Hardy–Sobolev Type Inequalities with Sharp Constants

Potential Anal

DOI10.1007/s11118-010-9190-0

Hardy–Sobolev Type Inequalities with Sharp Constants

in Carnot–Carathéodory Spaces

Donatella Danielli·Nicola Garofalo·

Nguyen Cong Phuc

Received:2November2008/Accepted:31May2010

©Springer Science+Business Media B.V.2010

Abstract We prove a generalization with sharp constants of a classical inequality due to Hardy to Carnot groups of arbitrary step,or more general Carnot–Carathéodory spaces associated with a system of vector fields of Hörmander type.Under a suitable additional assumption(see Eq.1.6below)we are able to extend such result to the nonlinear case p=2.We also obtain a sharp inequality of Hardy–Sobolev type.

Keywords Hardy type inequalities·Carnot groups·Carnot–Carathéodory spaces·Horizontal p-Laplacian

Mathematics Subject Classifications(2010)35H20·35H30

Donatella Danielli supported in part by NSF CAREER Award DMS-0239771

and by NSF Grant DMS-0801090.

Nicola Garofalo supported in part by NSF Grant DMS-0701001and by NSF Grant

DMS-1001317.

Nguyen Cong Phuc supported in part by NSF Grant DMS-0901083.

D.Danielli·N.Garofalo(B)

Department of Mathematics,Purdue University,West Lafayette,IN47907,USA

e-mail:garofalo@math.purdue.edu

D.Danielli

e-mail:danielli@math.purdue.edu

N.C.Phuc

Department of Mathematics,Louisiana State University,

303Lockett Hall,Baton Rouge,LA70803,USA

e-mail:pcnguyen@math.lsu.edu

D.Danielli et al. 1Introduction

In[22]the following Hardy type inequality was proved for the Heisenberg group H n

H n |φ|2

N2

|∇H N|2dg≤

2

Q−2

2

H n

|∇Hφ|2dg,φ∈C∞o(H n\{e}),(1.1)

where we have indicated with N=(|z|4+16t2)1/4the Koranyi–Folland non-isotropic gauge and with Q=2n+2the homogeneous dimension of H n.The constant in the right-hand side of Eq.1.1is optimal,see[29].Following ideas introduced in[23]for uniformly elliptic operators,the inequality1.1was used in[22] to establish some strong unique continuation properties for singular perturbations of the Kohn–Spencer sub-Laplacian on H n.The inequality1.1was extended to the nonlinear case p=2in[45].

On the other hand,when G is a Carnot group with homogeneous dimension Q, then Folland and Stein[20]proved the following basic result which constitutes a subelliptic Sobolev embedding theorem:let1<p<Q,and set p∗=pQ/(Q−p). There exists S p(G)>0such that

G |φ|p∗dg

1

p∗

≤S p(G)

G

|∇Hφ|p dg

1

p

,φ∈C∞0(G).(1.2)

In connection with Eq.1.2,we mention that Vassilev has shown in[49]that an

extremal function always exists,and thereby the sharp constant in Eq.1.2is attained.

Recently,J.Goldstein asked the second named author the question of whether

an inequality of Hardy type such as Eq.1.1be valid in the setting of Carnot groups.

Understanding such problem was the original motivation of this paper.In fact,we

will provide a general positive answer to Goldstein’s question in the more general

context of a system of smooth vector fields satisfying the finite rank condition rank

Lie[X1,...,X m]≡n,see the case p=2of Theorem1.2below.Furthermore,under the additional technical assumption Eq.1.6,Theorem1.2states that the result for

p=2can be extended when p=2to a nonlinear Hardy type inequality with sharp

constants.

We emphasize that,in our approach,the role of the hypothesis1.6is purely

instrumental.Its main function is that it allows us to apply the coarea formula in

the version established in[43].It is also worth stressing that Eq.1.6appears as a very

natural and plausible assumption which,as we remark in Proposition1.1below,for

general Carnot groups would follow from an equally plausible homogeneity property

of the fundamental solution.Furthermore,Eq.1.6represents the differentiated

version of the fundamental estimate obtained in[8],see Eq.1.4below.With all

this being said,in the nonlinear case p=2the hypothesis1.6is,to the best of our

knowledge,presently only known to be satisfied in groups of Heisenberg type.This

(almost trivially)follows from the explicit fundamental solutions discovered by two

of us and Capogna in[8],see Eq.1.23below(and also[31]for the case p=Q of such

result).Therefore,when p=2,the value of our Theorem1.2is presently confined to

providing an alternative proof of results already appeared in the papers[10,45]and

[28].

In fact,as far as we are aware of,that of groups of Heisenberg type is the most

general setting in which when p=2the Hardy inequality with sharp constants

Eq.1.25below is presently known.In this connection we should mention that in

Hardy–Sobolev Type Inequalities with Sharp Constants

[10]and [28]extensions of such nonlinear Hardy inequality have been stated for the class of the so-called polarizable groups .However,we note that there presently exist no known examples of polarizable groups,besides those of Heisenberg type.In our opinion our proof has the advantage of shifting the attention from special symmetry properties which can only be valid in groups of Heisenberg type (such as,for instance,the fact that the Folland–Kaplan gauge is a solution of the horizontal ∞-Laplacian),to working directly with the horizontal p -Laplacian.This is the spirit of the hypothesis 1.6.

As we have already mentioned,Theorem 1.2is new already for the linear case p =2.It is worth emphasizing here that,even in this linear setting,our approach allows to cover situations in which there is no underlying group or homogeneous structure,and which could not be dealt with by the existing techniques.This point will b …… 此处隐藏：32018字，全部文档内容请下载后查看。喜欢就下载吧 ……