Improved hardy inequalities on Riemannian manifolds

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Improved hardy inequalities on Riemannian manifolds © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group Published version Mohanta, Kaushik; Tyagi, Jagmohan Mohanta, K., & Tyagi, J. (2023). Improved hardy inequalities on Riemannian manifolds. Complex Variables and Elliptic Equations, Early online. https://doi.org/10.1080/17476933.2023.2247998 2023 Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=gcov20 Complex Variables and Elliptic Equations An International Journal ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/gcov20 Improved hardy inequalities on Riemannian manifolds Kaushik Mohanta & Jagmohan Tyagi To cite this article: Kaushik Mohanta & Jagmohan Tyagi (2023): Improved hardy inequalities on Riemannian manifolds, Complex Variables and Elliptic Equations, DOI: 10.1080/17476933.2023.2247998 To link to this article: https://doi.org/10.1080/17476933.2023.2247998 © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. Published online: 21 Aug 2023. Submit your article to this journal View related articles View Crossmark data COMPLEX VARIABLES AND ELLIPTIC EQUATIONS https://doi.org/10.1080/17476933.2023.2247998 Improved hardy inequalities on Riemannian manifolds Kaushik Mohantaaand Jagmohan Tyagib aDepartment of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland; bDiscipline of Mathematics, Indian Institute of Technology Gandhinagar, Gandhinagar, Gujarat, India ABSTRACT We study the following version of Hardy-type inequality on a domain in a Riemannian manifold (M,g):  |∇u|p gραdVg≥|p−1+β| pp |u|p|∇ρ|p g |ρ|pραdVg + V|u|pραdVg,∀u∈C∞ c(). We provide sufficient conditions on p,α,β,ρand Vfor which the above inequality holds. This generalizes earlier well-known works on Hardy inequalities on Riemannian manifolds. The functional setup covers a wide variety of particular cases, which are discussed briefly: for example, RNwith p<N,RN\{0}with p≥N,HN, etc. ARTICLE HISTORY Received 2 May 2023 Accepted 11 August 2023 COMMUNICATED BY H. Boas KEYWORDS Hardy inequality; manifold; reminder term; critical case AMS SUBJECT CLASSIFICATIONS 58J05; 35A23; 46E35 1. Introduction The Hardy inequality plays an important role in analysis and in the theory of partial differential equations. In RN, it reads as follows: RN |u|p |x|p≤cRN |∇u|p,∀u∈C∞ c(RN),(1) where the constant cis independent of u.Whenwerestrictuto be in the space W1,p 0(RN), with p<N,orinW1,p 0(RN\{0}),withp≥N, (1) holds; in that case the smallest possible choice for cbecomes p N−pp, and this constant is never achieved (see for example [1,2]). Hence there is a scope to get an improvement in the above inequality. Brezis-Vázquez [2] have shown for the case p=2, that we can add an L2-term in the left hand side of (1) even with the best constant. This was further generalized in many directions, for example, see [1,3–13]. There are many results in literature on this subject in the context of a complete Riemannian manifolds (M,g). An important result in this direction is due to Kombe-O¨ zaydin [14]. They proved that for a nonnegative function ρwith |∇ρ|g=1andρ ≥C ρ,thefollowing CONTACT Kaushik Mohanta kaushik[email protected] Kaushik Mohanta and Jagmohan Tyagi have contributed equally in this article. © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. 2K. MOHANTA AND J. TYAGI holds C+1+α−p ppM |u|pρα ρp≤M |∇u|p gραfor any u∈C∞ c(M). Moreover, for a bounded domain with smooth boundary and in the case p=2, they also provedthat the aboveinequalitystillholdsforanyuinC∞ c(), if we add a reminder term of the form C1(|∇u|qρqα/2)2/qin the left hand side of the equation, where 1<q<2and C=C(N,q,). D’Ambrosio-Dipierro [15] proved another version of Hardy inequality, where the restriction |∇ρ|=1 is not there. They showed that if is an open set in Mand there is a ρ:→[0, ∞)such that ρ∈W1,p loc () with pρ≤0 weakly, then |∇ρ| ρ∈Lp loc, and for any u∈C∞ c(),thefollowingholds: p−1 pp |u|p|∇ρ|p g ρp≤ |∇u|p g. They discussed, in details, the advantage and applications of this kind of Hardy inequality. For other related results, the reader may refer to [16–21] and the references therein. As of now, there is no known result regarding Hardy inequality with a reminder term as prescribed by D’Ambrosio-Dipierro [15].Wewishtoaddressthisprobleminthispaperby proving a slightly more general version of the result. An interesting feature of our method is that the results of [14,15]followimmediately. Also the proof becomes much simpler (see Theorem 1.1) provided no reminder term is expected. The formulation allows greater control over the constant, and, at least in the Euclidean case, we can get the inequality with the best constants. Here we mainly focus on the improved Hardy inequality in all its generality. We refrain ourselves from studying the special cases exclusively, although that is very much possible to use our results to achieve improvement of the inequalities obtained in [15]. Throughout the article, (M,g)stands for a fixed oriented Riemannian manifold. We shall often use the notation X,Yto denote g(X,Y)foranytwovectorfieldsXand Y.We use the symbol dVgtodenotethevolumeform,however,thesymbolwillbeoftendropped when there is no scope for confusion. Now,westatethemainresultsofthispaper,whichweshallproveinSection2.Thefirst one is the following Hardy inequality, which, in the particular case α=−β,isprovenin [15]. However, our proof is much more simpler. Theorem 1.1: Let 1<p<∞,α,β∈R, M be a complete Riemannian manifold, be a domain in M with boundary ∂(possiblyempty).Letthereexistafunctionρ:→(0, ∞) with |∇ρ|gρ −p+α p∈Lp loc() such that 1 p−1+β∇ξ,|∇ρ|p−2∇ρρ−p+1+α≥ |∇ρ|p ρp−αξ,∀ξ∈C1 c() with ξ≥0. (2) Then, for any u ∈W1,p 0(),wehavethefollowinginequality:  |∇u|p gραdVg≥|p−1+β| pp |u|p|∇ρ|p g ρpραdVg. COMPLEX VARIABLES AND ELLIPTIC EQUATIONS 3 Theorem 1.2: Let p,α,β,M,and ρbe as in Theorem 1.1.LettheconstantC=C(p)be as in Lemma 2.2 for p ≥2and in the case p <2,C(p):=2p−3p(p−1).Further,inthecase p<2, assume that for any ξ∈C1 c() mboxwith ξ≥0 ρ −p+1−β p∈W1,p loc (), (p−1+β)∇ξ,|∇ρ|p−2∇ρρα+β≥ |∇ρ|p ρρα+βξ.(3) If there exist functions V :→[0, ∞)and ϕ∈W1,p loc () such that for any ξ∈C1 c(), |∇ϕ|p−2∇ϕ,∇ξϕ−p+1ρα−(p−1) |∇ϕ|pξραϕ−p −(p−1+β)|∇ϕ|p−2∇ϕ,∇ρξϕ−p+1ρ−1+α ≥ V C(p)ξρα,(4) then for any u ∈W1,p 0(),wehavethefollowinginequality:  |∇u|p gραdVg≥|p−1+β| pp |u|p|∇ρ|p g |ρ|pραdVg+ V|u|pραdVg.(5) Before moving further, let us observe that the hypotheses (2), (3), (4) can be thought of as weak formulations of certain problems; we explain this in the following using the Green’s theorem (see Lemma 2.1). Consider the following condition on ρ: −1 p−1+βpρ≥α+β p−1+β |∇ρ|p ρ.(6) In the special case α=−β,thisisjustthep-superharmonicity condition. We rewrite this as −1 p−1+βρ−p+1+αpρ+p−1−α p−1+βρ−p+α|∇ρ|p≥|∇ρ|p ρp−α, and then multiply both sides by a test function ξ∈C1 c(), integrate over ,andthenapply the product rule of divergence operator to get −1 p−1+β div ρ−p+1+α|∇ρ|p−2∇ρξ≥ |∇ρ|p ρp−αξ. An application of Green’s theorem then gives (2). Thus (2) is an weak formulation of (6). Similarly, the second condition of (3) can be interpreted as the weak formulation of −1 p−1+βpρ≥p−1+α+2β p−1+β |∇ρ|p ρ, and (4) can be regarded as a weak formulation of pϕ+(p−1+α+β)|∇ϕ|p−2∇ϕ,∇ρ ρ+V C(p)ϕp−1≤0. 4K. MOHANTA AND J. TYAGI Now, we discuss some immediate consequences of Theorem 1.2. The following results says that if |p−1+β| ppis the best constant in (5), then (4) has no solution when V=|∇ρ|p ρp. Corollary 1.3: Let p,α,β,ρbe as in Theorem 1.2,andV=C|∇ρ| ρbe such that there exists some ϕfor which (4) is satisfied. Then the constant |p−1+β| ppin (5) is not sharp. Remark 1.4: The above result says nothing about when the constant |p−1+β| ppin (5) is sharp. However it is easy to see that the constant is sharp if and only if we can never have V=C∇ρ ρin (5) for any C>0. So the question of whether the constant is sharp is related to necessity of (4) in (5); whereas result Theorem 1.2 concerns with the sufficiency part. In some very particular case the condition may also be necessary as can be seen from [22, Theorem 1]. We discuss some special cases of Theorem 1.2. In the Euclidean setup, we get the following results as a corollary. Let us take M=RN,ρ=|x|d.Then∇ρ=d|x|d−2xand we get the following corollary: Corollary 1.5: Assume 1<p<∞,d= 0,α,β∈Rbe such that p −1+β= 0, α+β p−1+β≤0and (N+(d−1)(p−1)) d(p−1+β) ≤0, and (N+(d−1)(p−1)) d(p−1+β) ≤−1when p <2. Assume further that there exists a function V :RN→Rsuch that the following problem admits a weak solution ϕ∈W1,p loc (): pϕ+d(p−1+α+β) |x|2|∇ϕ|p−2∇ϕ·x+V C(p)ϕp−1≤0. (7) Consider the two cases: (i) ⊆RN,p<N+αdwith (p−1+β)d<N−pwhenp<2, and (ii) ⊆RN\{0},p≥N+αd. Then,inbothcases,wehavethefollowinginequality:  |∇u|p g|x|αddx ≥|d(p−1+β)| pp |u|p |x|p−αddx+ V|u|p|x|αd,∀u∈W1,p 0(). COMPLEX VARIABLES AND ELLIPTIC EQUATIONS 5 Remark that in the case α=β=0, d=−1and=RN,[23, Proposition 1.2] implies that (7) has no solution in RNfor any 1 <p<N. Now, let us consider the so-called critical case: p=N≥2. Set :=B1(0)⊂RN,ρ= −log |x|.Wehave Corollary 1.6: Let N ≥2,B 1(0)denote the unit ball in RN,α,β∈Rbe such that p −1+ β= 0and α+β N−1+β≤0. Assume further that there exists a function V :B1(0)⊂RN→Rsuch that the following problem admits a weak solution ϕ∈W1,p loc (B1(0)) : Nϕ−(N−1+α+β)|∇ϕ|N−2 |x|2log x∇ϕ·x+V C(N)≤0. Then for any u ∈W1,N 0(B1(0)),wehave B1(0) |∇u|N|log |x||αdx≥|N−1+β| NNB1(0) |u|N |x|N|log |x||N−αdx +B1(0) V|u|N|log |x||αdx. Considerthe caseM=HN:=RN−1×(0, ∞)and set ρ(x):=xN.Thisgives|∇ρ|=1 and pρ=0. This implies Corollary 1.7: Let 1<p<∞,α,β∈R. Let the constant C =C(p)be as in Theorem 1.2. Assume (α +β)(p−1+β) ≤0. Further,inthecasep<2, assume that (p−1+α+2β)(p−1+β) ≤0. If there exists a positive p-harmonic function ϕ∈W1,p loc (HN)such that (p−1+α+ β) ∂ϕ ∂xN≤0,thenforanyu∈W1,p 0(HN),wehavethefollowinginequality: HN |∇u|p gxα Ndx≥|p−1+β| ppHN |u|p xp−α N dx −C(p)(p−1+α+β)HN ∂ϕ ∂xN |∇ϕ|p−2 ϕp−1xα+1 N|u|pdx. Remark 1.8: The conditions (2) in Theorem 1.1 and (4) in Theorem 1.2 may seem artificially imposed at first glance. However, in the most commonly used form of Hardy inequality,thatisinthesetupofCorollary1.5withα=β=0, (2) holds automatically with 6K. MOHANTA AND J. TYAGI proper choice of d. In case of Riemannian manifolds validity of Equation (2) is not immediate, it is related to p-hyperbolicity of the underlying manifold. A discussion regarding this can be found in [15]. In the particular case p=2, =RN,andwhenVis radial, (4) is actually a necessary condition too for Equation (5) to hold. This can be seen from some symmetrization argument (to reduce the condition to its one-dimensional analogue) and [22, Theorem 1]. In Section 2,weshallprovesomepreliminaryresults,Theorem1.1followedbytheproof of Theorem 1.2. 2. Proof of the theorems The proof of the following lemma can be found in [24, Theorem III.7.6.] for the case p=2. The proof of this version can also be done similarly as the essence of the proof lies in the Stokes theorem and in the product rule of divergence operator: div(fX)=fdiv(X)+∇f,X. Lemma 2.1 (Green’s Formula): Let p>1, M be complete, oriented Riemannian manifold, adomaininMwithsmoothboundary.Letf ∈C2(),ξ∈C1 c(). Then, we have  <|∇f|p−2∇f,∇ξ>=− ξpf. The following result plays a key role in the proof of the theorem. Lemma 2.2: Let x ∈MandX x,Yx∈TxM be two tangent vectors. Then, for p ≥2,thereis a constant C =C(p)>0such that |Xx+Yx|p g−|Xx|p g≥C(p)|Yx|p g+p|Xx|p−2 g<Xx,Yx>, and for 1≤p≤2, |Xx+Yx|p g−|Xx|p g≥p(p−1) 2 |Yx|2 ||Xx|+|Yx||2−p+p|Xx|p−2<Xx,Yx>. In general, for 1<p<∞,wehave |Xx+Yx|p g−|Xx|p g≥p|Xx|p−2<Xx,Yx>. Proof: For the case p≥2, refer to [25, Chapter 12] and recall that any N-dimensional Hilbert space is isomorphic to RN.Wegiveaproofofthecase1≤p≤2, which is adapted from [26, Lemma 1]. In the following calculation, we omit the point xforbetterreadability.Considerthe function f(t):=|X+tY|p. It can be easily verified that, for t∈(0, 1),f(t)≥p(p−1)|Y|2|X+tY|p−2.Usingthis, Taylor’s theorem, the fact that p−2≤0, we get |X+Y|p=f(1) COMPLEX VARIABLES AND ELLIPTIC EQUATIONS 7 =f(0)+f(0)+1 0 (1−t)f(t)dt ≥|X|p+p|X|p−2X,Y +p(p−1)1 0 (1−t)|Y|2|X+tY|p−2dt ≥|X|p+p|X|p−2X,Y+p(p−1) 2|Y|2||X|+|Y||p−2. This proves the lemma.  Wenowpresenttheproofofourfirstmainresult. Proof of Theorem 1.1: By a density argument, we can always assume that u∈C1 c().Set w(x)=u(x)ρ −p+1−β p(x)in .Then ∇u=ρ p−1+β p∇w+p−1+β pρ −1+β pw∇ρ. Note that if |∇u(x)|p gραdVg=∞, then we have nothing to prove, so we assume it to be finite. In the following calculation, we need the term |u|p|∇ρ|p g ρp−αto be integrable over ;thisis indeed true, as u∈C1 c() and |∇ρ|p g ρp−α∈L1 loc() accordingtohypothesis.UsingLemma2.2 and (2), we have  |∇u(x)|p gραdVg−|p−1+β| pp |u(x)|p|∇ρ(x)|p g |ρ(x)|pραdVg =⎛ ⎝ ρ p−1+β p∇w+p−1+β p w∇ρ ρ 1−β p p g −|p−1+β| pp w∇ρ ρ 1−β p p g⎞ ⎠ρα ≥p|p−1+β| pp−2p−1+β p w∇ρ ρ 1−β p p−2ρ p−1+β p∇w,w∇ρ ρ 1−β pρα =|p−1+β| pp−2p−1+β p∇|w|p,|∇ρ|p−2∇ρρα+β =|p−1+β| pp−2p−1+β p∇(ρ−p+1−β|u|p),|∇ρ|p−2∇ρρα+β =|p−1+β| pp−2p−1+β p∇|u|p,|∇ρ|p−2∇ρρ−p+1+α −|p−1+β| pp p |∇ρ|p ρp−α|u|p ≥0.