain the desired bound for.

It remains to prove Lemma 3.4.

Proof of Lemma 3.4When m=1,the 1-dimensional hyperbolic space H1is isometric to R,and Lemma 3.4 follows from its counterpart on R(see,e.g.,[9]).Alternatively,it will follow from our treatment in the case m≥2,0<λ<1 below.

So assume from now on m ≥ 2.Suppose that ϕ is smooth with compact support on Hm,p>m and λ >0.We will construct our desired decomposition ϕ = ϕ1+ ϕ2.Recall that since p>m,the Morrey inequality on Hmimplies

To see this,let ζ∈(R)be a cut-o fffunction,such that ζ(s)=1 if|s|≤,and ζ(s)=0 if|s|≥ 1.Let x0:=(0,1)∈ Hm,and let ζx0(x):= ζ(d(x,x0)),where d is the hyperbolic distance on Hm.Consider the localization ζx0ϕof ϕ,to the unit ball centered at x0.It satisfies

where the left-hand side is a shorthand for

Here∇edenotes the Euclidean gradient of a function.(3.10)holds because by the support of ζx0,we have

and

where we have used Lemma 3.3 in the last inequalities(note that Lemma 3.3 applies now,since m ≥ 2).In particular,by Morrey’s inequality on Rm,from(3.10),we get

for all x,y∈Rm.Taking x=x0and y∈Hmsuch that d(y,x0)=2,we get

Since the isometry group of Hmacts transitively on Hm,and since the right-hand side of the above inequality is invariant under isometries,we obtain

for all x∈Hm,and hence(3.9).

In particular,in view of(3.9),when λ ≥ 1,it suffices to take ϕ1= ϕ and ϕ2=0.We then get the desired estimates for ϕ1and ϕ2trivially.

On the other hand,suppose now 0<λ<1.We fix a compactly supported smooth functionwith

For x=(x?,xm)∈Hm,we define

where we wrote v=(v?,vm)∈Rm−1×R,and define

Note that ϕ2is smooth with compact support on Hm,and hence so is ϕ1.Now for i=1,2,···,m − 1,we have

Since v?→ η(λ−1v)has support uniformly bounded with respect to 0< λ <1,we have e−vm≤ C on the support of the integral,where C is independent of 0< λ <1.Hence by H¨older’s inequality,we have

the last line following from the changes of variables zm=evm,and then z?=x?+zmv?.We thus see that

as desired.

Furthermore,when i=m,

Using that|vi| ≤ C on the support of the integrals(uniformly in 0< λ <1),and H¨older’s inequality as above,we see that

as desired.

Finally,to estimate ϕ1,note that

Hence

But

so we can plug this back in the equation for ϕ1,and integrate by parts in v.Now

and

Hence

When 0< λ <1,the integral in v in each term can now be estimated by H¨older’s inequality,yielding

This completes the proof of Lemma 3.4.

Appendix A Indirect Proof of Theorem 1.2

We will give an alternative proof of Theorem 1.2 from the following variant of Theorem 1.1,whose proof can be found,for instance,in[9](it can also be deduced by a small modification of the proof we gave above of Theorem 1.1).

Proposition A.1(see[9])Suppose that f is a smooth vector field on Rn(not necessarily divf=0).Then for any compactly supported smooth vector field φ on Rn,we have

where?·,·?is the pointwise Euclidean inner product of two vector fields in Rn.

To prove Theorem 1.2,we consider a function ζ∈(R)and we define for α ∈Hnthe function ζα:Hn→ R by

We assume that

Now given vector fields f and φ as in Theorem 1.2,we write

If α =(0,1)∈ Hn,then

whereis the Euclidean inner product of two vectors.Hence by Proposition A.1,this last integral is bounded by

where we write∇eto emphasize that the gradients are with respect to the Euclidean metric.Now on the support of ζα,we have|xn|1,so altogether,we get

This remains true even if α ?(0,1),since there is an isometry mapping αto(0,1),and since(A.2)is invariant under isometries of Hn.By integrating with respect to α∈Hn,we see that

We now use Lemma 3.3 to bound?φ?Ln(Hn)by?∇gφ?Ln(Hn).This concludes our alternative proof of Theorem 1.2.

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