We gave the Hardy type identities and inequalities for the divergence type operator Lf,V on smooth metric measure spaces. Additionally, we improved a Rellich type inequality by using the improved Hardy type inequality. Our results improved and included many previously known results as special cases.
Citation: Pengyan Wang, Jiahao Wang. Hardy type identities and inequalities with divergence type operators on smooth metric measure spaces[J]. AIMS Mathematics, 2024, 9(6): 16354-16375. doi: 10.3934/math.2024792
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We gave the Hardy type identities and inequalities for the divergence type operator Lf,V on smooth metric measure spaces. Additionally, we improved a Rellich type inequality by using the improved Hardy type inequality. Our results improved and included many previously known results as special cases.
Hardy type inequalities play crucial roles in analysis, probability, and partial differential equations. We first recall the classical L2- Hardy inequality:
∫Rn|∇ϕ|2dx≥(n−22)2∫Rn|ϕ|2|x|2dx, |
for n≥3 and ϕ∈C∞0(Rn∖{0}). The constant (n−22)2 is sharp and is never attained by nontrivial functions. The Rellich inequality is a natural generalization of the above Hardy inequality. In Rn, it reads as follows:
∫Rn|Δϕ|2dx≥(n(n−4)4)2∫Rn|ϕ|2|x|4dx, |
where n≥5 and ϕ∈C∞0(Rn∖{0}).
In recent years, there have been many results in the literature on the Hardy and Rellich type inequalities in the context of a complete Riemannian manifold. In particular, the following Hardy inequality has been first established on Riemannian manifold (M,g) by Carron [1]:
∫Mρα|∇ϕ|2dvg≥(C+α−1)24∫Mρα−2ϕ2dvg, |
where α∈R, C+α−1>0, ϕ∈C∞0(M∖ρ−1{0}). The weight function ρ is nonnegative and it satisfies |∇ρ|=1 and Δρ≥Cρ in the sense of distribution. Here, dvg, ∇, and Δ denote the volume element, the gradient, and the Laplace operator on M. Under the same geometric assumptions on the weight function ρ, Kombe and Özaydin [2] established the weighted Lp-Hardy inequality:
∫Mρα|∇ϕ|pdvg≥(C+1+α−pp)p∫Mρα−pϕpdvg, |
where 1≤p<∞, C+1+α−p>0, and ϕ∈C∞0(M∖ρ−1{0}). Kombe and Özaydin [3] also proved a new weighted Hardy-Poincaré inequality. They showed that if M is a complete non-compact Riemannian manifold of dimension n>1 and ρ is a nonnegative function on M such that |∇ρ|=1 and Δρ≥Cρ in the sense of distribution, where C>0, the following inequality holds:
∫Mρα+p|∇ρ⋅∇ϕ|pdvg≥(C+1+αp)p∫Mρα|ϕ|pdvg. |
Xia [4] proved the following Hardy type inequality on a complete non-compact Riemannian manifold. Let M be an n-dimensional complete non-compact Riemannian manifold, where n≥2, and let ρ be a nonnegative function on M such that |∇ρ|=1 and Δρ≥Cρ+H in the sense of distribution, where H is a continuous function on M and C is a constant. The result is the following: For any p,q∈R with q>1+C and any compactly supported smooth function ϕ∈C∞0(M∖ρ−1{0}), the following inequality holds:
(q−C−1)∫M|ϕ|pρqdvg≤|p|∫M|ϕ|p−1ρq−1|∇ϕ|dvg+∫MH|ϕ|pρq−1dvg. |
Huang and Ye [5] considered the first order Hardy inequalities using simple identities. This basic setting not only permits to derive quickly many well-known Hardy inequalities with optimal constants, but also supplies improved or new estimates in miscellaneous situations. We also refer the interested reader to [6,7,8,9,10,11,12,13,14,15,16], which are excellent monographs on the topic.
In this paper, we are interested in proving some Hardy type identities and inequalities on the smooth metric measure spaces related to the divergence type operator Lf,V. Before that, we would like to briefly introduce the smooth metric measure spaces and the divergence type operator Lf,V.
A smooth metric measure space (Mn,g,dμ) is a Riemannian manifold (Mn,g) equipped with a conformal Riemannian volume dμ=1Ve−fdvg, where dvg denotes the Riemannian volume measure on M, V is a positive twice differentiable function on M, and f is a real-valued smooth function on M.
On a smooth metric measure space (Mn,g,dμ), we can define the weighted Ricci curvature ^RicVf,m [17] given by
^RicVf,m=ΔfVVg−1V∇2V+Ricmf, |
where
Δf⋅:=efdiv(e−f∇⋅)=Δ⋅−∇f∇, | (1.1) |
is the f-Laplacian (see [18,19,20,21]),
Ricmf=Ric+∇2f−1m−n∇f⊗∇f, |
is the m-Bakry-Émery Ricci curvature, and m is a real constant. Here, ∇, Δ, ∇2, and div are the gradient, Laplacian, Hessian, and divergence operator on M with respect to g, respectively.
We introduce the divergence type operator Lf,V on M as follows (see [17,22] for detail):
Lf,Vφ=efdiv[e−fV2∇(φV)], | (1.2) |
where φ is a smooth function and V is a positive twice differentiable function defined on M. In general, the divergence type operator Lf,V is not self-adjoint with respect to the standard L2-inner product because of the first order term, but it is self-adjoint with respect to the weighted measure dμ=1Ve−fdvg. That is, for any smooth functions u,v∈C∞0(M), we have
∫MuLf,Vvdμ=−∫MV3∇uV∇vVdμ=∫MvLf,Vudμ. | (1.3) |
We observe that the divergence type operator Lf,V encompasses, as very special cases, many differential linear operators. In the case that V=1, i.e., Lf,1φ=Δfφ. In the case that f is constant, i.e., Lf,Vφ=LVφ=VΔφ−φΔV, we call LV by the generalized Schrödinger operator. From the viewpoint of geometry, the generalized Schrödinger operator LV plays an important role in the geometric understanding of the sub-static manifolds; see [23].
Du and Mao [24] proved that some Hardy and Rellich type inequalities on the smooth metric measure spaces related to the f-Laplace. Also, Li, Abolarinwa, Alkhaldi, and Ali [25] generalized some integral inequalities of Hardy type to the setting of the smooth metric measure spaces. These studies are without the curvature conditions. On the other hand, the case of additional curvature conditions has been studied. Kolesnikov and Milman [26] proved the Hardy-Poincaré inequality under the curvature condition Ricmf>0 on M:
mm−1∫Mφ2e−fdvg≤∫Me−f⟨(Ricmf)−1∇φ,∇φ⟩dvg, |
where 1m∈(−∞,1n] and φ∈C1(M). Huang and Zhu [22] gave the following Hardy-Poincaré inequality: Let M be a compact Riemannian manifold and ^RicVf,m>0, then
mm−1∫MVφ2e−fdvg≤∫Me−fV⟨(^RicVf,m)−1∇φ,∇φ⟩dvg, |
where φ∈C∞(M). Huang and Zhu [22] studied weighted L2-Hardy-Poincaré inequalities on a smooth metric measure space related to the divergence type operator Lf,V under the curvature condition.
The primary objectives of the present paper are twofold: First, we want to establish some L2 Hardy type identities and Lp Hardy type inequalities related to the divergence type operator Lf,V on the smooth metric measure spaces without curvature conditions. Second, as application we would like to show some Rellich type inequalities for the divergence type operator Lf,V.
The remainder of the paper is as follows: In Section 2, we will prove Hardy type identity related to the divergence type operator Lf,V and several related corollaries. Then, we will prove some Lp Hardy type inequalities in Section 3. In the last section, we will prove a Rellich type inequality related to the divergence type operator Lf,V.
In this section, we will prove some Hardy type identities related to the divergence type operator Lf,V and several related corollaries.
Theorem 2.1. Let (Mn,g,dμ) be an n-dimensional (n≥2) complete non-compact smooth metric measure space. Suppose that V is a positive twice differentiable function and f is a real-valued smooth function on M and W∈C1(M). Then, for any compactly supported smooth functions uV∈C10(M) and φV∈C20(M), the following identity holds:
∫MWV3|∇(uV)|2dμ=∫MWV3|∇(uV)−(uV)(φV)−1∇(φV)|2dμ−∫MefV(uV)2(φV)−1div(e−fV2W∇(φV))dμ, | (2.1) |
where dμ=1Ve−fdvg and dvg is the Riemannian volume element related to g.
Proof. As a consequence of the integration by parts (1.3), we have
∫MWV3|∇(uV)−(uV)(φV)−1∇(φV)|2dμ=∫Me−fV2W|∇(uV)|2dvg−2∫Me−fV2W⟨∇(uV),(uV)(φV)−1∇(φV)⟩dvg+∫Me−fV2W(uV)2(φV)−2|∇(φV)|2dvg=∫Me−fV2W|∇(uV)|2dvg−∫Me−fV2W⟨(φV)−1∇(uV)2,∇(φV)⟩dvg+∫Me−fV2W(uV)2(φV)−2|∇(φV)|2dvg=∫Me−fV2W|∇(uV)|2dvg+∫M(uV)2div(e−fV2W(φV)−1∇(φV))dvg+∫Me−fV2W(uV)2(φV)−2|∇(φV)|2dvg=∫Me−fV2W|∇(uV)|2dvg+∫M(φV)−1(uV)2div(e−fV2W∇(φV))dvg−∫M(φV)−2(uV)2⟨∇(φV),e−fV2W∇(φV)⟩dvg+∫Me−fV2W(uV)2(φV)−2|∇(φV)|2dvg=∫MWV3|∇(uV)|2dμ+∫MefV(uV)2(φV)−1div(e−fV2W∇(φV))dμ. |
Then, we can get
∫MWV3|∇(uV)|2dμ=∫MWV3|∇(uV)−(uV)(φV)−1∇(φV)|2dμ−∫MefV(uV)2(φV)−1div(e−fV2W∇(φV))dμ. |
This completes the proof of Theorem 2.1.
As a special case for W=1 in Theorem 2.1, we have the following.
Corollary 2.2. Let (Mn,g,dμ) be an n-dimensional (n≥2) complete non-compact smooth metric measure space. Suppose that V is a positive twice differentiable function and f is a real-valued smooth function on M. Then, for any uV∈C10(M) and φV∈C20(M), the following identity holds:
∫MV3|∇(uV)|2dμ=∫MV3|∇(uV)−(uV)(φV)−1∇(φV)|2dμ−∫MV(uV)2(φV)−1(Lf,Vφ)dμ, | (2.2) |
where dμ=1Ve−fdvg and dvg is the Riemannian volume element related to g.
By using Corollary 2.2, we have the following.
Corollary 2.3. Let (Mn,g,dμ) be an n-dimensional (n≥2) complete non-compact smooth metric measure space. Suppose that V is a positive twice differentiable function and f is a real-valued smooth function on M. Then, for any uV∈C10(M) and φV∈C20(M), the following inequality holds:
∫MV3|∇(uV)|2dμ≥−∫MV(uV)2(φV)−1(Lf,Vφ)dμ, | (2.3) |
where dμ=1Ve−fdvg and dvg is the Riemannian volume element related to g.
As the special case that φV=(ρV)1−C2 in the Corollary 2.3, we have the following.
Corollary 2.4. Let (Mn,g,dμ) be an n-dimensional (n≥2) complete non-compact smooth metric measure space. Suppose that V is a positive twice differentiable function and f is a real-valued smooth function on M. Let ρ be a nonnegative function on M such that |∇(ρV)|=1 and Lf,Vρ≥C(ρV)−1V2 in the sense of distribution, where C is a constant and C>1. Then, for any uV∈C10(M∖(ρV)−1{0}), the following inequality holds:
∫MV3|∇(uV)|2dμ≥(C−1)24∫MV3(uV)2(ρV)−2dμ, |
where dμ=1Ve−fdvg and dvg is the Riemannian volume element related to g.
Proof. Taking φV=(ρV)1−C2, we have
−(φV)−1(Lf,Vφ)=−ef(ρV)C−12div(e−fV2∇(ρV)1−C2)=−(ρV)C−12efdiv((1−C2)e−fV2(ρV)−C−12∇(ρV))=−(1−C2)(ρV)C−12ef⟨∇(ρV)−C−12,e−fV2∇(ρV)⟩−(1−C2)(ρV)C−12(ρV)−C−12efdiv(e−fV2∇(ρV))=1−C24V2(ρV)−2|∇(ρV)|2+C−12(ρV)−1Lf,Vρ. |
Then, we have
−(φV)−1(Lf,Vφ)=1−C24V2(ρV)−2|∇(ρV)|2+C−12(ρV)−1Lf,Vρ≥1−C24V2(ρV)−2+C2−C2V2(ρV)−2=C2−2C+14V2(ρV)−2=(C−1)24V2(ρV)−2, |
where we use the assumption |∇(ρV)|=1 and Lf,Vρ≥C(ρV)−1V2. As a result, we can get
−(φV)−1(Lf,Vφ)≥(C−1)24V2(ρV)−2. | (2.4) |
Substituting (2.4) into (2.3), we can get
∫MV3|∇(uV)|2dμ≥(C−1)24∫MV3(uV)2(ρV)−2dμ. |
This completes the proof of Corollary 2.4.
Remark 2.5. In the special case that f is a constant and V=1, Corollary 2.4 reduces to Carron's result in [1, Proposition 2.1].
In this section, we will prove some Lp-Hardy type inequalities on smooth metric measure spaces. Our first result is the following.
Theorem 3.1. Let (Mn,g,dμ) be an n-dimensional (n≥2) complete non-compact smooth metric measure space. Suppose that V is a positive twice differentiable function and f is a real-valued smooth function on M. Let ρ be a nonnegative function on M such that |∇(ρV)|=1 and Lf,Vρ≤(H+C(ρV)−1)V2 in the sense of distribution, where H is a continuous function on M and C is a constant. Then, for any p,q∈R,q>1+C, and any compactly supported smooth functions ϕV∈C∞0(M∖(ρV)−1{0}), the following inequality holds:
(q−C−1)∫MV3|ϕV|p(ρV)−qdμ≤|p|∫MV3|ϕV|p−1(ρV)1−q|∇(ϕV)|dμ+∫MHV3|ϕV|p(ρV)1−qdμ. | (3.1) |
Proof. For a vector field X on M, we denote by div X the divergence of X. Note that
div(e−fV2|ϕV|p(ρV)1−q∇(ρV))=pe−fV2|ϕV|p−1⟨∇|ϕV|,(ρV)1−q∇(ρV)⟩+|ϕV|pdiv(e−fV2(ρV)1−q∇(ρV))=pe−fV2|ϕV|p−1⟨∇|ϕV|,(ρV)1−q∇(ρV)⟩+|ϕV|p(ρV)1−qdiv(e−fV2∇(ρV))+(1−q)e−fV2|ϕV|p⟨(ρV)−q∇(ρV),∇(ρV)⟩=pe−fV2|ϕV|p−1⟨∇|ϕV|,(ρV)1−q∇(ρV)⟩+e−f|ϕV|p(ρV)1−q(Lf,Vρ)+(1−q)e−fV2|ϕV|p(ρV)−q|∇(ρV)|2. |
By direct computation, we have
div(e−fV2|ϕV|p(ρV)1−q∇(ρV))=pe−fV2|ϕV|p−1⟨∇|ϕV|,(ρV)1−q∇(ρV)⟩+e−f|ϕV|p(ρV)1−q(Lf,Vρ)+(1−q)e−fV2|ϕV|p(ρV)−q|∇(ρV)|2≤pe−fV2|ϕV|p−1⟨∇|ϕV|,(ρV)1−q∇(ρV)⟩+e−fV2|ϕV|p(ρV)1−q(H+C(ρV)−1)+(1−q)e−fV2|ϕV|p(ρV)−q=pe−fV2|ϕV|p−1⟨∇|ϕV|,(ρV)1−q∇(ρV)⟩+(1+C−q)e−fV2|ϕV|p(ρV)−q+e−fV2H|ϕV|p(ρV)1−q, | (3.2) |
where we use the assumption |∇(ρV)|=1 and Lf,Vρ≤(H+C(ρV)−1)V2.
Since q>1+C and |∇(ρV)|=1, we have
(q−C−1)∫Me−fV2|ϕV|p(ρV)−qdvg≤p∫Me−fV2|ϕV|p−1⟨∇|ϕV|,(ρV)1−q∇(ρV)⟩dvg+∫Me−fV2H|ϕV|p(ρV)1−qdvg≤|p∫Me−fV2|ϕV|p−1⟨∇|ϕV|,(ρV)1−q∇(ρV)⟩dvg|+∫Me−fV2H|ϕV|p(ρV)1−qdvg≤|p|∫Me−fV2|ϕV|p−1(ρV)1−q|∇(ϕV)||∇(ρV)|dvg+∫Me−fV2H|ϕV|p(ρV)1−qdvg=|p|∫Me−fV2|ϕV|p−1(ρV)1−q|∇(ϕV)|dvg+∫Me−fV2H|ϕV|p(ρV)1−qdvg. |
Then, we have
(q−C−1)∫MV3|ϕV|p(ρV)−qdμ≤|p|∫MV3|ϕV|p−1(ρV)1−q|∇(ϕV)|dμ+∫MHV3|ϕV|p(ρV)1−qdμ. | (3.3) |
This completes the proof of Theorem 3.1.
Remark 3.2. In the special case that V=1 and f is a constant, the inequality (3.1) reduces to Xia's result in [4, Theorem 2.1].
Theorem 3.3. Let (Mn,g,dμ) be an n-dimensional (n≥2) complete non-compact smooth metric measure space. Suppose that V is a positive twice differentiable function and f is a real-valued smooth function on M. Let ρ be a nonnegative function on M such that |∇(ρV)|=1 in the sense of distributions. Then, for any p,q∈R,1<p<+∞,0≤q≤p, and any compactly supported smooth function ϕV∈C∞0(M∖(ρV)−1{0}), we have
(i) When Lf,Vρ≤C(ρV)−1V2 in the sense of distributions, where C<q−1 is a constant, the following inequality holds:
∫MV3|ϕV|p(ρV)−qdμ≤(pq−C−1)q(∫MV3|ϕV|pdμ)p−qp(∫MV3|∇(ϕV)|pdμ)qp. | (3.4) |
(ii) When Lf,Vρ≥C(ρV)−1V2 in the sense of distributions, where C>q−1 is a constant, the following inequality holds:
∫MV3|ϕV|p(ρV)−qdμ≥(pC+1−q)q(∫MV3|ϕV|pdμ)p−qp(∫MV3|∇(ϕV)|pdμ)qp. | (3.5) |
Proof. (ⅰ) In the special case that H=0 in (3.1), we can get
(q−C−1)∫MV3|ϕV|p(ρV)−qdμ≤|p|∫MV3|ϕV|p−1(ρV)1−q|∇(ϕV)|dμ. | (3.6) |
It follows from the Hölder inequality that
∫Me−fV2|ϕV|p−1(ρV)1−q|∇(ϕV)|dvg=∫M(e−fV2)1p(e−fV2)p−1p|ϕV|p−1(ρV)1−q|∇(ϕV)|dvg≤(∫Me−fV2|∇(ϕV)|pdvg)1p(∫Me−fV2|ϕV|p(ρV)p(1−q)p−1dvg)p−1p≤(∫Me−fV2|∇(ϕV)|pdvg)1p(∫Me−fV2|ϕV|p(ρV)−qdvg)q−1q(∫Me−fV2|ϕV|pdvg)(1−p(q−1)q(p−1))p−1p. | (3.7) |
Substituting (3.7) into (3.6), we can get
(q−C−1)∫Me−fV2|ϕV|p(ρV)−qdvg≤|p|(∫Me−fV2|∇(ϕV)|pdvg)1p(∫Me−fV2|ϕV|p(ρV)−qdvg)q−1q(∫Me−fV2|ϕV|pdvg)(1−p(q−1)q(p−1))p−1p≤|p|(∫Me−fV2|∇(ϕV)|pdvg)1p(∫Me−fV2|ϕV|p(ρV)−qdvg)q−1q(∫Me−fV2|ϕV|pdvg)p−qpq. |
Then,
∫MV3|ϕV|p(ρV)−qdμ≤(pq−C−1)q(∫MV3|ϕV|pdμ)p−qp(∫MV3|∇(ϕV)|pdμ)qp. |
This completes the proof of (ⅰ).
(ⅱ) Similar to the derivation of (3.4) above, the inequality (3.5) can be obtained without any difficulty. This completes the proof of Theorem 3.3.
Remark 3.4. In the special case that V=1, Theorem 3.3 reduces to the result of Du and Mao in [24, Theorem 2.1].
Then, we prove the following Hardy-Poincaré type inequalities related to the divergence type operator Lf,V.
Theorem 3.5. Let (Mn,g,dμ) be an n-dimensional (n≥2) complete non-compact smooth metric measure space. Suppose that V is a positive twice differentiable function and f is a real-valued smooth function on M. Let ρ be a nonnegative function on M such that |∇(ρV)|=1 and Lf,Vρ≥(C(ρV)−1+G)V2 in the sense of distribution, where C>0 is a constant and G is a continuous function. Then, for any p,q,α∈R, φ∈C∞0(M∖(ρV)−1{0}), p∈(1,∞), Aα=(C+α+1)p with C+α+1>0, we have the following inequality:
∫MV3(ρV)α+p|⟨∇(ρV),∇(φV)⟩|pdμ≥Apα∫MV3(ρV)α|φV|pdμ+Ap−1α∫MV3G(ρV)α+1|φV|pdμ. | (3.8) |
Proof. It follows from |∇(ρV)|=1 and Lf,Vρ≥(C(ρV)−1+G)V2 that
efdiv(e−fV2(ρV)∇(ρV))=⟨∇(ρV),V2∇(ρV)⟩+ef(ρV)div(e−fV2∇(ρV))=V2|∇(ρV)|2+(ρV)Lf,Vρ≥V2(1+C+G(ρV)). | (3.9) |
Multiplying (3.9) by (ρV)α|φV|p and integrating both sides over M gives
(1+C)∫Me−fV2(ρV)α|φV|pdvg+∫Me−fGV2(ρV)α+1|φV|pdvg≤∫M(ρV)α|φV|pdiv(e−fV2(ρV)∇(ρV))dvg=−∫M⟨(ρV)∇(ρV),e−fV2∇((ρV)α|φV|p)⟩dvg=−α∫Me−fV2(ρV)α−1|φV|p(ρV)⟨∇(ρV),∇(ρV)⟩dvg−p∫Me−fV2|φV|p−2(φV)(ρV)α(ρV)⟨∇(φV),∇(ρV)⟩dvg=−α∫Me−fV2(ρV)α|φV|pdvg−p∫Me−fV2(ρV)α+1|φV|p−2(φV)⟨∇(ρV),∇(φV)⟩dvg, | (3.10) |
which implies
(1+C+α)∫Me−fV2(ρV)α|φV|pdvg+∫Me−fV2G(ρV)α+1|φV|pdvg≤−p∫Me−fV2(ρV)α+1|φV|p−2(φV)⟨∇(ρV),∇(φV)⟩dvg. | (3.11) |
It follows from the Hölder inequality that
−p∫Me−fV2(ρV)α+1|φV|p−2(φV)⟨∇(ρV),∇(φV)⟩dvg≤|p∫Me−fV2(ρV)α+1|φV|p−2(φV)⟨∇(ρV),∇(φV)⟩dvg|≤p∫Me−fV2(ρV)α+1|φV|p−1|⟨∇(ρV),∇(φV)⟩|dvg=p∫M(e−fV2(ρV)α|φV|p)p−1p(e−fV2(ρV)α+p|⟨∇(ρV),∇(φV)⟩|p)1pdvg≤p(∫Me−fV2(ρV)α|φV|pdvg)p−1p(∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(φV)⟩|pdvg)1p. |
Thus,
−p∫Me−fV2(ρV)α+1|φV|p−2(φV)⟨∇(ρV),∇(φV)⟩dvg≤p(∫Me−fV2(ρV)α|φV|pdvg)p−1p(∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(φV)⟩|pdvg)1p. | (3.12) |
Then, we use the Young inequality in (3.12), which is described as follows: Denoting
Φ=:(∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(φV)⟩|pdvg)1pandΨ=:(∫Me−fV2(ρV)α|φV|pdvg)p−1p, |
then for any ϵ>0,
ΦΨ=ϵΦΨϵ≤1p(ϵΦ)p+1q(Ψϵ)qwithq=pp−1relating top, |
we can get
(∫Me−fV2(ρV)α|φV|pdvg)p−1p(∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(φV)⟩|pdvg)1p≤(p−1)pϵpp−1∫Me−fV2(ρV)α|φV|pdvg+1pϵp∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(φV)⟩|pdvg, |
namely,
−p∫Me−fV2(ρV)α+1|φV|p−2(φV)⟨∇(ρV),∇(φV)⟩dvg≤(p−1)ϵpp−1∫Me−fV2(ρV)α|φV|pdvg+ϵp∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(φV)⟩|pdvg. | (3.13) |
Hence, putting (3.11) and (3.13) together, we can get
∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(φV)⟩|pdvg≥ϵ−p(1+C+α−p−1ϵpp−1)∫Me−fV2(ρV)α|φV|pdvg+ϵ−p∫Me−fV2G(ρV)α+1|φV|pdvg. | (3.14) |
We take Y(ϵ) to be the function Y(ϵ)=ϵ−p(1+C+α−p−1ϵpp−1). By direct computation, we can conclude that Y(ϵ) reaches its maximum value when ϵ=(p1+C+α)p−1p(for details, see [25, Theorem 3]).
Finally, the required inequality can be determined by substituting ϵ=(p1+C+α)p−1p into (3.14) as follows:
∫MV3(ρV)α+p|⟨∇(ρV),∇(φV)⟩|pdμ≥Apα∫MV3(ρV)α|φV|pdμ+Ap−1α∫MV3G(ρV)α+1|φV|pdμ. |
This completes the proof of Theorem 3.5.
Using Theorem 3.5, we can obtain the following applications:
Theorem 3.6. Let (Mn,g,dμ) be an n-dimensional complete noncompact smooth metric measure space. Suppose that V is a positive twice differentiable function and f is a real-valued smooth function on M. Let ρ be a nonnegative function on M such that |∇(ρV)|=1 in the sense of distributions. Then for any p,q∈R,1<p<+∞,0≤q≤p, and any compactly supported smooth function ϕV∈C∞0(M∖(ρV)−1{0}), we have
(i) When Lf,Vρ≤C(ρV)−1V2 in the sense of distributions, where C>0 is a constant and C+α<−1, the following inequality holds:
∫MV3(ρV)α+p|⟨∇(ρV),∇(ϕV)⟩|pdμ≥(|C+α+1|p)p∫MV3(ρV)α|ϕV|pdμ. | (3.15) |
(ii) When Lf,Vρ≥C(ρV)−1V2 in the sense of distributions, where C>0 is a constant and C+α>−1, the following inequality holds:
∫MV3(ρV)α+p|⟨∇(ρV),∇(ϕV)⟩|pdμ≥(C+α+1p)p∫MV3(ρV)α|ϕV|pdμ. | (3.16) |
Proof. (ⅰ) It follows from |∇(ρV)|=1 and Lf,Vρ≥C(ρV)−1V2 that
div(e−fV2(ρV)∇(ρV))=e−fV2|∇(ρV)|2+(ρV)div(e−fV2∇(ρV))≥(1+C)e−fV2. |
Thus,
div(e−fV2(ρV)∇(ρV))≥(1+C)e−fV2. | (3.17) |
Multiplying (3.17) by (ρV)α|ϕV|pand integrating over M yields
(1+C)∫Me−fV2(ρV)α|ϕV|pdvg≤∫M(ρV)α|ϕV|pdiv(e−fV2(ρV)∇(ρV))dvg=−∫M⟨(ρV)∇(ρV),e−fV2∇((ρV)α|ϕV|p)⟩dvg=−α∫Me−fV2(ρV)α|ϕV|pdvg−p∫Me−fV2|ϕV|p−2(ϕV)(ρV)α+1⟨∇(ρV),∇(ϕV)⟩dvg. | (3.18) |
Since C+α+1<0, by using the Hölder inequality, we can infer from (3.18) that
|C+α+1|∫Me−fV2(ρV)α|ϕV|pdvg≤−p∫Me−fV2|ϕV|p−2(ϕV)(ρV)α+1⟨∇(ρV),∇(ϕV)⟩dvg≤|p∫Me−fV2|ϕV|p−2(ϕV)(ρV)α+1⟨∇(ρV),∇(ϕV)⟩dvg|≤p∫Me−fV2|ϕV|p−1(ρV)α+1|⟨∇(ρV),∇(ϕV)⟩|dvg≤p∫M(e−fV2(ρV)α|ϕV|p)p−1p(e−fV2(ρV)α+p|⟨∇(ρV),∇(ϕV)⟩|p)1pdvg≤p(∫Me−fV2(ρV)α|ϕV|pdvg)p−1p(∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(ϕV)⟩|pdvg)1p. | (3.19) |
It follows from the Young inequality that
|C+α+1|∫Me−fV2(ρV)α|ϕV|pdvg≤p(∫Me−fV2(ρV)α|ϕV|pdvg)p−1p(∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(ϕV)⟩|pdvg)1p≤(p−1)[ϵ−1(∫Me−fV2(ρV)α|ϕV|pdvg)p−1p]pp−1+[ϵ(∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(ϕV)⟩|pdvg)1p]p=(p−1)ϵ−pp−1∫Me−fV2(ρV)α|ϕV|pdvg+ϵp∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(ϕV)⟩|pdvg. | (3.20) |
Thus, for any ϵ>0, from (3.20), we have
∫Me−fV2(ρV)α+p|⟨∇(ρV),∇(ϕV)⟩|pdvg≥ϵ−p(|C+α+1|−(p−1)ϵ−pp−1)∫Me−fV2(ρV)α|ϕV|pdvg. |
Taking
ϵ=(p|C+α+1|)p−1p, |
in the above inequality, we can get
∫MV3(ρV)α+p|⟨∇(ρV),∇(ϕV)⟩|pdμ≥(|C+α+1|p)p∫MV3(ρV)α|ϕV|pdμ. |
(ⅱ) Similar to the proof of (3.15) above, the inequality (3.16) can be obtained without any difficulty. This completes the proof of Theorem 3.6.
Remark 3.7. In the special case that V=1, Theorem 3.6 reduces to the result of Du and Mao in [24, Theorem 4.1].
Then, we will prove the weighted Lp-Hardy type inequality.
Theorem 3.8. Let (Mn,g,dμ) be an n-dimensional (n≥2) complete non-compact smooth metric measure space. Suppose that V is a positive twice differentiable function and f is a real-valued smooth function on M. Let ρ be a nonnegative function on M such that (ρV)Lf,Vρ≥CV2|∇(ρV)|2 in the sense of distribution, where C>0 is a constant. Then, the following inequality holds for any ϕV∈C∞0(M∖ρ−1{0}):
∫MV3(ρV)α(sinhβ(ρV))|∇(ρV)|2−p|∇(ϕV)|pdμ≥(C+1+α+β−pp)p∫MV3(ρV)α−p(sinhβ(ρV))|ϕV|p|∇(ρV)|2dμ, | (3.21) |
where p,α,β∈R,1≤p<∞, and C+1+α+β−p>0.
Proof. By direct computation, we have
∫Me−fV2(ρV)α−psinhβ(ρV)|ϕV|p|∇(ρV)|2dvg≤1C∫Me−f(ρV)α−p+1sinhβ(ρV)|ϕV|p(Lf,Vρ)dvg=1C∫M(ρV)α−p+1|ϕV|psinhβ(ρV)div(e−fV2∇(ρV))dvg=−1C∫M⟨∇(sinhβ(ρV)|ϕV|p(ρV)α−p+1),e−fV2∇(ρV)⟩dvg=−∫M(α−p+1+β(ρV)coth(ρV))Ce−fV2(ρV)α−psinhβ(ρV)|ϕV|p|∇(ρV)|2dvg−pC∫Me−fV2(ρV)α−p+1sinhβ(ρV)|ϕV|p−2(ϕV)⟨∇(ρV),∇(ϕV)⟩dvg, | (3.22) |
where we have used the assumption (ρV)Lf,Vρ≥CV2|∇(ρV)|2 in the first line of the inequality.
Observing that (ρV)coth(ρV)≥1, we can rewrite the inequality (3.22) as
∫Me−fV2(ρV)α−psinhβ(ρV)|ϕV|p|∇(ρV)|2dvg≤−(α−p+1+β)C∫Me−fV2(ρV)α−psinhβ(ρV)|ϕV|p|∇(ρV)|2dvg−pC∫Me−fV2(ρV)α−p+1sinhβ(ρV)|ϕV|p−2(ϕV)⟨∇(ρV),∇(ϕV)⟩dvg. |
Then,
(C+α−p+1+β)∫Me−fV2(ρV)α−psinhβ(ρV)|ϕV|p|∇(ρV)|2dvg≤p∫Me−fV2(ρV)α−p+1sinhβ(ρV)|ϕV|p−1⟨∇(ρV),∇(ϕV)⟩dvg. | (3.23) |
It follows from the Hölder inequality that
(C+α−p+1+β)∫Me−fV2(ρV)α−psinhβ(ρV)|ϕV|p|∇(ρV)|2dvg≤p∫Me−fV2(ρV)α−p+1sinhβ(ρV)|ϕV|p−1⟨∇(ρV),∇(ϕV)⟩dvg≤|p∫Me−fV2(ρV)α−p+1sinhβ(ρV)|ϕV|p−1⟨∇(ρV),∇(ϕV)⟩dvg|≤p∫M(e−fV2(ρV)α−psinhβ(ρV)|ϕV|p|∇(ρV)|2)p−1p(e−fV2(ρV)αsinhβ(ρV)|∇(ρV)|2−p|∇(ϕV)|p)1pdvg≤p(∫Me−fV2(ρV)α−psinhβ(ρV)|ϕV|p|∇(ρV)|2dvg)p−1p(∫Me−fV2(ρV)αsinhβ(ρV)|∇(ρV)|2−p|∇(ϕV)|pdvg)1p. | (3.24) |
namely,
∫MV3(ρV)αsinhβ(ρV)|∇(ρV)|2−p|∇(ϕV)|pdμ≥(C+1+α+β−pp)p∫MV3(ρV)α−psinhβ(ρV)|ϕV|p|∇(ρV)|2dμ. |
This completes the proof of Theorem 3.8.
In this section, by applying Theorem 3.8 of Section 3, we can give the following Rellich type inequality.
Theorem 4.1. Let (Mn,g,dμ) be an n-dimensional (n≥2) complete non-compact smooth metric measure space. Suppose that V is a positive twice differentiable function and f is a real-valued smooth function on M. Let ρ be a nonnegative function on M such that |∇(ρV)|=1 and (ρV)Lf,Vρ≥CV2 in the sense of distributions, where C>0 is a constant and max{3−C,(7−C)/3}≤α≤2. Then, the following inequality holds for ϕV∈C∞0(M∖(ρV)−1{0}):
∫MV3(ρV)α|Lf,Vϕ|2dμ≥(C+1−α2)2∫MV3(ρV)α−2|∇(ϕV)|2dμ, | (4.1) |
where dμ=1Ve−fdvg and dvg is the Riemannian volume element related to g.
Proof. As a consequence of integration by parts (1.3), we compute and estimate the righthand side:
∫Me−fV2(ρV)α−2|∇(ϕV)|2dvg=∫Me−fV2(ρV)α−2⟨∇(ϕV),∇(ϕV)⟩dvg=∫M⟨∇(ϕV),e−fV2(ρV)α−2∇(ϕV)⟩dvg=−∫M(ϕV)div(e−fV2(ρV)α−2∇(ϕV))dvg=−∫M(ϕV)(e−fV2⟨∇(ρV)α−2,∇(ϕV)⟩+(ρV)α−2div(e−fV2∇(ϕV)))dvg=−∫Me−fV2(ϕV)⟨∇(ρV)α−2,∇(ϕV)⟩dvg−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg=−(α−2)∫Me−fV2(ϕV)(ρV)α−3⟨∇(ρV),∇(ϕV)⟩dvg−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg=−(α−2)2∫Me−fV2(ρV)α−3⟨∇(ρV),∇(ϕV)2⟩dvg−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg=−(α−2)2∫M(ρV)α−3⟨e−fV2∇(ρV),∇(ϕV)2⟩dvg−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg=(α−2)2∫M(ϕV)2div(e−fV2(ρV)α−3∇(ρV))dvg−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg. | (4.2) |
It follows from |∇(ρV)|=1 and (ρV)Lf,Vρ≥CV2 that
∫Me−fV2(ρV)α−2|∇(ϕV)|2dvg=(α−2)2∫M(ϕV)2div(e−fV2(ρV)α−3∇(ρV))dvg−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg=−(2−α)2∫M((α−3)e−fV2(ρV)α−4(ϕV)2|∇(ρV)|2+(ρV)α−3(ϕV)2div(e−fV2∇(ρV)))dvg−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg=−(2−α)2∫M((α−3)e−fV2(ρV)α−4(ϕV)2+e−f(ρV)α−3(ϕV)2(Lf,Vρ))dvg−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg≤−(2−α)(C+α−3)2∫Me−fV2(ρV)α−4(ϕV)2dvg−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg. |
Thus,
∫Me−fV2(ρV)α−2|∇(ϕV)|2dvg≤−(2−α)(C+α−3)2∫Me−fV2(ρV)α−4(ϕV)2dvg−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg. | (4.3) |
Applying the Hölder inequality to the above, we have
−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg≤|∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg|≤(∫Me−fV2(ρV)α|Lf,Vϕ|2dvg)12(∫Me−fV−2(ϕV)2(ρV)α−4dvg)12. |
Thus,
−∫Me−f(ρV)α−2(ϕV)(Lf,Vϕ)dvg≤(∫Me−fV2(ρV)α|Lf,Vϕ|2dvg)12(∫Me−fV−2(ϕV)2(ρV)α−4dvg)12. | (4.4) |
Then, substituting (4.4) into (4.3),
(∫Me−fV2(ρV)α|Lf,Vϕ|2dvg)12(∫Me−fV−2(ϕV)2(ρV)α−4dvg)12≥∫Me−fV2(ρV)α−2|∇(ϕV)|2dvg+(2−α)(C+α−3)2∫Me−fV2(ρV)α−4(ϕV)2dvg. |
We denote
A=∫Me−fV2(ρV)α−2|∇(ϕV)|2dvg,B=∫Me−fV2(ρV)α−4(ϕV)2dvg,D=∫Me−fV2(ρV)α|Lf,Vϕ|2dvg, |
and then can get
√D≥√A(√AB+(2−α)(C+α−3)2√BA). |
Denoting further S=AB,
√D≥√A(√S+(2−α)(C+α−3)2√1S):=√Af(S). |
Now, we compute a lower bound for the function f(S)=√S+((2−α)(C+α−3)/2)√1/S. To begin, we use Theorem 3.8 (in the special case thatβ=0,p=2,|∇(ρV)|=1) and then can get
A=∫Me−fV2(ρV)α−2|∇(ϕV)|2dvg≥(C+α−32)2∫Me−fV2(ρV)α−4(ϕV)2dvg=(C+α−32)2B. |
We have the following condition on S:
S=AB≥(C+α−32)2. |
Next, the function f(S) is increasing for S∈(0,+∞) and it attains its minimum at S=(2−α)(C+α−3)/2. However, (2−α)(C+α−3)/2≤((C+α−3)/2)2 when α≥(7−C)/3, so f(S) attains its minimum at S=((C+α−3)/2)2, and this minimum is equal to ((C+1−α)/2)2. Finally, we obtain the following inequality:
D=∫MV3(ρV)α|Lf,Vϕ|2dμ≥(C+1−α2)2A=(C+1−α2)2∫MV3(ρV)α−2|∇(ϕV)|2dμ. |
This completes the proof of Theorem 4.1.
In this paper, we have established some Hardy type identities and inequalities for the divergence type operator Lf,V on smooth metric measure spaces. First, we have established some L2 Hardy type identities. As their corollary, we have obtained a L2 Hardy type inequality. Second, we have established some Lp Hardy type inequalities. As their corollary, we have obtained a Lp Rellich type inequality. From the proof of the above results, we see that our method does not work for us to obtain the sharp constants. Hence, we shall further pursue sharp Hardy type inequalities for the divergence type operator Lf,V on smooth metric measure spaces in the subsequent papers.
Pengyan Wang: Methodology, writing-original draft; Jiahao Wang: validation, writing-review and editing. All authors have read and agreed to the published version of the manuscript.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The research of authors is supported by NSFC (No.12101530), the Science and Technology Project of Henan Province (No.232102310321), Nanhu Scholars Program for Young Scholars of XYNU (No.2023), and Xinyang Normal University Graduate Student Innovation Fund Project (2024KYJJ060).
The authors declare no conflict of interest.
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