SEMIGROUP DOMINATION ON A RIEMANNIAN MANIFOLD WITH BOUNDARY

ICHIRO SHIGEKAWA

Dedicated to Professor Takeyuki Hida on his 70th birthday

Abstract. We discuss the semigroup domination on a Riemannian manifold with boundary. Our main interest is the Hodge-Kodaira Laplacian for differential forms. We consider two kinds of boundary conditions; the absolutely boundary condition and the relative boundary condition. Our main tool is the square field operator. We also develop a general theory of semigroup commutation.

1. Introduction

In this paper, we discuss a theory of semigroup domination. To be precise, let (X, B, m) be a σ-finite measure space. Suppose we are given a semigroup {T_{t}} on L^{2} = L^{2}(m). We assume that {T_{t}} is positivity preserving. Besides, we are given a semigroup {T⃗_{t}} acting on Hilbert space valued square integrable functions. We denote the norm of the Hilbert space by | · |. If we have

|T⃗_{t}u| ≤ T_{t}|u|, ∀u ∈ L^{2},  (1.1)

we say that the semigroup {T⃗_{t}} is dominated by {T_{t}}. We are interested in when this inequality holds.

A necessary and sufficient condition for (1.1) is given by the abstract Kato theorem due to B. Simon [18, 19]. Later E. Ouhabaz [13] gave a necessary and sufficient condition in terms of bilinear form under the sector condition. In my previous paper [16], we discuss this problem in the framework of square field operator. Typical example to which our theorem is applicable is the Hodge-Kodaira Laplacian for differential forms on a Riemannian manifold (see also [10] in this direction).

In this paper, we consider the Hodge-Kodaira Laplacian on a Riemannian manifold with boundary. We consider two kinds of boundary condition: the relative boundary condition and the absolute boundary condition. In this case, we cannot apply the result in [16] and so we generalize the notion of Γ⃗ that corresponds to the Bakry-Emery Γ_{2}. So far, Γ⃗ is an L^{1} function. But in our formulation, it is no more a function; it is a smooth measure in the sense of Dirichlet form. The positivity of the smooth measure is essential. Using this notion, we give a sufficient condition to (1.1).

We remark that this kind of problem was also discussed by Donnelly-Li [6]. They proved the heat kernel domination. We take a different approach. Méritet [12] also proved the cohomology vanishing theorem.

1991 Mathematics Subject Classification. 60J60, 58G32.

Key words and phrases. semigroup domination, Hodge-Kodaira Laplacian, absolute boundary condition, relative boundary condition.

The organization of this paper is as follows. In Section 2, we give a generalization of Γ⃗ and prepare the general theory for the semigroup domination. To do this, we use the Ouhabaz criterion for the semigroup domination. We also discuss the theory of semigroup commutation in Section 3. Combining the semigroup domination with the semigroup commutation, we can reformulate the Bakry-Emery criterion for the logarithmic Sobolev inequality. In Section 4, we consider the Hodge-Kodaira Laplacian and apply our theory to it. To give a sufficient condition for the domination, the second fundamental form on the boundary is crucial.

2. Square field operator and the contraction semigroup

In this section, we discuss the contraction semigroup in the framework of the square field operator, which is called “opérateur carré du champ” in the French literature. Our main interest is the semigroup domination on a Riemannian manifold with boundary in Section 4, but we prepare a general theory in this section.

Let (X, B) be a measurable space and m be a σ-finite measure. Suppose we are given a strongly continuous contraction symmetric semigroup {T_{t}} on L^{2}. We further assume that T_{t} is Markovian, i.e., if f ∈ L^{2} satisfies 0 ≤ f ≤ 1, then 0 ≤ T_{t}f ≤ 1. Here these inequalities hold a.e. But we do not specify ‘a.e.’ either in the sequel. We denote the generator by A and the associated Dirichlet form by E.

We note that we can regard {T_{t}} as a semigroup on L^{p}(m), p ≥ 1 by the Riesz-Thorin interpolation theorem. We denote the generator of {T_{t}} on L^{p}(m) by A_{p}.

To introduce the square field operator, we assume the following condition (see Bouleau-Hirsch [3, Chapter 1, §4] for details): (Γ) For f, g ∈ Dom(A_{2}), we have f · g ∈ Dom(A_{1}).

Under the above assumption, we set

Γ(f, g) = \frac{1}{2}\{A_{1}(f · g) − A_{2}f · g − f · A_{2}g\}.  (2.1)

Here we remark that our definition of Γ is different from that of [3] up to a constant.

Furthermore we suppose that E has the local property in the following sense (see [3, Definition I.5.1.2]): (L) For any real valued function f ∈ Dom(E), F, G ∈ C_{0}^{∞}(ℝ),

supp F ∩ supp G = ∅ ⟹ E(F_{0}(f), G_{0}(f) = 0,  (2.2)

where F_{0}(x) = F (x) − F (0), G_{0}(x) = G(x) − G(0).

The above condition is satisfied as soon as it is satisfied for each element of a dense subset of Dom(E). In particular, the following identity is most commonly used: For f, g ∈ Dom(E) ∩ L^{∞},

Γ(fg, h) = fΓ(g, h) + gΓ(f, h) for ∀h ∈ Dom(E).  (2.3)

In addition, we are given a strongly continuous semigroup {T⃗_{t}} on L^{2}(m; K) where L^{2}(m; K) is the set of all square integrable K-valued functions, K being a separable Hilbert space. We denote the generator by A⃗ (or A⃗_{2} to specify the space L^{2}(m; K)). We also assume that {T⃗_{t}} is symmetric and is associated with a bilinear form E⃗. Of course, E⃗ is bounded from below.

Our interest is the following semigroup domination:

|T⃗_{t}u| ≤ T_{t}|u|, ∀u ∈ L^{2},  (2.4)

To give a sufficient condition for (2.4), we now define the square field operator for {T⃗_{t}}. To do this, we imposed the following condition in [16]: (Γ⃗_{λ}) For u, v ∈ Dom(A⃗_{2}), we have (u|v)_{K} ∈ Dom(A_{1}) and there exists λ ∈ ℝ such that

A_{1}|u|^{2} − 2(A⃗_{2}u|u)_{K} + 2λ|u|^{2} ≥ 0.  (2.5)

Under the above condition we define Γ⃗ by

Γ⃗(u, v) = \frac{1}{2}\{A_{1}(u|v)_{K} − (A⃗_{2}u|v)_{K} − (u|A⃗_{2}v)_{K}\}.  (2.6)

The condition (Γ⃗_{λ}) is rather restrictive. In particular we have to assume that (u|v)_{K} ∈ Dom(A_{1}) for u, v ∈ Dom(A⃗_{2}). We can define Γ⃗ without this condition. In fact, under the above definition, a formal calculation leads to

−E((u, v), f) + E⃗(fu, v) + E⃗(u, fv) = 2 ∫_{X} Γ⃗(u, v)fdm.  (2.7)

In the above expression it is only necessary to assume that (u, v) ∈ Dom(E), fu, fv ∈ Dom(E⃗).

Keeping in mind this observation, we modify the condition (Γ⃗_{λ}) as follows:

(Γ⃗′_{λ}-1) For u, v ∈ Dom(E⃗) ∩ L^{∞} and f ∈ Dom(E) ∩ L^{∞}, we have (u, v) ∈ Dom(E), fu, fv ∈ Dom(E⃗) and moreover Dom(E⃗) ∩ L^{∞} is dense in Dom(E⃗).

(Γ⃗′_{λ}-2) There exists Γ⃗: Dom(E⃗) ∩ L^{∞} × Dom(E⃗) ∩ L^{∞} → L^{1}, a smooth measure σ, and a non-negative symmetric tensor γ⃗ (i.e, γ⃗ is an H^{∗} ⊗ H^{∗}-valued function on X) such that

−E((u, v), f) + E⃗(fu, v) + E⃗(u, fv) = 2 ∫_{X} Γ⃗(u, v)fdm + 2 ∫_{X} γ⃗(ũ, ṽ)f̃dσ
∀f ∈ Dom(E) ∩ L^{∞}, ∀u, v ∈ Dom(E⃗) ∩ L^{∞}.  (2.8)

Here ũ, ṽ, f̃ are quasi-continuous modification of u, v, f, respectively. Of course, we have assumed that each element of Dom(E⃗) admits a quasi-continuous modification.

(Γ⃗′_{λ}-3) For the real constant λ, it holds that

Γ⃗(u, u) + λ|u|^{2} ≥ 0 for u ∈ Dom(A⃗_{2}).  (2.9)

(Γ⃗′_{λ}-4) It holds that

E⃗(u, v) = ∫_{X} Γ⃗(u, v)dm + ∫_{X} γ⃗(ũ, ṽ)dσ ∀u, v ∈ Dom(E⃗) ∩ L^{∞}.  (2.10)

In the sequel, (Γ⃗′_{λ}) refers to these four conditions. We emphasize that we assumed the positivity of γ⃗ and σ. Since Dom(E⃗) ∩ L^{∞} is dense in Dom(E⃗), it is easy to see that Γ⃗ is well-defined on Dom(E⃗) × Dom(E⃗) as a continuous bilinear mapping into L^{1} and (2.10) holds for u, v ∈ Dom(E⃗)

Next we introduce the following condition on Γ⃗ and E⃗. It was already appeared in [16]. It is related to the derivation property of Γ⃗. (D⃗) For u, v ∈ Dom(E⃗) ∩ L^{∞}, f ∈ Dom(E) ∩ L^{∞}, it holds that

2fΓ⃗(u, v) = −Γ(f, (v|u)_{K}) + Γ⃗(u, fv) + Γ⃗(fu, v),  (2.11)

From this, we have

2fΓ⃗(u, u) = −Γ(f, |u|^{2}) + 2Γ⃗(u, fu).  (2.12)

Further, by (2.8), we have

−E((u, u), f) + 2E⃗(fu, u) = 2 ∫_{X} Γ⃗(u, u)fdm + 2 ∫_{X} γ⃗(ũ, ũ)f̃dσ.  (2.13)

Substituting (2.12) into this equation, we have

−E((u, u), f) + 2E⃗(fu, u) = − ∫_{X} Γ(f, |u|^{2})dm + 2 ∫_{X} Γ⃗(fu, u)dm + 2 ∫_{X} γ⃗(ũ, ũ)f̃dσ
= −E((u, u), f) + 2 ∫_{X} Γ⃗(fu, u)dm + 2 ∫_{X} γ⃗(ũ, ũ)f̃dσ

Thus we have

E⃗(fu, u) = ∫_{X} Γ⃗(fu, u)dm + ∫_{X} γ⃗(ũ, ũ)f̃dσ.  (2.14)

In particular, for f ≥ 0, it holds that

E⃗(fu, u) ≥ ∫_{X} Γ⃗(fu, u)dm.  (2.15)

We remark that if 1 ∈ Dom(E), (2.10) follows from (2.14). Here 1 denotes the function identically equal to 1.

We can give a sufficient condition for (2.4) in terms of square field operator. To do this, we need the following Ouhabaz’ criterion. For the semigroup domination (2.4), the following condition is necessary and sufficient:

1. If u ∈ Dom(E), then |u| ∈ Dom(E) and

E(|u|, |u|) ≤ E⃗(u, u) + λ∥u∥_{2}^{2}.  (2.16)

2. If u ∈ Dom(E⃗) and f ∈ Dom(E) with 0 ≤ f ≤ |u|, then f sgn u ∈ Dom(E⃗) and

E(f, |u|) ≤ E⃗(u, f sgn u) + λ(f, |u|)  (2.17)

where sgn u = u/|u|. Now we have the following theorem.

Theorem 2.1. Assume conditions (Γ), (L), (Γ⃗_{λ}) and (D⃗). Then, for u ∈ Dom(E⃗) we have |u| ∈ Dom(E) and

Γ(|u|, |u|) ≤ Γ⃗(u, u) + λ|u|^{2}.  (2.18)

In addition, for f ∈ Dom(E) ∩ L^{∞} and u ∈ Dom(E⃗) ∩ L^{∞}, we have

{Γ⃗(fu, fu) + λ|fu|^{2}}^{1/2} ≤ |f|{Γ⃗(u, u) + λ|u|^{2}}^{1/2} + |u|Γ(f, f)^{1/2}.  (2.19)

Furthermore we have the following semigroup domination:

|T⃗_{t}u| ≤ e^{λt}T_{t}|u| for u ∈ L^{2}(X; K)  (2.20)

Proof. Many parts are the same as in [16]. We give a proof for completeness. For simplicity, we give a proof in the case λ = 0. Take u ∈ Dom(E⃗) ∩ L^{∞} and f ∈ Dom(E) ∩ L^{∞}. From the assumption, v = fu ∈ Dom(E⃗) and we substitute v in (2.11):

Γ⃗(fu, fu) + Γ⃗(u, |f|^{2}u) = 2fΓ⃗(u, fu) + Γ(f, f|u|^{2}).

Hence

Γ⃗(fu, fu) = −Γ⃗(u, |f|^{2}u) + 2fΓ⃗(u, fu) + Γ(f, f|u|^{2})

= −|f|^{2}Γ⃗(u, u) − \frac{1}{2}Γ(|f|^{2}, |u|^{2}) + f\{2fΓ⃗(u, u) + Γ(f, |u|^{2})\} + Γ(f, f|u|^{2})

= −|f|^{2}Γ⃗(u, u) − \frac{1}{2}fΓ(f, |u|^{2}) − \frac{1}{2}fΓ(f, |u|^{2}) + 2|f|^{2}Γ⃗(u, u)

+ fΓ(f, |u|^{2}) + fΓ(f, |u|^{2}) + |u|^{2}Γ(f, f)

= |f|^{2}Γ⃗(u, u) + \frac{1}{2}fΓ(f, |u|^{2}) + \frac{1}{2}fΓ(f, |u|^{2}) + |u|^{2}Γ(f, f).

In particular, if we take f = |u|^{2}, we get

Γ⃗(|u|^{2}u, |u|^{2}u) = |u|^{4}Γ⃗(u, u) + 2|u|^{2}Γ(|u|^{2}, |u|^{2}).  (2.21)

On the other hand, substituting v = u and f = |u|^{2} in (2.11), we have

2|u|^{2}Γ⃗(u, u) + Γ(|u|^{2}, |u|^{2}) = 2Γ⃗(|u|^{2}u, u).

Taking square,

4|u|^{4}Γ⃗(u, u)^{2} + 4|u|^{2}Γ⃗(u, u)Γ(|u|^{2}, |u|^{2}) + Γ(|u|^{2}, |u|^{2})^{2}
= 4Γ⃗(|u|^{2}u, u)^{2}
≤ 4Γ⃗(|u|^{2}u, |u|^{2}u)Γ⃗(u, u) (by the Schwarz inequality)
= 4{|u|^{4}Γ⃗(u, u) + 2|u|^{2}Γ(|u|^{2}, |u|^{2})}Γ⃗(u, u) (∵ (2.21))
= 4|u|^{4}Γ⃗(u, u)^{2} + 8|u|^{2}Γ(|u|^{2}, |u|^{2})Γ⃗(u, u).

Thus we have

Γ(|u|^{2}, |u|^{2}) ≤ 4|u|^{2}Γ⃗(u, u).  (2.22)

Now for ε > 0, set φ_{ε}(t) = \sqrt{t + ε^{2}} − ε. Then by the derivation property and (2.22),

Γ(φ_{ε}(|u|^{2}), φ_{ε}(|u|^{2})) ≤ \frac{1}{4(|u|^{2} + ε^{2})}Γ(|u|^{2}, |u|^{2}) ≤ \frac{4|u|^{2}}{4(|u|^{2} + ε^{2})}Γ⃗(u, u) ≤ Γ⃗(u, u).

From this we can show that {φ_{ε}(|u|^{2})}_{ε>0} is a bounded set in Dom(E). In fact

E(φ_{ε}(|u|^{2}), φ_{ε}(|u|^{2})) = ∫_{X} Γ(φ_{ε}(|u|^{2}), φ_{ε}(|u|^{2}))dm

≤ ∫_{X} Γ⃗(u, u)dm

≤ E⃗(u, u).

Hence we can take a sequence {ε_{j}} tending to 0 such that {φ_{ε_{j}}(|u|^{2})}_{j} converges weakly in Dom(E). The limit is |u| since it converges to |u| strongly in L^{2}. This means that |u| ∈ Dom(E). Further, taking a subsequence if necessary, we may assume that the Cesaro mean converges strongly in Dom(E), i.e.,

\frac{1}{n} ∑_{j=1}^{n} φ_{ε_{j}}(|u|^{2}) → |u| strongly in Dom(E).

By the continuity of Γ, we have

Γ(\frac{1}{n} ∑_{j=1}^{n} φ_{ε_{j}}(|u|^{2}), \frac{1}{n} ∑_{j=1}^{n} φ_{ε_{j}}(|u|^{2})) → Γ(|u|, |u|) strongly in L^{1}.

On the other hand, by the Minkowski inequality, it follows that

Γ(\frac{1}{n} ∑_{j=1}^{n} φ_{ε_{j}}(|u|^{2}), \frac{1}{n} ∑_{j=1}^{n} φ_{ε_{j}}(|u|^{2}))^{1/2} ≤ \frac{1}{n} ∑_{j=1}^{n} Γ(φ_{ε_{j}}(|u|^{2}), φ_{ε_{j}}(|u|^{2}))^{1/2} ≤ Γ⃗(u, u)^{1/2}.

Therefore we have

Γ(|u|, |u|) ≤ Γ⃗(u, u).  (2.23)

In fact, it is enough to take a sequence {ε_{n}} such that the Cesaro mean of {φ_{ε_{n}}(|u|^{2})} converges to |u| in Dom(E).

Now we return to Γ⃗(fu, fu):

Γ⃗(fu, fu) = |f|^{2}Γ⃗(u, u) + \frac{1}{2}fΓ(f, |u|^{2}) + \frac{1}{2}fΓ(f, |u|^{2}) + |u|^{2}Γ(f, f)
≤ |f|^{2}Γ⃗(u, u) + |f|Γ(|u|^{2}, |u|^{2})^{1/2}Γ(f, f)^{1/2} + |u|^{2}Γ(f, f)
≤ |f|^{2}Γ⃗(u, u) + 2|f||u|Γ⃗(u, u)^{1/2}Γ(f, f)^{1/2} + |u|^{2}Γ(f, f)
= {|f|Γ⃗(u, u)^{1/2} + |u|Γ(f, f)^{1/2}}^{2}

which shows (2.19).

We show that for g ∈ Dom(E) ∩ L∞_{+} and u ∈ Dom(E⃗) ∩ L^{∞} we have

E⃗(u, gu) ≥ E(g|u|, |u|).  (2.24)

To see this,

E⃗(u, gu) ≥ ∫_{X} Γ⃗(u, gu)dm (∵ (2.15))

= ∫_{X}\{gΓ⃗(u, u) + \frac{1}{2}Γ(g, |u|^{2})\}dm

≥ ∫_{X}{gΓ(|u|, |u|) + |u|Γ(g, |u|)}dm

= ∫_{X} Γ(g|u|, |u|)dm

= E(g|u|, |u|).

Now we have to check Ouhabaz’ condition. Since we have assumed that Dom(E⃗) ∩ L^{∞} is dense in Dom(E⃗), for any u ∈ Dom(E⃗), we take a sequence {u_{n}} ⊆ Dom(E⃗) ∩ L^{∞} that converges to u in Dom(E⃗). Set

ψ_{ε}(t) = \frac{1}{1 + εt}, t ≥ 0.

Then ψ_{ε}(|u_{n}|)|u_{n}| ∈ Dom(E) and by (2.19) we have

Γ⃗(ψ_{ε}(|u_{n}|)u_{n}, ψ_{ε}(|u_{n}|)u_{n})^{1/2} ≤ ψ_{ε}(|u_{n}|)Γ⃗(u_{n}, u_{n})^{1/2} + |u_{n}|Γ(ψ_{ε}(|u_{n}|), ψ_{ε}(|u_{n}|))^{1/2}

≤ ψ_{ε}(|u_{n}|)Γ⃗(u_{n}, u_{n})^{1/2} + |u_{n}||ψ′_{ε}|Γ(|u_{n}|, |u_{n}|)^{1/2}

≤ \{ψ_{ε}(|u_{n}|) + \frac{ε|u_{n}|}{(1 + ε|u_{n}|)^{2}} \}Γ⃗(u_{n}, u_{n})^{1/2}

≤ 2ψ_{ε}(|u_{n}|)Γ⃗(u_{n}, u_{n})^{1/2}

≤ 2Γ⃗(u_{n}, u_{n})^{1/2}.

Now

E⃗(ψ_{ε}(|u_{n}|)u_{n}, ψ_{ε}(|u_{n}|)u_{n})
= ∫_{X} Γ⃗(ψ_{ε}(|u_{n}|)u_{n}, ψ_{ε}(|u_{n}|)u_{n})dm + ∫_{X} γ⃗(ψ_{ε}(|u_{n}|)u_{n}, ψ_{ε}(|u_{n}|)u_{n})dσ
≤ ∫_{X} Γ⃗(u_{n}, u_{n})dm + ∫_{X} γ⃗(u_{n}, u_{n})dσ
= E⃗(u_{n}, u_{n}).

Thus we have {ψ_{ε}(|u_{n}|)u_{n}} is E⃗_{1} bounded and converges to ψ_{ε}(|u|)u strongly in L^{2} by taking a subsequence if necessary. Therefore we have ψ_{ε}(|u|)u ∈ Dom(E⃗) ∩ L^{∞}.

For any f ∈ Dom(E) and u ∈ Dom(E⃗) with 0 ≤ f ≤ |u|. We further assume that u is bounded. Set

φ_{ε}(t) = \frac{1}{t + ε}

and

g = fφ_{ε}(|u|) = \frac{f}{|u| + ε}.

Clearly g ∈ Dom(E) ∩ L^{∞}. Now substituting g into (2.24), we have

E(|u|, \frac{f}{|u| + ε}|u|) ≤ E⃗(u, \frac{f}{|u| + ε}u).

We note that the boundedness of u is necessary to obtain this inequality. Letting ε → 0, we get the desired result. To do this, we note

Γ⃗(fφ_{ε}(|u|)u, fφ_{ε}(|u|)u)^{1/2} ≤ fφ_{ε}(|u|)Γ⃗(u, u) + |u|Γ(fφ_{ε}(|u|), fφ_{ε}(|u|))^{1/2} and further

Γ⃗(fφ_{ε}(|u|), fφ_{ε}(|u|))^{1/2} ≤ fΓ(φ_{ε}(|u|), φ_{ε}(|u|))^{1/2} + φ_{ε}(|u|)Γ(f, f)
≤ f|φ′_{ε}(|u|)|Γ(|u|, |u|)^{1/2} + φ_{ε}(|u|)Γ(f, f)
≤ φ_{ε}(|u|){Γ(|u|, |u|)^{1/2} + Γ(f, f)}.

Combining both of them, we have

Γ⃗(fφ_{ε}(|u|)u, fφ_{ε}(|u|)u)^{1/2} ≤ fφ_{ε}(|u|)Γ⃗(u, u) + |u|φ_{ε}(|u|){Γ(|u|, |u|)^{1/2} + Γ(f, f)}
≤ 2Γ⃗(u, u)^{1/2} + Γ(f, f)^{1/2}.

Now we have {fφ_{ε}(|u|)u; ε > 0} is bounded in Dom(E⃗). Clearly fφ_{ε}(|u|)u → f sgn u in L^{2} and hence we have f sgn u ∈ Dom(E⃗). Moreover, we can take a sequence whose Cesaro mean converges strongly in Dom(E⃗), and hence we have

Γ⃗(f sgn u, f sgn u)^{1/2} ≤ 2Γ⃗(u, u)^{1/2} + Γ(f, f)^{1/2}  (2.25)

and

E(|u|, f) ≤ E⃗(u, f sgn u).  (2.26)

Lastly we remove the restriction of the boundedness of u. So now we do not assume that u is bounded. So first we notice that

ψ_{ε}(|u|)|u| ≥ ψ_{ε}(f)f

and both are bounded. We take a sequence ε_{j} converging to 0 such that

u_{N} = \frac{1}{N} ∑_{j=1}^{N} ψ_{ε_{j}}(|u|)u → u strongly in Dom(E⃗),

f_{N} = \frac{1}{N} ∑_{j=1}^{N} ψ_{ε_{j}}(f)f → f strongly in Dom(E).

Now u_{N} is bounded and f_{N} ≤ |u_{N}| and hence, by using (2.26) we have

E(|u_{N}|, f_{N}) ≤ E⃗(u_{N}, f_{N} sgn u_{N}).  (2.27)

By taking a subsequence if necessary, we may assume that {|u_{N}|} converges to |u| weakly in Dom(E) and {f_{N} sgn u_{N}} converges to f sgn u weakly in Dom(E⃗). Hence, by taking limit, we have

E(|u|, f) ≤ E⃗(u, f sgn u)  (2.28)

which is the desired result. This completes the proof.

As before, it is sufficient to assume (D⃗) for elements of a core. We state it as a proposition. We also include an analogue of (Γ′).

Proposition 2.2. Assume that the assumptions (Γ) and (L) hold. Furthermore assume that there exist an algebra C ⊆ Dom(E) ∩ L^{∞} and a subspace D ⊆ Dom(A⃗_{2}) ∩ L^{∞} such that fu ∈ D (i.e, D is a C-module), (u, v) ∈ C for f ∈ C, u, v ∈ D and further (Γ⃗′_{λ}) and (D⃗) hold for f ∈ C, u, v ∈ D, e.g., (Γ⃗′_{λ}-2) means that any element in D admits a quasi-continuous modification and (2.8) holds. We also suppose that C is a core for E and D is a core for E⃗.

Then (Γ⃗_{λ}) and (D⃗) are satisfied for f ∈ Dom(E) ∩ L^{∞}, u, v ∈ Dom(E⃗) ∩ L^{∞}.

Proof. We may suppose λ = 0. Take u ∈ D and f ∈ C. From the assumption, v = fu ∈ D. Now by the same argument as in the proof of Theorem 2.1, we have

Γ⃗(fu, fu) = f^{2}Γ⃗(u, u) + \frac{1}{2}fΓ(f, |u|^{2}) + \frac{1}{2}fΓ(f, |u|^{2}) + |u|^{2}Γ(f, f)
≤ f^{2}Γ⃗(u, u) + |f|Γ(f, f)^{1/2}Γ(|u|^{2}, |u|^{2})^{1/2} + |u|^{2}Γ(f, f).

Clearly this shows that

E⃗(fu, fu) = ∫_{X} Γ⃗(fu, fu)dm + ∫_{X} γ⃗(fu, fu)dσ
≤ ∥f∥_{∞}^{2} ∫_{X} Γ⃗(u, u)dm + ∥f∥_{∞} ∫_{X} Γ(f, f)^{1/2}Γ(|u|^{2}, |u|^{2})^{1/2}dm
+ ∥u∥_{∞}^{2} ∫_{X} Γ(f, f)dm + ∥f∥_{∞}^{2} ∫_{X} γ⃗(u, u)dσ

≤ ∥f∥_{∞}^{2} ∫_{X} Γ⃗(u, u)dm + ∥f∥_{∞}{∫_{X} Γ(f, f)dm}^{1/2}{∫_{X} Γ(|u|^{2}, |u|^{2})dm}^{1/2}

+ ∥u∥_{∞}^{2} ∫_{X} Γ(f, f)dm + ∥f∥_{∞}^{2} ∫_{X} γ⃗(u, u)dσ
= ∥f∥_{∞}^{2}E⃗(u, u) + ∥f∥_{∞}E(f, f)^{1/2}E(|u|^{2}, |u|^{2})^{1/2} + ∥u∥_{∞}^{2}E(f, f).

We claim that for f ∈ Dom(E) ∩ L^{∞}, u ∈ D, we have fu ∈ Dom(E⃗) and (2.11), (2.8) hold. To show this, take any real valued function f ∈ C. Then, for any C^{1}-function F : ℝ → ℝ, we take a sequence of polynomials {P_{n}} such that P_{n} → F uniformly on any compact sets up to the first derivative. Since E⃗(P_{n}(f)u, P_{n}(f)u) is bounded and P_{n}(f)u → F (f)u in L^{2}. Thus we have F (f)u ∈ Dom(E⃗) and

2F (f)Γ⃗(u, v) = Γ(F (f), (u|v)_{K}) − Γ⃗(u, F (f)v) − Γ⃗(F (f)u, v)  (2.29)
E⃗(F (f)u, F (f)u) ≤ 2∥F (f)∥_{∞}^{2}E⃗(u, u) + 2∥F (f)∥_{∞}E(F (f), F (f))^{1/2}E(|u|^{2}, |u|^{2})^{1/2}
+ 2∥u∥_{∞}^{2}E(F (f)f, F (f)).  (2.30)

Now for any f ∈ Dom(E) ∩ L^{∞}, we take a bounded C^{1} function φ such that φ(t) = t for |t| ≤ ∥f∥_{∞}. Let {f_{n}} ⊆ C be a sequence converging to f in Dom(E). Clearly g_{n} = φ(f_{n}) → f weakly in Dom(E) and {g_{n}} is uniformly bounded. Moreover, by (2.30) we have sup_{n} E⃗(g_{n}u, g_{n}u) < ∞. Hence we can extract a subsequence {g_{n_{j}}u} whose Cesaro mean converges strongly in Dom(E⃗). Together with the fact that g_{n}u → fu in L^{2}, we can see that for f ∈ Dom(E) ∩ L^{∞} and u ∈ D, we have fu ∈ Dom(E⃗) and (2.11) hold.

Since (2.11) holds for u, v ∈ D and f ∈ Dom(E) ∩ L^{∞}, we can take f = |u|^{2}. By repeating the argument in the proof of Theorem 2.1, for u ∈ D and f ∈ Dom(E) ∩ L^{∞}, we have fu ∈ Dom(E⃗) and

Γ⃗(fu, fu)^{1/2} ≤ |f|Γ⃗(u, u)^{1/2} + |u|Γ(f, f)^{1/2}  (2.31)
Γ(|u|, |u|) ≤ Γ⃗(u, u)  (2.32)

Since D is dense in Dom(E⃗), it is easy to see that (2.32) holds for u ∈ Dom(E⃗).

Now we can prove (2.11) for f ∈ Dom(E)∩L^{∞} and u ∈ Dom(E⃗)∩L^{∞}. Take any u ∈ D. For ε > 0, set

ψ_{ε}(t) = \frac{1}{1 + εt}, t ∈ [0, ∞).

Note that ψ′_{ε} = −\frac{ε}{(1+εt)^{2}} . Since ψ_{ε}(|u|) ∈ Dom(E), we have ψ_{ε}(|u|)u ∈ Dom(E⃗), and

Γ⃗(fψ_{ε}(|u|)u, fψ_{ε}(|u|)u)^{1/2}
≤ |f|ψ_{ε}(|u|)Γ⃗(u, u)^{1/2} + |u|Γ(fψ_{ε}(|u|), fψ_{ε}(|u|))^{1/2}
≤ |f|ψ_{ε}(|u|)Γ⃗(u, u)^{1/2} + |u|{ψ_{ε}(|u|)^{2}Γ(f, f) + 2ψ_{ε}(|u|)ψ′_{ε}(|u|)fΓ(|u|, f)
+ f^{2}ψ′_{ε}(|u|)^{2}Γ(|u|, |u|)}^{1/2}

≤ |f|ψ_{ε}(|u|)Γ⃗(u, u)^{1/2} + |u|{ψ_{ε}(|u|)^{2}Γ(f) + 2ψ_{ε}(|u|)ψ′_{ε}(|u|)|f|Γ(|u|, |u|)^{1/2}Γ(f, f)^{1/2}

+ f^{2}ψ′_{ε}(|u|)^{2}Γ(|u|, |u|)}^{1/2}
≤ |f|ψ_{ε}(|u|)Γ⃗(u, u)^{1/2} + |u|ψ_{ε}(|u|)Γ(f, f)^{1/2}
+ 2ψ_{ε}(|u|)|ψ′_{ε}(|u|)||f|Γ(|u|, |u|)^{1/2}Γ(f, f)^{1/2} + f^{2}|ψ′_{ε}(|u|)|^{2}Γ(|u|, |u|)}^{1/2}
≤ |f|ψ_{ε}(|u|)Γ⃗(u, u)^{1/2} + |u|ψ_{ε}(|u|)Γ(f, f)^{1/2} + |u||f||ψ′_{ε}(|u|)|Γ⃗(u, u)^{1/2}.

Now for any v ∈ Dom(E⃗) ∩ L^{∞}, we take a sequence {u_{n}} ⊆ D converging to v in Dom(E⃗). Then by the above estimate, we can see that

sup_{n} E⃗(fψ_{ε}(|u_{n}|)u_{n}, fψ_{ε}(|u_{n}|)u_{n}) < ∞.

Hence, by virtue of Banach-Saks theorem, we have, by taking a subsequence if necessary, the Cesaro mean of {fψ_{ε}(|u_{n}|)u_{n}} converges to fψ_{ε}(|u|)u in Dom(E⃗). Moreover, by noting that

Γ⃗(\frac{1}{n} ∑_{k=1}^{n} fψ_{ε}(|u_{k}|)u_{k}, \frac{1}{n} ∑_{k=1}^{n} fψ_{ε}(|u_{k}|)u_{k})^{1/2}

≤ \frac{1}{n} ∑_{k=1}^{n} Γ⃗(fψ_{ε}(|u_{k}|)u_{k}, fψ_{ε}(|u_{k}|)u_{k})^{1/2}

≤ \frac{1}{n} ∑_{k=1}^{n}\{|f|ψ_{ε}(|u_{k}|)Γ⃗(u_{k}, u_{k})^{1/2} + |u_{k}|ψ_{ε}(|u_{k}|)Γ(f, f)^{1/2} + |u_{k}||f||ψ′_{ε}(|u_{k}|)|Γ⃗(u_{k}, u_{k})^{1/2}\}.

Letting n → ∞, we have for v ∈ Dom(E⃗) ∩ L^{∞},

Γ⃗(fψ_{ε}(|u|)v, fψ_{ε}(|v|)v)^{1/2} ≤ |f|ψ_{ε}(|v|)Γ⃗(v, v)^{1/2} + |v|ψ_{ε}(|v|)Γ(f, f)^{1/2}
+ |v||f||ψ′_{ε}(|v|)|Γ⃗(v, v)^{1/2}.

Next, letting ε → 0, we eventually obtain that fv ∈ Dom(E⃗) and

Γ⃗(fv, fv)^{1/2} ≤ |f|Γ⃗(v, v)^{1/2} + |v|Γ(f, f)^{1/2}.

Next, we see that (2.11) holds for u, v ∈ Dom(E⃗) ∩ L^{∞} and f ∈ Dom(E) ∩ L^{∞}. To see this, we first note the following identity: for u, v ∈ D and f, ψ ∈ Dom(E),

2fΓ⃗(ψu, v) = Γ(f, (v|ψu)_{K}) − Γ⃗(ψu, fv) − Γ⃗(fψu, v).

This identity is clear from the assumption if ψ ∈ C. By an approximating argument, we can see it for ψ ∈ Dom(E) ∩ L^{∞}. Using this, we can repeat the above argument and get the desired result.

So far, we have obtained that Dom(E⃗) ∩ L^{∞} is a Dom(E) ∩ L^{∞}-module. We can also prove (2.8). Here we remark that every element of Dom(E⃗) admits a quasi-continuous modification. To see this, we note that for u ∈ D,

Cap_{1}(|u| ≥ ε) ≤ \frac{1}{ε^{2}} E_{1}(|u|, |u|)

≤ \frac{1}{ε^{2}} ∫_{X} Γ(|u|, |u|)dm + ∫_{X} |u|^{2}dm

≤ \frac{1}{ε^{2}} ∫_{X} Γ⃗(u, u)dm + ∫_{X} |u|^{2}dm

≤ \frac{1}{ε^{2}} E⃗_{1}(u, u).

The rest is easy by a standard Borel-Cantelli argument.

This completes the proof.

3. Commutation relation

In this section, we discuss the commutation relation between two semigroups. Combining this with semigroup domination, we can discuss the logarithmic Sobolev inequality along the Bakry-Emery argument.

Let B and B̂ be Banach spaces. We assume that B̂ is reflexive. Suppose we are given strongly continuous semigroups {T_{t}} on B and {T̂_{t}} on B̂. We denote their generators by A and Â, respectively. We also denote the resolvents of A and Â by G_{α} and Ĝ_{α}, respectively. Further, we are given a closed linear operator D from B to B̂ with the dense domain Dom(D). The dual operator of D is denoted by D^{∗}. We assume that Dom(A) ⊆ Dom(D). Under this condition, we have the following theorem.

Theorem 3.1. Then the following conditions are equivalent to each other.

(i) Dom(Â^{∗}) ⊆ Dom(D^{∗}) and

⟨Au, D^{∗}θ⟩ = ⟨Du, Â^{∗}θ⟩, ∀u ∈ Dom(A), ∀θ ∈ Dom(Â^{∗}).  (3.1)

(ii) For sufficiently large α,

DG_{α}u = Ĝ_{α}Du, ∀u ∈ Dom(D).  (3.2)

(iii) For any t ≥ 0, T_{t} Dom(D) ⊆ Dom(D) and

DT_{t}u = T̂_{t}Du, ∀u ∈ Dom(D).  (3.3)

Proof. First we show the implication (i) ⇒ (ii). We note that {T̂_{t}^{∗}} is also a strongly continuous semigroup with the generator Â^{∗} from the reflexivity of B̂. From (i), we have, for u ∈ Dom(A) and θ ∈ Dom(Â^{∗}),

⟨(α − A)u, D^{∗}θ⟩ = ⟨Du, (α − Â^{∗})θ⟩.

Substituting u = G_{α}v, θ = Ĝ_{α}^{∗}ξ, we have

⟨v, D^{∗}Ĝ_{α}^{∗}ξ⟩ = ⟨DG_{α}v, ξ⟩.

If v ∈ Dom(D), it follows

⟨Ĝ_{α}Dv, ξ⟩ = ⟨DG_{α}v, ξ⟩

which implies (ii).

To show (ii) ⇒ (iii), we note that for any integer n,

DG_{α}^{n}u = Ĝ_{α}^{n}Du

which can be easily shown by the induction on n. Since the semigroup can be expresses in terms of the resolvents as

T_{t}u = lim_{n→∞}(\frac{n}{t}G_{nt})^{n}u,

we have for u ∈ Dom(D),

T̂_{t}Du = lim_{n→∞}(\frac{n}{t}Ĝ_{nt})^{n}Du

= lim_{n→∞} D(\frac{n}{t}Ĝ_{nt})^{n}u.

This, combined with the closeness of D, implies T_{t}u ∈ Dom(D) and T̂_{t}Du = DT_{t}u. The implication (iii) ⇒ (ii) can be similarly shown by noting

G_{α}u = ∫_{0}^{∞} e^{−αt}T_{t}udt.

Lastly we show (ii) ⇒ (i). From assumption Dom(A) ⊆ Dom(D), the closed operator S = DG_{α} is defined whole space B and hence bounded. Hence, for u ∈ Dom(D), θ ∈ B̂^{∗}, we have

⟨u, S^{∗}θ⟩ = ⟨Su, θ⟩ = ⟨DG_{α}u, θ⟩ = ⟨Ĝ_{α}Du, θ⟩ = ⟨Du, Ĝ_{α}^{∗}θ⟩.

Now, setting θ = (α − Â^{∗})ξ, ξ ∈ Dom(Â^{∗}), we have

⟨u, S^{∗}(α − Â^{∗})ξ⟩ = ⟨Du, ξ⟩.

From this, it follows ξ ∈ Dom(D^{∗}). Thus, for ξ ∈ Dom(Â^{∗}), it holds that

⟨Su, (α − Â^{∗})ξ⟩ = ⟨u, D^{∗}ξ⟩, ∀u ∈ Dom(D).  (3.4)

Since Dom(D) is dense in B, the above identity holds for all u ∈ B. In particular, if we set u = (α − A)v, v ∈ Dom(A), we have

⟨S(α − A)v, (α − Â^{∗})ξ⟩ = ⟨(α − A)v, D^{∗}ξ⟩.

By noting S(α − A)v = DG_{α}(α − A)v = Dv, we get (i). This completes the proof.

Next we give an sufficient condition for the above theorem. We consider the same situation as in Theorem 3.1. In the sequel, any domain of an operator is equipped with a topology given by the graph norm.

Proposition 3.2. Suppose that there exist a subspace D ⊂ Dom(A) satisfying one of the following conditions:

(i) AD ⊆ Dom(D), DD ⊆ Dom(Â), D is dense in Dom(A) and Dom(Â^{∗}) ⊆ Dom(D^{∗}), Further for any u ∈ D, it holds that

DAu = ÂDu.  (3.5)

(ii) AD ⊆ Dom(D), DD ⊆ Dom(Â) and (α − A)D is dense in Dom(D) for sufficiently large α. Further (3.5) holds for any u ∈ D.

Then, one of (and hence all) three conditions in Theorem 3.1 holds.

Proof. First we assume (i). Then, for any u ∈ D and θ ∈ Dom(Â^{∗}), we have

⟨Au, D^{∗}θ⟩ = ⟨DAu, θ⟩ = ⟨ÂDu, θ⟩ = ⟨Du, Â^{∗}θ⟩.

Since D is dense in Dom(A), we can see that ⟨Au, D^{∗}θ⟩ = ⟨Du, Â^{∗}θ⟩ holds for u ∈ Dom(A) and θ ∈ Dom(Â^{∗}). Here, we use that D is a bounded operator from Dom(A) to B̂. Hence (i) in Theorem 3.1 holds.

Next assume (ii). From the assumption, we have for u ∈ D and θ ∈ Dom(Â^{∗}),

⟨D(α − A)u, θ⟩ = ⟨(α − Â)Du, θ⟩ = ⟨Du, (α − Â^{∗})θ⟩.

If we see v = (α − A)u ∈ (α − A)D, it follows that

⟨Dv, θ⟩ = ⟨DG_{α}v, (α − Â^{∗})θ⟩.

Now, using the denseness of (α − A)D in Dom(D), the above identity holds for all v ∈ Dom(D). Further, as was seen in the proof of Theorem 3.1, S = DG_{α} is a bounded operator on B, we have

⟨Dv, θ⟩ = ⟨v, S^{∗}(α − Â^{∗})θ⟩ ∀v ∈ Dom(D).

Then it follows θ ∈ Dom(D^{∗}) which implies Dom(Â^{∗}) ⊆ Dom(D^{∗}). Hence, this case is reduce to the case (i).

Now, we take Hilbert spaces H and Ĥ in place of Banach spaces B and B̂. Moreover generators A and Â are self-adjoint and associated with the closed quadratic forms E and Ê, respectively. Of course, E and Ê are bounded from below. Then we have the following;

Proposition 3.3. Suppose that there exist a subspace D ⊂ Dom(A^{3/2}) satisfying the following conditions. Dom(E) ⊆ Dom(D), DD ⊆ Dom(Â), D is dense in Dom(A^{3/2}), and it holds that

DAu = ÂDu, ∀u ∈ D.  (3.6)

Then, one of (and hence all) three conditions in Theorem 3.1 holds.

Proof. From the assumption D ⊆ Dom(A^{3/2}), we have (α−A)D ⊆ Dom(A^{1/2}) = Dom(E). Moreover (α − A)D is dense in Dom(E) by virtue of the denseness of D in Dom(A^{3/2}). By an argument similar to the proof of Proposition 3.2 (ii), we can get the conclusion.

We can also give another sufficient condition.

Proposition 3.4. We assume the following conditions:

D ⊆ Dom(A) ⊆ Dom(E) ⊆ Dom(D),
D̂ ⊆ Dom(Â) ⊆ Dom(Ê) ⊆ Dom(D^{∗}),
D Dom(A) ⊆ Dom(Ê),
D^{∗}D̂ ⊆ Dom(E).

D is a dense subspace of Dom(E) and D̂ is a dense subspace of Dom(Ê). Further, we assume that it holds

(Au, D^{∗}θ) = (Du, Âθ), for u ∈ D, θ ∈ D̂.  (3.7)

Then, one of (and hence all) three conditions in Theorem 3.1 holds.

Proof. Take any u ∈ D, θ ∈ D̂. From the assumption, D^{∗}θ ∈ Dom(E) and hence

E(u, D^{∗}θ) = (Au, D^{∗}θ) = (Du, Âθ).

Now using the denseness of D in Dom(E), we have

E(u, D^{∗}θ) = (Du, Âθ), for u ∈ Dom(E), θ ∈ D̂.

In particular, Du ∈ Dom(Ê) for u ∈ Dom(Ê), we have, for θ ∈ D̂

(Au, D^{∗}θ) = E(u, D^{∗}θ) = (Du, Âθ), = Ê(Du, θ).

Since D̂ is dense in Dom(Ê), we obtain

(Au, D^{∗}θ) = Ê(Du, θ) for u ∈ Dom(A), θ ∈ Dom(Ê).

Taking θ from Dom(Â) ⊆ Dom(E), it holds that

(Au, D^{∗}θ) = Ê(Du, θ) = (Du, Âθ), for u ∈ Dom(A), θ ∈ Dom(Â),

which is the desired result.

Remark 3.1. The condition (3.7) can be replaced by DAu = ÂDu for u ∈ D. Of course in this case we assume that they are well-defined.

In many applications, E is given by

E(u, v) = (Du, Dv)Ĥ.  (3.8)

Hence the assumption Dom(D) ⊆ Dom(E) is automatic, i.e., the identity holds.

Now we formulate the Bakry-Emery criterion for the logarithmic Sobolev inequality in our setting. Let the notations be as before and we impose the following conditions. The Hilbert space H is L^{2}(X, m) on a measure space (X, m). We assume that m is a probability measure. The quadratic form E is a local Dirichlet form with 1 ∈ Dom(E) and E(1, 1) = 0. D is a closed operator (usually first order differential operator) from L^{2}(X, m) to the space of L^{2} sections of a vector bundle E over X. We assume, for the notational simplicity, that E is a trivial bundle with a fiber K that is a Hilbert space. We denote ∇ instead of D. We assume that the square field operator is given by

Γ(f, g)(x) = (∇f(x), ∇g(x))_{K}  (3.9)

The Dirichlet form E is written as

E(f, g) = ∫_{X}(∇f(x), ∇g(x))_{K}dm  (3.10)

Suppose further that we are given another symmetric semigroup {T⃗_{t}} on L^{2}(m; K). We assume the following semigroup domination for which we have given a sufficient condition in the previous section.

|T⃗_{t}θ| ≤ e^{−λt}T_{t}|θ|  (3.11)

Lastly we assume that the following commutativity condition:

T⃗_{t}∇f = ∇T_{t}f.  (3.12)

As we have seen, we already have sufficient conditions for this.

Under the above conditions, we can show the logarithmic Sobolev inequality for the Dirichlet form E. Before that, let us see the relation to the Bakry-Emery Γ_{2}-criterion. Suppose the following commutativity condition on a space D,

∇Af = A⃗∇f, ∀f ∈ D.  (3.13)

Of course, we have to choose a suitable space D. We ignore the technical detail for a while. We also assume the condition (Γ⃗_{λ}). Hence Γ⃗ is realized as an L^{1} function. Under these conditions,

2Γ⃗(∇f, ∇g) = A_{1}(∇f, ∇g) − (A⃗∇f, ∇g) − (∇f, A⃗∇g)
= A_{1}(∇f, ∇g) − (∇Af, ∇g) − (∇f, ∇Ag)
= A_{1}Γ(f, g) − Γ(Af, g) − Γ(f, Ag).

The right hand side is usually called the Bakry-Emery Γ_{2}. So our condition (2.9) corresponds to

Γ_{2}(f, f) ≥ λΓ(f, f)

which is the famous Bakry-Emery criterion.

Now we can state the Bakry-Emery theorem in our formulation as follows:

Theorem 3.5. Under assumptions (3.11) and (3.12) for λ > 0, the following logarithmic Sobolev inequality holds:

∫_{X} f^{2} log(f^{2}/∥f∥_{2}^{2})dm ≤ \frac{2}{λ}E(f, f),  ∀f ∈ Dom(E).  (3.14)

Proof. From our assumption, it holds that

|T⃗_{t}ω| ≤ e^{−λt}T_{t}|ω|.

Therefore, taking ω = ∇f, we have

|∇T_{t}f| ≤ e^{−λt}T_{t}|∇f|.

Now the rest is the standard argument (see e.g., Deuschel-Stroock [4, Proof of Theorem 6.2.42]).

4. Riemannian manifold with boundary

In this section, we consider a Riemannian manifold with boundary.

Let M be a d-dimensional compact Riemannian manifold with a boundary ∂M. We assume that everything is smooth. As usual, the Hodge-Kodaira Laplacian is defined as follows:

□ = −(dδ + δd).  (4.1)

Here d is the exterior differentiation and δ is its (formal) dual. To specify that it acts p-forms, we denote it by □_{(p)}. Moreover we suppose that □ is defined on a set of all C^{∞} differential forms. We always assume that differential forms are always smooth, i.e., C^{∞}. Later we consider symmetric operators but they are all essentially self-adjoint on smooth differential forms (to be precise, under a suitable boundary condition) and hence it is enough to achieve formal calculation on smooth forms. We denote the p-th exterior bundle of T^{∗}M by ⋀^{p} T^{∗}M. The set of all smooth differential p-forms, i.e., smooth sections of ⋀^{p} T^{∗}M, will be denoted by A^{p}(M).

Since the manifold M has a boundary, we have to impose boundary condition. Several boundary conditions are known but we consider the two conditions: the absolute boundary condition and the relative boundary condition. To describe boundary condition, we introduce the following local coordinate (x^{1}, x^{2}, …, x^{d−1}, r) near the boundary: The local coordinate gives an diffeomorphism between a neighborhood in M and U ×[0, ε) ⊂ ℝ^{d−1} ×R_{+}. We identify the neighborhood in M with U × [0, ε) through the local coordinate. We assume that, on the boundary ∂M, r = 0 and \frac{∂}{∂x^{j}} ⊥ \frac{∂}{∂r}, j = 1, …, d−1. We denote the inner normal vector by N, i.e., N = \frac{∂}{∂r}. Further we assume that for fixed (x_{0}^{1}, … x_{0}^{d−1}) ∈ U, γ_{t} = (x_{0}^{1}, … x_{0}^{d−1}, t) is a geodesic with the velocity 1. For any 1-form θ on M, we can decompose it as

θ = θ_{t} + θ_{n} ∧ dr  (4.2)

where neither θ_{n} nor θ_{t} contains dr. This decomposition is well-defined on the boundary. Using this decomposition, we define

B_{r}(θ) = θ_{t}  (4.3)
B_{a}(θ) = θ_{n}.  (4.4)

Recall that θ_{t} and θ_{n} are defined only on ∂M. The suffix r stands for the relative boundary condition and a for the absolute boundary condition. We use these operators to define the boundary condition.

In the sequel, B stands for either B_{r} or B_{a}. Now we restrict the domain of □ according to boundary condition. Set

Dom(□_{(p)}^{B}) = {ϕ ∈ A^{p}(M); Bϕ = 0, B(d + δ)ϕ = 0}.  (4.5)

By an integration by parts formula for differential forms, we can see that □_{(p)}^{B} is symmetric and moreover it is essentially self-adjoint on Dom(□_{(p)}^{B}). We can think of the associated bilinear form E_{(p)}^{B}, which is given as follows:

Dom(E_{(p)}^{B}) = {ϕ ∈ A^{p}(M); Bϕ = 0}  (4.6)

and

E_{(p)}^{B}(ω, η) = (dω, dη) + (δω, δη).  (4.7)

Let see why this is so. We recall the integration by parts formula for differential forms. Take any ω ∈ A^{p}(M) and η ∈ A^{p+1}(M), then it holds that

∫_{M}(dω, η)dm = ∫_{M}(ω, δη)dm + ∫_{∂M} ω ∧ ∗η.  (4.8)

Here ∗ is the Hodge star operator which sends p-form to d − p-form. Now we have

−(□_{(p)}^{B_{a}}ω, η) = − ∫_{M}(□_{(p)}^{B_{a}}ω, η)dm
= ∫_{M}(dδω + δdω, η)dm
= ∫_{M}(δω, δη)dm + ∫_{∂M} δω ∧ ∗η + ∫_{M}(dω, dη)dm − ∫_{∂M} η ∧ ∗dω
= ∫_{M}(δω, δη)dm + ∫_{M}(dω, dη)dm

= E_{(p)}^{B_{a}}(ω, η).

Here dσ is the surface element of ∂M. We used the boundary condition B_{a} to get δω∧∗η = 0 on ∂M. To see this, ∗η contains dr since B_{a}η = 0. On the other hand, δω does not contain dr since B_{a}δω = 0. Hence δω ∧ ∗η contains dr. Therefore δω ∧ ∗η = 0 on ∂M since dr = 0 on ∂M. η ∧ ∗dω also vanishes on ∂M.

We can show similar identity for □_{(p)}^{B_{r}}

Next we see the commutation relation. Formally the commutation relation d□ = □d holds i.e., at least in the interior of M. But the boundary condition is involved and we need some arguments.

Take ω ∈ Dom(□_{(p)}^{B_{a}}), η ∈ Dom(□_{(p+1)}^{B_{a}})

− ∫_{M}(□ω, δη)dm = ∫_{M}(dδω + δdω, δη)dm
= ∫_{M}(δω + δdω, δδη)dm + ∫_{∂M} δω ∧ ∗δη
+ ∫_{M}(dω, dδη)dm − ∫_{∂M} δη ∧ ∗dw
= ∫_{M}(dω, dδη)dm + ∫_{M}(ddω, dη)dm
= ∫_{M}(dω, dδη)dm + ∫_{M}(dω, δdη)dm + ∫_{∂M} dω ∧ ∗dη
= − ∫_{M}(dω, □η)dm.

We used the boundary condition B_{a}. Absolutely same argument works for B_{r}.

Now invoking the essential self-adjointness of □_{(p)}^{B_{a}} and □_{(p)}^{B_{r}}, we can get the semigroup commutation. Of course, we must choose the same boundary condition.

We proceed to the issue of semigroup domination. For 0-forms, i.e., scalar functions, the boundary condition B_{a} corresponds to the Neumann condition: \frac{∂f}{∂r} = 0 on ∂M. On the other hand, B_{r} corresponds to the Dirichlet boundary condition: f = 0 on ∂M. From now on, for scalar functions, we always consider the Neumann Laplacian □_{(0)}^{B_{a}} and we denote it by Δ for simplicity. We also denote the associated Dirichlet form by E, i.e., E = E_{(0)}^{B_{a}}. We do not use the Dirichlet form (for scalar functions) with Dirichlet boundary condition. The reason is that the semigroup generated by Hodge-Kodaira Laplacian can not be dominated by the Dirichlet semigroup since the diffusion dies after hitting the boundary.

In order to apply our theorem, we have to check condition (Γ⃗_{λ}). In particular, to show (2.8), we have to calculate −E((θ, η), f) + E⃗(fθ, η) + E⃗(θ, fη).

First we note that

− ∫_{M} Δ(θ, η)fdm = ∫_{M} ∇^{∗}∇(θ, η)fdm
= ∫_{M} δd(θ, η)fdm
= ∫_{M}(d(θ, η), df)dm + ∫_{∂M} f⟨d(θ, η), N⟩dσ

(∵ Green-Stokes formula)

= E((θ, η), f) + ∫_{∂M} f∇_{N}(θ, η)dσ.

Here dσ denotes the surface element of ∂M and we recall that N is the inner normal vector. Hence

E((θ, η), f) = − ∫_{M} Δ(θ, η)fdm − ∫_{∂M} f∇_{N}(θ, η)dσ.

On the other hand, it holds that

Δ(θ, η) + (∇^{∗}∇θ, η) + (θ, ∇^{∗}∇η) = 2(∇θ, ∇η).

By the Weitzenböck formula −□_{(p)}^{B} = ∇^{∗}∇ + R_{(p)}, we have

Δ(θ, η) − (□_{(p)}^{B}θ, η) − (θ, □_{(p)}^{B}η) = 2(∇θ, ∇η) + 2R_{(p)}(θ, η).

We do not give the explicit form of R_{(p)}, but we just say that R_{(p)} can be written in terms of the curvature. Now by combining these identities, we can obtain the following identity:

−E((θ, η), f) + E_{(p)}^{B}(fθ, η) + E_{(p)}^{B}(θ, fη)
= 2 ∫_{M}(∇θ, ∇η)fdm + 2 ∫_{M} R_{(p)}(θ, η)fdm + 2 ∫_{∂M} ∇_{N}(θ, η)fdσ  (4.9)

In fact

−E((θ, η), f) + E_{(p)}^{B}(fθ, η) + E_{(p)}^{B}(θ, fη)
= ∫_{M} Δ(θ, η) · fdm + 2 ∫_{∂M} ∇_{N}(θ, η)fdσ − ∫_{M}{(□_{(p)}^{B}θ, η)f − (θ, □_{(p)}^{B}η)f}dm
= 2 ∫_{M}(∇θ, ∇η)fdm + 2 ∫_{M} R_{(p)}(θ, η)fdm + 2 ∫_{∂M} ∇_{N}(θ, η)fdσ

Therefore Γ⃗ in (Γ⃗′_{λ}-2) is given as

Γ⃗(θ, η) = (∇θ, ∇η) + R_{(p)}(θ, η).  (4.10)

We can easily check the condition (D⃗) in this case. The rest is to show that the third part of L.H.S. in (4.9) corresponds to γ⃗dσ in (2.8).

We take a local frame {ω^{1}, …, ω^{d−1}, dr} in T^{∗}M so that it forms a orthonormal basis and ω^{j} is parallel along the geodesic t → (x^{1}, …, x^{d−1}, t). We also denote the dual basis of {ω^{1}, …, ω^{d−1}, dr} by {X_{1}, …, X_{d−1}, N}. Note that On the other hand,

dω^{k}[X_{j}, N] = ⟨∇_{N}ω^{k}, X_{j}⟩ − ⟨∇_{X_{j}}ω^{k}, N⟩
= −⟨∇_{X_{j}}ω^{k}, N⟩

Here we used that ∇_{N}ω^{k} = 0 since ω^{k} is parallel along the path t ↦ (x^{1}, …, x^{d−1}, t). Moreover we note that ⟨∇_{X_{j}}ω^{k}, N⟩ = (α(X_{j}, X_{k}), N) where α is the second fundamental form on ∂M. Thus we have

dω^{k}[X_{j}, N] = −(α(X_{j}, X_{k}), N).  (4.11)

Using the second fundamental form, we introduce an operator A as follows:

Aω^{i} = ∑_{j=1}^{d−1}(α(X_{i}, X_{j}), N)ω^{j}.  (4.12)

A can be extended to a linear operator from A^{1}(∂M) to A^{1}(∂M) and it is independent of the choice of {ω^{j}}. Moreover we define dΓ(A): A^{p}(∂M) → A^{p}(∂M) as

dΓ(A)(θ_{1} ∧ ⋯ ∧ θ_{p}) = ∑_{j=1}^{p} θ_{1} ∧ ⋯ ∧ Aθ_{j} ∧ ⋯ ∧ θ_{p}.  (4.13)

Then we can have the follolwing. Lemma 4.1. On the boundary ∂M, it holds that for I = {i_{1} < ⋯ < i_{p}} ⊆ {1, …, d−1},

(dω^{I})_{n} = −(−1)^{|I|}dΓ(A)ω^{I}.  (4.14)

Proof. From (4.11) and (4.12), it holds that

∑_{k=1}^{d−1} dω^{j}[X_{k}, N]ω^{k} = −Aω^{j}.

Hence

dω^{I} = d(ω^{i_{1}} ∧ ⋯ ∧ ω^{i_{p}})

= ∑_{μ=1}^{p}(−1)^{μ}ω^{i_{1}} ∧ ⋯ ∧ dω^{i_{μ}} ∧ ⋯ ∧ ω^{i_{p}}

= ∑_{μ=1}^{p} ∑_{1≤k≤l≤d−1}(−1)^{μ}ω^{i_{1}} ∧ ⋯ ∧ dω^{i_{μ}}[X_{k}, X_{l}](ω^{k} ∧ ω^{l}) ∧ ⋯ ∧ ω^{i_{p}}

+ ∑_{μ=1}^{p} ∑_{k=1}^{d−1}(−1)^{μ}ω^{i_{1}} ∧ ⋯ ∧ dω^{i_{μ}}[X_{k}, N](ω^{k} ∧ dr) ∧ ⋯ ∧ ω^{i_{p}}

= ∑_{μ=1}^{p} ∑_{1≤k≤l≤d−1}(−1)^{μ}ω^{i_{1}} ∧ ⋯ ∧ dω^{i_{μ}}[X_{k}, X_{l}](ω^{k} ∧ ω^{l}) ∧ ⋯ ∧ ω^{i_{p}}

− (−1)^{p} ∑_{μ=1}^{p} ∑_{k=1}^{d−1} ω^{i_{1}} ∧ ⋯ ∧ Aω^{i_{μ}} ∧ ⋯ ∧ ω^{i_{p}} ∧ dr.

Now, recalling the definition of dΓ(A), we have

(dω^{I})_{n} = −(−1)^{p}dΓ(A)ω^{I}

which is the desired result.

Now we can deal with the absolute boundary condition. Theorem 4.2. For p-forms θ, η ∈ Dom(E_{(p)}^{B_{a}}) and f ∈ C^{∞}(M), it holds that

−E((θ, η), f) + E_{(p)}^{B_{a}}(fθ, η) + E_{(p)}^{B_{a}}(θ, fη)

= 2 ∫_{M}(∇θ, ∇η)fdm + 2 ∫_{M} R_{(p)}(θ, η)fdm + 2 ∫_{∂M}(dΓ(A)θ_{t}, η_{t})fdσ (4.15)

Proof. First we write the boundary condition in terms of local coordinate. We decompose θ as follows:

θ = ∑_{I:|I|=p} f_{I}ω^{I} + ∑_{J:|J|=p−1} g_{J}ω^{J} ∧ dr.

Noting that g_{J} = 0 on ∂M, we have,

dθ = ∑_{I:|I|=p} df_{I} ∧ ω^{I} + ∑_{I:|I|=p} f_{I}dω^{I}

+ ∑_{J:|J|=p−1} dg_{J} ∧ ω^{J} ∧ dr + ∑_{J:|J|=p−1} g_{J}dω^{J} ∧ dr

= ∑_{I:|I|=p} df_{I} ∧ ω^{I} + ∑_{I:|I|=p} f_{I}dω^{I}.

Since (dθ)_{n} = 0, we have

0 = ∑_{I:|I|=p}⟨df_{I}, N⟩dr ∧ ω^{I} + ∑_{I:|I|=p} f_{I}(dω^{I})_{n}

= ∑_{I:|I|=p}(−1)^{|I|}⟨df_{I}, N⟩ω^{I} ∧ dr − ∑_{I:|I|=p}(−1)^{|I|}f_{I}dΓ(A)ω^{I} ∧ dr

= ∑_{I:|I|=p}(−1)^{|I|}⟨df_{I}, N⟩ω^{I} ∧ dr − (−1)^{|I|}dΓ(A)θ_{t} ∧ dr.

Now we have

∑_{I:|I|=p}⟨df_{I}, N⟩ω^{I} = dΓ(A)θ_{t}.

Now we calculate (∇_{N}θ, η).

(∇_{N}θ, η) = (∇_{N}{ ∑_{I:|I|=p} f_{I}ω^{I} + ∑_{J:|J|=p−1} g_{J}ω^{J} ∧ dr}, η_{t})

= ( ∑_{I:|I|=p}(∇_{N}f_{I})ω^{I}, η_{t}) (∵ ∇_{N}ω^{I} = 0)

= (dΓ(A)θ_{t}, η_{t}).

This completes the proof

Nest we consider the relative boundary condition. This can be easily done by noting that the Hodge operation gives an isometry which interchanges the absolute boundary condition and the relative boundary condition. We denote the Hodge operation on ∂M by ∗. Then we have

Theorem 4.3. For p-forms θ, η ∈ Dom(E_{(p)}^{B_{r}}) and f ∈ C^{∞}(M), it holds that

−E((θ, η), f) + E_{(p)}^{B_{r}}(fθ, η) + E_{(p)}^{B_{r}}(θ, fη)
= 2 ∫_{M}(∇θ, ∇η)fdm + 2 ∫_{M} R_{(p)}(θ, η)fdm + 2 ∫_{∂M}(∗^{−1}dΓ(A)∗θ_{n}, η_{n})fdσ. (4.16)

Now we can apply Theorem 2.1. For example, if Ric ≥ λI and A is non-negative definite, then we have

|T⃗_{t}θ| ≤ e^{−λt}|θ|, for θ ∈ A^{1}(M)

where T⃗_{t} is a semigroup generated by □_{(1)}^{B_{a}}. Moreover we can give an estimate of the constant for the logarithmic Sobolev inequality.

Taking f = 1 in (4.15), we have

E_{(p)}^{B_{a}}(θ, η) = ∫_{M}(∇θ, ∇η)dm + ∫_{M} R_{(p)}(θ, η)dm + ∫_{∂M}(dΓ(A)θ_{t}, η_{t})dσ. (4.17)

This identity is already known (see e.g., Schwarz [17, Thm 2.1.5]).

In particular, if p = 1, it holds that

E_{(1)}^{B_{a}}(θ, η) = ∫_{M}(∇θ, ∇η)dm + ∫_{M} Ric(θ, η)dm + ∫_{∂M}(Aθ_{t}, η_{t})dσ.  (4.18)

Here Ric denotes the Ricci curvature. Using this identity, we can show the Lichnèrowicz theorem. Let Δ be the Laplacian with Neumann boundary condition on M. We assume that Ric ≥ (d − 1)ρI with ρ > 0 and A is non-negative definite. Then we have

λ_{1} ≥ ρd  (4.19)

where λ_{1} is the first non-zero eigenvalue of Δ.

To see this, note that

(Δf, Δf) = −(ddf, ddf) − (δdf, δdf)
= E_{(1)}^{B_{a}}(∇f, ∇f)
= ∫_{M}(∇^{2}f, ∇^{2}f)dm + ∫_{M} Ric(∇f, ∇f)dm + ∫_{∂M}(A(∇f)_{t}, (∇f)_{t})dσ.

Now, by the standard argument, we easily have

\frac{d − 1}{d}(Δf, Δf) ≥ ∫_{M} Ric(∇f, ∇f)dm ≥ (d − 1)ρE(f, f).

Now we can get the desired result.

Moreover it is known that the identity holds if and only if M is isomorphic to the hemisphere (Xia [20]). Similar result holds for the Laplacian with Dirichlet boundary condition, see Reilly [14].

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Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

E-mail address: ichiro@kusm.kyoto-u.ac.jp
URL: http://www.kusm.kyoto-u.ac.jp/~ichiro/