Hardy inequalities on Riemannian manifolds and applications

Ann. I. H. Poincaré – AN 31 (2014) 449–475

www.elsevier.com/locate/anihpc

Hardy inequalities on Riemannian manifolds and applications

Lorenzo D’Ambrosio

, Serena Dipierro

aDipartimento di Matematica, via E. Orabona, 4, I-70125, Bari, Italy bSISSA, Sector of Mathematical Analysis, via Bonomea, 265, I-34136, Trieste, Italy Received 6 April 2012; received in revised form 25 March 2013; accepted 12 April 2013

Available online 23 May 2013

Abstract

We prove a simple sufficient criterion to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second order differential operator_pu:=div(|∇u|^p−2∇u). Namely, ifρis a nonnegative weight such that−_pρ0, then the Hardy inequality

c M

|u|^p

ρ^p |∇ρ|^pdvg M

|∇u|^pdvg, u∈C^∞₀ (M),

holds. We show concrete examples specializing the functionρ.

Our approach allows to obtain a characterization ofp-hyperbolic manifolds as well as other inequalities related to Caccioppoli inequalities, weighted Gagliardo–Nirenberg inequalities, uncertain principle and first order Caffarelli–Kohn–Nirenberg interpola- tion inequality.

©2013 Elsevier Masson SAS. All rights reserved.

MSC:58J05; 31C12; 26D10

Keywords:Hardy inequality; Riemannian manifolds; Parabolic manifolds; Caccioppoli inequality; Weighted Gagliardo–Nirenberg inequality;

Interpolation inequality

1. Introduction

AnN-dimensional generalization of the classical Hardy inequality asserts that for everyp >1 c

Ω

|u|^p w^p dx

Ω

|∇u|^pdx, u∈C^∞₀ (Ω),

whereΩ⊂R^Nis an open set, and the weightwis, for instance,w(x):= |x|andp < N, orw(x):=dist(x, ∂Ω)and Ωis convex (see for example[5,10,21,27,41–43,45]and references therein).

* Corresponding author.

E-mail addresses:dambros@dm.uniba.it(L. D’Ambrosio),serydipierro@yahoo.it(S. Dipierro).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2013.04.004

The preeminent role of Hardy inequalities and the knowledge of the best constants involved is a well known fact, as the reader can recognize from the wide literature that uses such a tool in Euclidean or in subelliptic setting as well as on manifolds ([4,6,9,10,12,22,37,46]just to cite a few).

On the other hand, the knowledge of the validity of a Hardy or Gagliardo–Nirenberg or Sobolev or Caffarelli–

Kohn–Nirenberg inequality on a manifoldM and their best constants allows to obtain qualitative properties on the manifoldM. For instance in[2,14,50]it was shown that ifMis a complete open Riemannian manifold with nonnega- tive Ricci curvature in which a Hardy- or Gagliardo–Nirenberg- or Caffarelli–Kohn–Nirenberg-type inequality holds, thenMis in some suitable sensecloseto the Euclidean space.

One of our aims is to prove some Hardy inequalities on Riemannian manifolds. In 1997, Carron in[16]studies weighted L²-Hardy inequalities on a Riemannian manifold M under some geometric assumptions on the weight functionρ, obtaining, among other results, the following inequality

c

M

u² ρ²dvg

M

|∇u|²dvg, u∈C^∞₀ (M), (1.1)

whereρ is a nonnegative function such that|∇ρ| =1,ρ^γ_ρ,ρ⁻¹{0}is a compact set of zero capacity and c= (^γ−₂¹)². In[16]the author applies this result to several explicit examples of Riemannian manifolds. Under the same hypotheses on the function ρ, Kombe and Özaydin in [38]extend Carron’s result to the casep=2 for functions in C^∞₀ (M\ρ⁻¹{0}), and the authors present an application to the punctured manifoldBⁿ\ {0}withBⁿthe Poincaré ball model of the hyperbolic space andρthe distance from the point 0 andp=2.

Li and Wang in[40]prove that ifMis a hyperbolic manifold (i.e. there exists a symmetric positive Green function Gx(·)for the Laplacian with pole atx), then

1 4

M

|∇yG_x(y)|²

G²_x(y) u²(y) dv_g

M

|∇u|²(y) dv_g, u∈C^∞₀

M\ {x} .

We also mention Miklyukov and Vuorinen, which in[44]prove that the inequality

M

α (x)

u(x)^qdvg

1/q

λ

M

β(x)∇u(x)^pdvg

1/p

, u∈W₀^1,p(M),

holds forqpprovided some conditions related to the isoperimetric profile ofMare satisfied.

In [1], Adimurthi and Sekar use the fundamental solution of a general second order elliptic operator to derive Hardy-type inequalities and then they extend their arguments to Riemannian manifolds using the fundamental solution ofp-Laplacian.

Bozhkov and Mitidieri in[7]prove the validity of(1.1)also forp=2 (1< p < N), provided there exists onMa C¹conformal Killing vector fieldKsuch that divK=μwithμa positive constant andρ= |K|.

Letp >1 and letρbe a nonnegative function. Our principal result is a simple criterion to establish if there holds a Hardy inequality involving the weightρ. Namely, ifρisp-superharmonic inΩ, that is−_pρ0, then

c

M

|u|^p

ρ^p |∇ρ|^pdv_g

M

|∇u|^pdv_g, u∈C^∞₀ (M), (1.2)

holds (see Theorem 2.1). Such a kind of criteria is already established in [20]for a quite general class of second order operators containing, among other examples, the subelliptic operators on Carnot groups. For this goal we shall mainly use a technique introduced by Mitidieri in[45]and developed in[18–20]and in[7,8]. The proof is based on the divergence theorem and on the careful choice of a vector field.

Let us point out some interesting outcomes of our approach. A first issue is that, since it is quite general, our approach includes Hardy inequalities already studied in[1,7,16,38,40]in the casep=2 as well as their generalization forp >1. Indeed, in all these cited papers, the authors assume extra conditions on the functionρor on the manifold.

Furthermore, in concrete cases, our result yields an explicit value of the constantc. Moreover, in several cases, this value is also the best constant (see[20]). To this regard, we discuss if the best constant is achieved or not and, in the latter case, we study the possibility to add a remainder term.

Another aspect of our technique is that it allows to characterize the p-hyperbolic manifolds. We remind that a manifoldMis calledp-hyperbolicif there exists a symmetric positive Green functionG_x(·)for thep-Laplacian with pole atx. We prove thatMisp-hyperbolic if and only if there exists a nonnegative nontrivial functionf ∈L¹_loc(M) such that

M

f|u|^pdv_g

M

|∇u|^pdv_g, u∈C^∞₀ (M).

Notice that one of the implications of this characterization forp=2 is the result proved in[40]. During the review process of this work, we have received the paper of Devyver, Fraas and Pinchover[23]. In[23]a general linear second order differential operatorP in the Euclidean framework is studied. The authors find a profound relation between the existence of positive supersolutions of P u=0, Hardy-type inequalities involving P and a weight W and the characterization of the spectrum of the weighted operator. We refer the interested reader to[23,24].

We also obtain a generalization of(1.2). Namely, for a nonnegative functionρ, the inequality |p−1−α|

p

p

M

|u|^p

ρ^p ρ^α|∇ρ|^pdv_g

M

|∇u|^pρ^αdv_g, u∈C^∞₀ (M), (1.3)

holds, provided−(p−1−α)_pρ0 (seeTheorem 3.1). The above inequality contains, as special case, the Cac- cioppoli inequality. Indeed, ifρis ap-subharmonic function, that is_pρ0, then(1.3)holds forα > p−1 and, in particular, forα=pwe have

1 p^p

M

|u|^p|∇ρ|^pdv_g

M

|∇u|^pρ^pdv_g, u∈C^∞₀ (M).

This is the so-called Caccioppoli inequality (see for instance[47]and the references therein for the versionp=2 on manifolds).

Another advantage of our approach is that it allows to obtain also other new and known results, like wighted Gagliardo–Nirenberg inequalities and the uncertain principle.

Finally we show that if(1.3)and a Sobolev-type inequality (that isc|u|L^p∗ |∇u|L^p) hold onM, then we obtain an interpolation inequality involving, as weights,ρand its gradient. As particular case, our results contain inequalities on manifolds related to the celebrated Caffarelli–Kohn–Nirenberg inequality.

The paper is organized as follows. We present the proof of (1.2)in Section 2, where important consequences and observations are derived. In Section3we show natural extensions of(1.2), obtaining also Hardy inequality with weights, Caccioppoli-type inequalities, weighted Gagliardo–Nirenberg inequalities and the uncertain principle. Some remarks on the best constant and if it is attained are discussed in Section 4. In Section5 we present a first order interpolation inequality. Finally Section6is devoted to present some concrete examples of Hardy-type inequalities on manifolds.

Notation. In what follows(M, g)is a complete RiemannianN-dimensional manifold,Ω⊂Mis an open set,dv_gis the volume form associated to the metricg,∇uand divhstand respectively for the gradient of a functionuand the divergence of a vector fieldhwith respect to the metricg(see[3]for further details). Throughout this paperp >1.

2. Hardy inequalities

In order to state Hardy inequalities involving a weight ρ, the basic assumption we made on ρ is that ρ is p-superharmonic in weak sense. Namely, we assume that ρ∈L¹_loc(Ω), |∇ρ| ∈L^p_loc⁻¹(Ω), and −_pρ0 on Ω in weak sense, that is for every nonnegativeϕ∈C¹₀(Ω), we have

Ω

|∇ρ|^p−2(∇ρ· ∇ϕ) dv_g0. (2.4)

The main result on Hardy inequalities is the following:

Theorem 2.1.Letρ∈W_loc^1,p(Ω)be a nonnegative function onΩ such that

−pρ0 onΩin weak sense.

Then ^|∇_ρ^ρp^|p ∈L¹_loc(Ω), and the following inequality holds:

p−1 p

p

Ω

|u|^p

ρ^p |∇ρ|^pdv_g

Ω

|∇u|^pdv_g, u∈C^∞₀ (Ω). (2.5)

Before provingTheorem 2.1, we shall present some immediate consequences and extensions of the main result.

Definition 2.2.LetΩ⊂Mbe an open set. We denote byD^1,p(Ω)the completion of C^∞₀ (Ω)with respect to the norm

|u|D^1,p =

Ω

|∇u|^pdvg

1/p

.

It is possible to extend the validity of(2.5)to functionu∈C^∞₀ (M). This extension is based on the inclusion

D^1,p(M)⊂D^1,p(Ω). (2.6)

The above inclusion is satisfied, for instance, whenM\Ωis a compact set of zerop-capacity (seeAppendix A).

Corollary 2.3.Letρ∈L¹_loc(M)be a function satisfying the assumptions ofTheorem2.1. If (2.6)holds, then ^|∇_ρ^ρp^|p∈ L¹_loc(M), and the following inequality holds:

p−1 p

p

M

|u|^p

ρ^p |∇ρ|^pdv_g

M

|∇u|^pdv_g, u∈C^∞₀ (M). (2.7)

Proof. The inequality (2.5)holds for every u∈C^∞₀ (Ω), then it holds for every u∈D^1,p(Ω). Since C^∞₀ (M)⊂ D^1,p(M), by using(2.6)we conclude the proof. 2

In order to illustrate further consequences ofTheorem 2.1we give the following:

Definition 2.4.A manifoldMis saidp-hyperbolic¹if there exists a symmetric positive Green functionG_x(·)for the p-Laplacian with pole atx,²if it is not the case we call itp-parabolic.

Several equivalent definitions ofp-parabolic manifolds can be given. For instance in[48]there is the following (see also the literature therein and[35])

Proposition 2.5.Letp >1. The following statements are equivalent:

a) Misp-parabolic;

b) there exists a compact setK⊂Mwith nonempty interior such that cap_p(K, M)=0;

c) there is no nonconstant positivep-superharmonic function onM;

d) there exists a sequence of functions u_j ∈C^∞₀ (M) such that0u_j 1, u_j →1 uniformly on every compact subset ofMand

M|∇u_j|^pdv_g→0.

1 Many authors call these manifolds non-p-parabolic.

2 That is−pGx=δxwhereδxis the Dirac measure concentrated at pointx.

Other characterizations of p-parabolic manifolds are based on several properties, for instance on the volume growth, on the isoperimetric profile of the manifold, on some properties of some cohomology, on the recurrence of the Brownian motion. See[28–31,39,48]and the references therein.

FromTheorem 2.1we deduce the following characterization ofp-hyperbolicity.

Theorem 2.6.A manifoldMisp-hyperbolic if and only if there exists a nonnegative nontrivial functionf ∈L¹_loc(M) such that

M

f|u|^pdv_g

M

|∇u|^pdv_g, u∈C^∞₀ (M). (2.8)

Proof. If M is p-hyperbolic then inequality (2.8) holds withf =(^p−_p¹)^p|∇Gx|p

G^px . Indeed G_x is nonnegative and satisfies the hypotheses ofTheorem 2.1(seeTheorem 6.4for further details).

Conversely, assume thatM isp-parabolic and that(2.8)is valid for a functionf 0. Then from d) ofProposi- tion 2.5there exists a sequence of functionsu_j∈C¹₀(M)such that 0u_j 1,u_j →1 uniformly on every compact subset ofMand

M

|∇u_j|^pdv_g→0 (asj→ +∞).

It implies that

Df dv_g=0 for every compact subsetDofMand thenf ≡0. This concludes the proof. 2 Remark 2.7.Since thep-hyperbolicity ofMis equivalent to the existence of a nonconstant positivep-superharmonic functionρonM, then byTheorem 2.1we obtain that inequality(2.8)holds withf =(^p−_p¹)^p|∇_ρ^ρp^|p.

Remark 2.8.OurTheorem 2.6implies that if the manifold M admits a C¹ conformal Killing vector fieldK (see i.e.[7]for the definition) such that divK=μ=0 withμconstant and|K|^−p∈L¹_loc(M), thenMisp-hyperbolic.

This follows combiningTheorem 2.6and Theorem 4 of[7](see alsoRemark 2.11ii) below).

In order to proveTheorem 2.1we fix some notation. Leth∈L¹_loc(Ω)be a vector field. We remind that the distri- bution divhis defined as

Ω

ϕdivh dv_g= −

Ω

(∇ϕ·h) dv_g, (2.9)

for everyϕ∈C¹₀(Ω).

Leth∈L¹_loc(Ω)be a vector field and letA∈L¹_loc(Ω)be a function. In what follows we writeAdivhmeaning that the inequality holds in distributional sense, that is for everyϕ∈C¹₀(Ω)such thatϕ0, we have

Ω

ϕA dvg

Ω

ϕdivh dvg= −

Ω

(∇ϕ·h) dvg. (2.10)

Remark 2.9.Letf∈C¹(R)be a real function such thatf (0)=0. Takingϕ=f (u)withu∈C¹₀(Ω)in(2.9), we have

Ω

f (u)divh dvg= −

Ω

f(u)(∇u·h) dvg.

In particular, choosingf (u)= |u|^p, withp >1, we get

Ω

|u|^pdivh dvg= −p

Ω

|u|^p−2u(∇u·h) dv_g, u∈C¹₀(Ω). (2.11)

Lemma 2.10.Leth∈L¹_loc(Ω)be a vector field and letA_h∈L¹_loc(Ω)be a nonnegative function such that i) Ahdivh,

ii) ^|h|p

A^p_h⁻¹ ∈L¹_loc(Ω).

Then for everyu∈C¹₀(Ω)we have

Ω

|u|^pA_hdv_gp^p

Ω

|h|^p

A^p_h⁻¹|∇u|^pdv_g. (2.12)

Proof. We note that the right hand side of(2.12)is finite sinceu∈C¹₀(Ω). Using the identity(2.11)and the Hölder inequality we obtain

Ω

|u|^pA_hdv_g

Ω

|u|^pdivh dv_g

p

Ω

|u|^p−1|h||∇u|dv_g

=p

Ω

|u|^p−1A^(p_h^−1)/p |h|

A^(p_h ^−1)/p|∇u|dv_g

p

Ω

|u|^pAhdvg

(p−1)/p

Ω

|h|^p

A^p_h⁻¹|∇u|^pdvg

1/p

.

This completes the proof. 2

Specializing the vector fieldhand the functionA_h, we shall deduce from(2.12)Hardy-type inequalities on Rie- mannian manifolds.

Remark 2.11.Letting us to point out a strategy to get Hardy inequalities at least in some special cases. Under the hypotheses ofLemma 2.10, ifA_h=divh, then(2.12)reads as

Ω

|u|^pdivh dv_gp^p

Ω

|h|^p

|divh|^p−1|∇u|^pdv_g. (2.13)

i) LetV be a function inL¹_loc(Ω)such that its weak partial derivatives of order up to two are inL¹_loc(Ω). IfV 0, choosingh= ∇V, we obtainA_h=divh=div(∇V )=V 0. Then from(2.13)

Ω

|u|^p|V|dv_gp^p

Ω

|∇V|^p

|V|^p−1|∇u|^pdv_g. (2.14)

This kind of inequalities for the Euclidean settingΩ=R^Nare already found by Davies and Hinz in[21].

At this point, in order to deduce from(2.14)an inequality like c

Ω

|u|^p ρ^p dv_g

Ω

|∇u|^pdv_g, (2.15)

we have to choose a suitable functionV. Let us consider the case whenρis the distance from a pointo∈Mand

|∇ρ| =1. A suitable choice forV isV =ρ^2−pif 1< p <2,V =lnρifp=2 andV = −ρ^2−p if 2< p < N. Following[16], if we require thatρ^γ_ρ the above choices yield the inequality(2.15)withc=(^γ−_p^p+1)^p. The success of this strategy is deeply linked to the hypothesis|∇ρ| =1. Indeed, it seems that such a strategy does not work even in the subelliptic setting, where the analogous of the hypothesis|∇ρ| =1 does not hold.

Furthermore, the fact that the hypothesis |∇ρ| =1 is sometimes restrictive even in the Euclidean case, can be seen in the following example. In the Euclidean unit ballB₁⊂R^Nthe inequality

c

B1

|u|^p

||x|ln|x||^pdx

B1

|∇u|^pdx (2.16)

holds for 1< pN(see Section6.3, Section6.6and[20]). If we wish to deduce(2.16)from(2.15)we are forced to chooseρ= −|x|ln|x|. However|∇ρ| =1.

ii) Letp < N. Assume that there exists a C¹conformal Killing vector fieldK(see i.e.[7]for the definition) such that divK=^N₂μ >0. Choosingh=_|K^K_|p, we haveAh:=divh=^N−₂^p_|K^μ_|p (see Lemma 3 in[7]) and the inequality (2.13)reads as

N−p Np

p

Ω

divK

|K|^p|u|^pdv_g

Ω

(divK)^1−p|∇u|^pdv_g. (2.17)

Therefore, by Lemma 2.10,(2.17) holds for every u∈C¹₀(Ω) provided|K|^−p∈L¹_loc(Ω). This last fact was obtained in[7, Theorem 4].

Proof of Theorem 2.1. Let 0< δ <1, andρ_δ:=ρ+δ. In order to applyLemma 2.10, we definehandA_has h:= −|∇ρδ|^p−2∇ρδ

ρ_δ^p−1 and A_h:=(p−1)|∇ρδ|^p

ρ_δ^p . (2.18)

Since _ρ¹

δ ¹_δ, the fact thatρ∈W_loc^1,p(Ω)implies thath∈L¹_loc(Ω)andAh∈L¹_loc(Ω). Moreover, by computation we have

|h|^p

A^p_h⁻¹=|∇ρδ|^p(p−1) ρ_δ^p(p−1)

ρδp(p−1)

|∇ρ_δ|^p(p−1) 1

(p−1)^(p−1) = 1

(p−1)^p−1∈L¹_loc(Ω), that is ii) ofLemma 2.10is fulfilled.

The hypothesis i) ofLemma 2.10is satisfied provided (p−1)

Ω

|∇ρ_δ|^p

ρ_δ^p ϕ dv_g

Ω

|∇ρ_δ|^p−2∇ρ_δ ρ_δ^p−1 · ∇ϕ

dv_g (2.19)

holds for every nonnegative functionϕ∈C¹₀(Ω). Then, for a fixedϕ∈C¹₀(Ω)nonnegative, we have to prove(2.19).

LetK=suppϕ⊂Ω and letU be a neighborhood ofKwith compact closure inΩ. We note that both integrals in (2.19)are finite since _ρ¹

δ ¹_δ andρ∈W_loc^1,p(Ω). Since

|∇lnρδ| =|∇ρ_δ| ρ_δ |∇ρ|

δ ∈L^p_loc(Ω), (2.20)

and lnρ_δ ∈L^p_loc(Ω), we have that lnρ_δ ∈W^1,p(U ). Thus, for every n∈N there exists φ_n∈C^∞(U ) such that

|φn−lnρδ|W^1,p <1/n,φn→lnρδpointwise a.e. and lnδφn.³

3 Reminding that the Sobolev spaceW^1,p(Ω)is the completion of the set u∈C^∞(Ω):

Ω

|u|^pdvg<∞and Ω

|∇u|^pdvg<∞

with respect to the norm

|u|_W1,p= Ω

|u|^pdvg+ Ω

|∇u|^pdvg 1/p

,

the approximation result follows by slight modification of classical arguments that the reader can find, for instance, in[3].

Settingψ_n=e^φn we have thatψ_n∈C^∞(U ),δψ_n,ψ_n→ρ_δa.e. and

K

|lnψ_n−lnρ_δ|^pdv_g→0,

K

∇ψn

ψn −∇ρδ

ρδ

^pdv_g→0 (asn→ +∞). (2.21)

For everyn∈N, the functionϕ_n defined asϕ_n:= ^ϕ

ψ_n^p−1 belongs to C¹₀(Ω)and it is nonnegative sinceϕ∈C¹₀(Ω) is nonnegative andψ_n>0. Usingϕ_nas a test function in(2.4)we have

0

Ω

|∇ρ|^p−2(∇ρ· ∇ϕ_n) dv_g=

Ω

|∇ρ|^p−2

∇ρ· ∇ ϕ

ψ_n^p−1

dv_g, (2.22)

which, since by computation∇( ^ϕ

ψ_n^p−1)= ^∇ϕ

ψ_n^p−1 −(p−1)^∇_ψ^ψpⁿ

n ϕ, implies (p−1)

Ω

|∇ρ|^p−2∇ρ· ∇ψ_n

ψ_n^p ϕ dv_g

Ω

|∇ρ|^p−2∇ρ ψ_n^p−1 · ∇ϕ

dv_g. (2.23)

Now, lettingn→ +∞we obtain by dominated convergence:

Ω

|∇ρ|^p−2∇ρ ψ_n^p−1 · ∇ϕ

dvg→

Ω

|∇ρ|^p−2∇ρ ρ^p_δ⁻¹ · ∇ϕ

dvg,

because|^{|∇ρ|p−2∇ρ·∇ϕ}

ψ_n^p−1 |^C|∇_δp^ρ−^|p−1 ¹ ∈L¹(U ). Now we claim that

Ω

|∇ρ|^p−2∇ρ· ∇ψ_n

ψ_n^p ϕ dvg=

Ω

|∇ρ|^p−2∇ρ ψ_n^p−1

∇ψ_n

ψ_n ϕ dvg→

Ω

|∇ρ|^p ρ_δ^p ϕ dvg. Indeed,

|∇ρ|^p−2∇ρ ψn^p−1

→ |∇ρ|^p−2∇ρ ρ_δ^p−1

pointwise a.e.

and, since ^{|∇ρ|p−2∇ρ}

ψ_n^{p−1 |∇}_δ^ρp^|−^p−11 ∈L^p(U ), by Lebesgue dominated convergence theorem we have that

|∇ρ|^p−2∇ρ

ψ_n^p−1 → |∇ρ|^p−2∇ρ

ρ_δ^p−1 inL^p(U ).

From this and the fact that

∇ψ_n ψ_n →∇ρ

ρ_δ inL^p(U )

we get the claim. Therefore, lettingn→ +∞in(2.23), we have (p−1)

Ω

|∇ρ|^p

ρ_δ^p ϕ dv_g

Ω

|∇ρ|^p−2∇ρ ρ_δ^p−1 · ∇ϕ

dv_g,

which is exactly(2.19), since∇ρ_δ= ∇ρ. An application ofLemma 2.10gives

p−1 p

p

Ω

|u|^p

ρ_δ^p |∇ρδ|^pdvg

Ω

|∇u|^pdvg. (2.24)

Finally, lettingδ→0 in(2.24)and using Fatou’s Lemma, we conclude the proof. 2

3. Further inequalities

In this section we shall present some slight but natural extensions ofTheorem 2.1andLemma 2.10. As byproducts of these generalizations we shall obtain Hardy inequalities with a weight in the right hand side, Caccioppoli-type inequalities, weighted Gagliardo–Nirenberg inequalities and the uncertain principle.

A first example of a possible generalization ofTheorem 2.1is the following:

Theorem 3.1.Letα∈R, and letρ∈W_loc^1,p(Ω)be a nonnegative function satisfying the following properties:

i) −(p−1−α)pρ0onΩin weak sense, ii) ^|∇_ρp^ρ−^|α^p, ρ^α∈L¹_loc(Ω).

Then the following Hardy inequality holds |p−1−α|

p

p

Ω

ρ^α|u|^p

ρ^p |∇ρ|^pdvg

Ω

ρ^α|∇u|^pdvg, u∈C^∞₀ (Ω). (3.25)

The proof of the above theorem is similar to the one ofTheorem 2.1and it is based on a careful choice of the vector fieldhand of the functionA_hinLemma 2.10.

Proof of Theorem 3.1. Let 0< δ <1, andρδ:=ρ+δ. In order to applyLemma 2.10we choose the vector fieldh and the functionA_has

h:= −(p−1−α)|∇ρ_δ|^p−2∇ρ_δ ρ_δ^p−1−α

, A_h:=(p−1−α)²|∇ρ_δ|^p

ρ_δ^p−α . (3.26)

Arguing as in the proof ofTheorem 2.1, we have to show that (p−1−α)²

Ω

|∇ρ_δ|^p ρ_δ^p−α

ϕ dv_g(p−1−α)

Ω

|∇ρ_δ|^p−2∇ρ_δ ρ_δ^p−1−α · ∇ϕ

dv_g, (3.27)

for every nonnegative functionϕ∈C¹₀(Ω). LetK:=suppϕ ⊂Ω and letU ⊂⊂Ω be a neighborhood ofK. Let k > δ, and defineρkδ:=inf{ρδ, k}. Arguing as in the proof ofTheorem 2.1, we have that there exists a sequence{ψn} such thatδψnk, and

K

|lnψ_n−lnρkδ|^pdv_g→0,

K

∇ψ_n

ψ_n −∇ρ_kδ ρ_kδ

^pdv_g→0 (asn→ +∞). (3.28)

Then we useϕ_n:= ^ϕ

ψn^p−1−α

as a test function in the hypothesis i), obtaining (p−1−α)²

Ω

|∇ρ|^p−2∇ρ· ∇ψn

ψ_n^p−α ϕ dv_g(p−1−α)

Ω

|∇ρ|^p−2∇ρ ψ_n^p−1−α · ∇ϕ

dv_g. (3.29)

In the caseα < p−1 we obtain(3.27)from(3.29)by slight modifications of the proof ofTheorem 2.1, so we will omit the proof.

Letα > p−1. We claim that, lettingn→ +∞in(3.29), and eventually taking a subsequence, we get (p−1−α)²

Ω

|∇ρ|^p−2∇ρ· ∇ρ_kδ

ρ_kδ^p−α ϕ dvg(p−1−α)

Ω

|∇ρ|^p−2∇ρ ρ^p_kδ^−1−α · ∇ϕ

dvg. (3.30)

In fact, for the right hand side the limit follows by dominated convergence, since |∇ρ|^p−2∇ρ

ψ_n^p−1−α · ∇ϕ

= |∇ρ|^p−2|∇ρ· ∇ϕ|ψ_n^α−p+1C|∇ρ|^p−1k^α−p+1∈L¹(U ).

Dealing with the left hand side of(3.29), we set

|∇ρ|^p−2∇ρ· ∇ψ_n

ψ_n^p−α =|∇ρ|^p−2∇ρ ψn^p−1

·∇ψ_n

ψ_n , ψ_n^α=:f_n·g_n. As in the proof ofTheorem 2.1, we have

f_n→ |∇ρ|^p−2∇ρ ρ_kδ^p−1

inL^p(U ), (3.31)

while from the relations

|gn| ∇ψ_n

ψ_n

·k^α→ ∇ρ_kδ

ρ_kδ

·k^α inL^p(U )

we obtain that the sequence g_n is bounded inL^p(U ). Therefore, up to a subsequence,g_n is weakly convergent in L^p(U ). Since

g_n→∇ρ_kδ

ρ_kδ ·ρ_kδ^α pointwise a.e.,

we have that the convergence is in the weak sense. This fact with(3.31)concludes the claim.

Next step is lettingk→ +∞in(3.30). Let us rewrite the integrand in the right hand side as |∇ρ|^p−2∇ρ

ρ^p_kδ^−1−α · ∇ϕ

=|∇ρ|^p−2∇ρ(ρkδ)

α−p

p (ρkδ)^{α−pp +1}· ∇ϕC|∇ρ|^p−1(ρδ)

α−p p (ρδ)^αp,

which is in L¹(U ), since C|∇ρ|^p−1(ρ_δ)

α−p

p ∈L^p(U ) and(ρ_δ)^αp ∈L^p(U ) by hypothesis ii). Thus we can use the dominated convergence to obtain the limit for the right hand side of(3.30).

In order to pass to the limit fork→ +∞in the left hand side of(3.30), we rewrite the integrand as

|∇ρ|^p−2∇ρ· ∇ρ_kδ

ρ_kδ^p−α ϕ=|∇ρ|^p−2∇ρ· ∇ρ_δ

ρ_kδ^p−α χ_{ρδk}ϕ=|∇ρ|^p

ρ_kδ^p−α χ_{ρδk}ϕ, (3.32)

where we have used the fact that∇ρ_δ= ∇ρ. Now, ifαpwe apply the dominated convergence, since the term in (3.32)is dominated by the functionC^|∇_δp^ρ−^|α^p ∈L¹(U ). Whereas, ifα > p we use the monotone convergence, since

|∇ρ|^pρ_kδ^α−pχ_{|ρδ|k}ϕis an increasing sequence of nonnegative functions.

Thus, lettingk→ +∞in(3.30), we get (p−1−α)²

Ω

|∇ρ|^p ρ_δ^p−α

ϕ dv_g(p−1−α)

Ω

|∇ρ|^p−2∇ρ ρ_δ^p−1−α · ∇ϕ

dv_g, (3.33)

which is exactly(3.27), since∇ρ_δ= ∇ρ.

As inTheorem 2.1, an application ofLemma 2.10gives α−p+1

p

p

Ω

ρ_δ^α|u|^p

ρ_δ^p |∇ρ_δ|^pdv_g

Ω

ρ_δ^α|∇u|^pdv_g. (3.34)

Finally, lettingδ→0 in(3.34), we conclude the proof. Indeed we can use the dominated convergence for the right hand side, sinceρ_δ^α|∇u|^pC(ρ+δ)^αC(ρ+1)^α∈L¹_loc(U ), and apply Fatou’s Lemma for the left hand side. 2 Remark 3.2.Ifαp, then the hypothesis ii) in the aboveTheorem 3.1, can be avoided. Indeed, sinceρ∈W_loc^1,p(Ω) we get thatρ^α∈L¹_loc(Ω)and from the proof it follows that^|∇_ρp^ρ−^|α^p ∈L¹_loc(Ω).

As a consequence ofTheorem 3.1(it suffices to take α=p+q), we obtain the following Caccioppoli-type in- equality forp-subharmonic functions, which is worth of mention: