L^{p} CONTRACTION SEMIGROUPS FOR VECTOR VALUED FUNCTIONS

ICHIRO SHIGEKAWA

Abstract. We discuss semigroups acting on vector valued functions. We consider a comparison theorem between two semigroups: a semigroup acting on scalar valued functions and a semigroup acting on vector valued functions. The criterion is given by the abstract Kato theorem. By using this, we give a sufficient condition for the criterion in the setting of square field operator. We also consider the essential self-adjointness of a perturbed semigroup.

1. Introduction

In this paper, we discuss a theory of L^{p} contraction semigroups for vector valued functions. The theory of Markovian semigroup is well-developed in terms of Dirichlet forms. By virtue of the interpolation theory, the symmetric Markovian semigroup is not only an L^{2} contraction semigroup but also an L^{p} contraction semigroup for any p ∈ [1, ∞). L^{p} setting of the semigroup is useful in applications. But the semigroup acts on scalar valued functions. The Markovian property makes no sense for semigroups acting on vector valued functions. To deal with semigroups acting on vector valued functions, we need a comparison theorem.

To be precise, let (X, B, m) be a σ-finite measure space. Suppose we are given a Markovian semigroup {T_{t}} on L^{2} = L^{2}(X, m). Further we suppose that {T_{t}} is a contraction semigroup in L^{p} for any p ∈ [1, ∞). Besides, we are given a semigroup {T⃗_{t}} acting on Hilbert space valued functions. We denote the norm of the Hilbert space by | · |. If we have

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

then we can see that {T⃗_{t}} defines a semigroup on L^{p} for p ∈ [1, ∞). A necessary and sufficient condition for (1.1) is given by the abstract Kato theorem due to B. Simon [34, 35] (see also [19]). In this paper, we give sufficient conditions for (1.1) in terms of the square field operator.

The organization of this paper is as follows. In Section 2, we consider two semigroups {T_{t}}, {T⃗_{t}} and review the abstract Kato theorem. In Section 3, we formulate the problem in the framework of the square field operator. We give sufficient conditions for (1.1). The positivity and the derivation property of the square field operator play an important role. As an application, we discuss a perturbed semigroup and consider the essential self-adjointness in Section 4. We give examples in Section 5. Our main interest is in the infinite dimensional case and we consider a submanifold of the Wiener space as an example.

1991 Mathematics Subject Classification. 60J60, 58G32.

2. Comparison theorem

Let (X, B) be a measurable space and m be a σ-finite measure. Suppose we are given a strongly continuous contraction semigroup {T_{t}} on L^{2}. We suppose 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, for simplicity. The generator of {T_{t}} is denoted by A and its resolvents by G_{α}, α > 0. In addition, we assume that

∫_{X} T_{t}f dm ≤ ∫_{X} f dm, f ∈ L^{1} ∩ L2_{+},

where + stands for the non-negative functions. Note that if {T_{t}} is symmetric, the above property follows from the Markovian property. Thus {T_{t}} becomes a contraction semigroup on L^{1}.

Now, by the Riesz-Thorin interpolation theorem, {T_{t}} is a contraction semigroup on L^{p} for p ∈ [1, ∞]. To be precise, for f ∈ L^{2} ∩ L^{p}, T_{t}f ∈ L^{2} ∩ L^{p} and

∥T_{t}f∥_{p} ≤ ∥f∥_{p}.

Therefore, for p ∈ [1, ∞), T_{t} can be uniquely extended to a bounded linear operator on L^{p}. But, for p = ∞, T_{t} is defined only on \overline{L^{2} ∩ L^{∞}}^{∥·∥_{∞}}, the completion of L^{2} ∩ L^{∞} with respect to ∥·∥_{∞}. We regard {T_{t}} as a contraction semigroup on L^{p} throughout the paper.

Let {T_{t}^{∗}} be the dual semigroup of {T_{t}}. From the assumption, {T_{t}^{∗}} is Markovian. Moreover, by duality, we can see that {T_{t}^{∗}} is a contraction semigroup on L^{p} for p ∈ [1, ∞] as well. In this case, T_{t} : L^{∞} → L^{∞} can be defined as a dual operator of T_{t}^{∗} : L^{1} → L^{1}. In the sequel, we suppose that T_{t} : L^{∞} → L^{∞} is defined in this manner. We denote the associated generator and resolvents by A^{∗} and G_{α}^{∗}, respectively.

Later, we will consider complex Hilbert spaces and so complex coefficient will be necessary. From now on we think that {T_{t}} acts on complex valued functions in a natural way. It is easy to see that for a complex valued function f,

|T_{t}f| ≤ T_{t}|f|.

In fact, when f = ∑ a_{i}1_{A_{i}}, A_{i} ∩ A_{j} = ∅ if i ≠ j, we have

|T_{t}f| = |T_{t}(∑ a_{i}1_{A_{i}})| = | ∑ a_{i}T_{t}1_{A_{i}}| ≤ ∑ |a_{i}|T_{t}1_{A_{i}} ≤ T_{t}(∑ |a_{i}|1_{A_{i}}) = T_{t}|f|.

Hence {T_{t}} is also a contraction semigroup on L^{p}(X, m; ℂ). As usual, the inner product in L^{2}(X, m; ℂ) is defined by

(f|g) = ∫_{X} fḡdm

where ¯ denotes the complex conjugate. We also use this notation as a paring between L^{p} and L^{q}, \frac{1}{p} + \frac{1}{q} = 1. Note that (·|·) is anti-linear in the second variable.

The following is a natural generalization of the Hölder inequality:

Lemma 2.1. For p ∈ (1, ∞), let q be the conjugate exponent of p: \frac{1}{p} + \frac{1}{q} = 1. Then it holds that

|T_{t}(fg)| ≤ T_{t}|fg| ≤ {T_{t}|f|^{p}}^{1/p}{T_{t}|g|^{q}}^{1/q} for f ∈ L^{p} and g ∈ L^{q}.  (2.1)

For p = 1, it holds that

|T_{t}(fg)| ≤ T_{t}|fg| ≤ (T_{t}|f|)∥g∥_{∞} for f ∈ L^{1} and g ∈ L^{∞}.  (2.2)

Furthermore we have, for 1 ≤ p ≤ p′ ≤ ∞,

{T_{t}|f|^{p}}^{1/p} ≤ {T_{t}|f|^{p′}}^{1/p′} for f ∈ L^{p} ∩ L^{p′}.  (2.3)

Proof. We recall the following inequality:

ab ≤ \frac{a^{p}}{p} + \frac{b^{q}}{q} a, b ≥ 0.  (2.4)

We fix versions of T_{t}|f|^{p} and T_{t}|g|^{q} and consider on the set {x : (T_{t}|f|^{p})(x) = 0 or (T_{t}|g|^{q})(x) = 0}. Then, for x ∈ {T_{t}|f|^{p} = 0} and n ∈ ℕ,

T_{t}|fg|(x) = T_{t}|nf(g/n)|(x) ≤ \frac{n^{p}T_{t}|f|^{p}(x)}{p} + \frac{T_{t}|g|^{q}(x)}{n^{q}q} ≤ \frac{T_{t}|g|^{q}(x)}{n^{q}q} a.e.

Since n is arbitrary, we have

T_{t}|fg| = 0 a.e. on {T_{t}|f|^{p} = 0}.

Similar result holds on the set {T_{t}|g|^{q} = 0}. Thus we have

T_{t}|fg| ≤ {T_{t}|f|^{p}}^{1/p}{T_{t}|g|^{q}}^{1/q} a.e. on {T_{t}|f|^{p} = 0 or T_{t}|g|^{q} = 0}.

Next for x ∈ {T_{t}|f|^{p} ≠ 0, T_{t}|g|^{q} ≠ 0},

\frac{|f|}{\{T_{t}|f|^{p}(x)\}^{1/p}} · \frac{|g|}{\{T_{t}|g|^{q}(x)\}^{1/q}} ≤ \frac{|f|^{p}}{pT_{t}|f|^{p}(x)} + \frac{|g|^{p}}{qT_{t}|g|^{q}(x)}.

Hence we have

\frac{T_{t}|fg|(x)}{\{T_{t}|f|^{p}(x)\}^{1/p}\{T_{t}|g|^{q}(x)\}^{1/q}} ≤ \frac{T_{t}|f|^{p}(x)}{pT_{t}|f|^{p}(x)} + \frac{T_{t}|g|^{p}(x)}{qT_{t}|g|^{q}(x)} = \frac{1}{p} + \frac{1}{q} = 1

and thereby, we have

T_{t}|fg|(x) ≤ {T_{t}|f|^{p}(x)}^{1/p}{T_{t}|f|^{p}(x)}^{1/q} a.e. on {T_{t}|f|^{p} ≠ 0, T_{t}|g|^{q} ≠ 0}.

Combining them, we obtain (2.1).

To show (2.3), let r be the conjugate exponent of p′/p. We take g ∈ L^{r} such that 0 ≤ g ≤ 1. Then by (2.1) and the Markov property, we have

T_{t}(|f|^{p}g) ≤ {T_{t}|f|^{p′}}^{p/p′}{T_{t}|g|^{r}}^{1/r} ≤ {T_{t}|f|^{p′}}^{p/p′}.

By letting g ↑ 1, we obtain (2.3).

We have seen that {T_{t}} is a contraction semigroup. Let us see the strong continuity.

Proposition 2.2. {T_{t}} (resp. {T_{t}^{∗}}) is a strongly continuous semigroup in L^{p} for p ∈ [1, ∞).

Proof. First we show the case p = 1. Take any f ∈ L1_{+}. Here, + means the positive functions and we use this convention without mentioning. Then by Lemma 2.1, we have

|T_{t}\sqrt{f}|^{2} ≤ T_{t}f

and therefore

∥T_{t}\sqrt{f}∥_{2}^{2} ≤ ∥T_{t}f∥_{1} ≤ ∥f∥_{1} = ∥\sqrt{f}∥_{2}^{2}.

Since \{T_{t}\} is a strongly continuous in L^{2}, we have lim_{t→0} ∥T_{t}\sqrt{f}∥_{2} = ∥\sqrt{f}∥_{2}. Combing this with the above fact, we have

lim_{t→0} ∥T_{t}f∥_{1} = ∥f∥_{1}.  (2.5)

Now if we take f ∈ L1_{+} ∩ L2_{+}, then there exists a sequence {t_{n}} such that t_{n} → 0 and T_{t_{n}}f → f a.e. This combined with (2.5) implies

lim_{t→0} ∥T_{t}f − f∥_{1} = 0  (2.6)

(see, e.g. [21, Theorem (13.47)]). Now it is easy to see that (2.6) holds for any f ∈ L^{1}.

To show the general case p ∈ (1, ∞), we take r > p and use the following inequality (an easy application of the Hölder inequality, see, e.g. [21, Theorem (13.19)]): for f ∈ L^{1} ∩L^{r},

∥T_{t}f − f∥_{q}^{q} ≤ ∥T_{t}f − f∥_{1}^{α}∥T_{t}f − f∥_{r}^{r(1−α)}.  (2.7)

Here α is taken so that p = α + (1 − α)r. Now the strong continuity follows from (2.6) and the contraction property in L^{r}.

Remark 2.1. Other proofs of the above strong continuity can be found in [28, Theorem X.55] when m is a finite measure and in [13, Proposition I.2.4.2] when p = 1.

For later use, we give a basic fact on A. We denote the generator in L^{p} by A_{p} to specify the acting space. Dom(A_{p}) denotes the domain of A_{p}. We always assume that Dom(A_{p}) is equipped with the graph norm. It is a Banach space with respect to the graph norm. Similarly, Dom(E) is equipped with the inner product E_{1}(·, ·) = E(·, ·) + (·|·)_{L^{2}}.

Further the inner product in L^{2} is denoted by (·|·). We use it as a paring of L^{p} and L^{q} where \frac{1}{p} + \frac{1}{q} = 1. Note that (·|·) is ℂ-linear in the first variable but anti ℂ-linear in the second variable.

Lemma 2.3. We have the followings:

(1) G_{α}(L^{1} ∩ L^{∞}) ⊆ ⋂_{1≤p<∞} Dom(A_{p}) and G_{α}(L^{1} ∩ L^{∞}) is independent of α and dense in L^{p} for any p ∈ [1, ∞).

(2) For 1 ≤ p < ∞ and f ∈ L^{p}, suppose that there exists g ∈ L^{p} such that

(f|αG_{α}^{∗}h − h) = (g|G_{α}^{∗}h), ∀h ∈ L^{1} ∩ L^{∞}.

Then f ∈ Dom(A_{p}) and A_{p}f = g.

(3) For 1 ≤ p < ∞ and f ∈ L^{p}, suppose that there exists g ∈ L^{p} such that

(f|A^{∗}h) = (g|h), ∀h ∈ G_{α}^{∗}(L^{1} ∩ L^{∞}).

Then f ∈ Dom(A_{p}) and A_{p}f = g.

(4) For 1 ≤ p, q < ∞, u ∈ Dom(A_{p}) ∩ Dom(A_{q}) ⇒ A_{p}u = A_{q}u.

(5) For 1 ≤ p, q < ∞, u ∈ Dom(A_{p}) ∩ L^{q} and A_{p}u ∈ L^{q} ⇒ u ∈ Dom(A_{q})

(6) Let q, q′ ∈ (1, ∞] be conjugate exponents of p, p′ ∈ [1, ∞), respectively: \frac{1}{p} + \frac{1}{q} = 1, \frac{1}{p′} + \frac{1}{q′} = 1. Then, if u ∈ Dom(A_{p}^{∗}) ∩ L^{q′} and v ∈ Dom(A_{p′}) ∩ L^{q}, we have (A_{p}^{∗}u|v) = (u|A_{p′}v).

Proof. (1) is trivial, e.g., the denseness follows from the fact that for u ∈ L^{p}, αG_{α}u → u in L^{p} as α → ∞.

Next we show (2). By the assumption, we have

(αG_{α}f − f|h) = (G_{α}g|h).

This implies αG_{α}f −f = G_{α}g and therefore G_{α}(αf −g) = f. Hence we have f ∈ Dom(A_{p}) and

A_{p}f = A_{p}G_{α}(αf − g) = −(αf − g) + αG_{α}(αf − g) = −(αf − g) + αf = g

which is the desired result.

(3) easily follows from (2). Set h = G_{α}^{∗}u, u ∈ L^{1} ∩ L^{∞}. Then,

(g|G_{α}^{∗}u) = (f|A^{∗}G_{α}^{∗}u) = (f| − u + αG_{α}^{∗}u).

By (2), we have A_{p}f = g.

To see (4), it is enough to note that

A_{p}u = lim_{t↓0} \frac{T_{t}u − u}{t} in L^{p}.

As for (5), since u = G_{α}(αu − A_{p}u) and αu − A_{p}u ∈ L^{q}, it follows that u ∈ Dom(A_{q}).

(6) can be shown as follows:

(A_{p}^{∗}u|v) = (lim_{t↓0} \frac{T_{t}^{∗}u − u}{t}|v) = (u| lim_{t↓0} \frac{T_{t}v − v}{t} ) = (u|A_{p′}v).

This completes the proof.

So far, we have considered the semigroup acting on scalar valued functions. Now we consider another semigroup {T⃗_{t}} that acts on vector valued functions. Sometimes we have to consider a space of sections of a vector bundle. But only the measurable structure is relevant and so we can regard it as a trivial bundle. We therefore consider the L^{p}-space of vector valued functions. Let K be a real or complex Hilbert space. We denote the inner product and the norm on K by (·|·)_{K} and | · |_{K}, respectively. For simplicity, we usually write | · | in place of | · |_{K}. We assume that K is separable. L^{p} = L^{p}(X, m; K) denotes the space of all K-valued measurable functions u with

∥u∥_{p} = { ∫_{X} |u(x)|^{p}dm}^{1/p} < ∞.

As usual, we identify two functions if they coincide m-a.e. Suppose we are given a strongly continuous contraction semigroup {T⃗_{t}} on L^{2}(X, m; K). We denote the associated generator, resolvents by A⃗ and G⃗_{α}, respectively.

We are interested in when the following relation holds:

|T⃗_{t}u(x)| ≤ T_{t}|u|(x) m-a.e. x, ∀u ∈ L^{2}(X, m; K).

As for this problem, the following comparison theorem is due to B. Simon [34, 35]:

Theorem 2.4. The following three conditions are equivalent to each other:

(1) |T⃗_{t}u| ≤ e^{λt}T_{t}|u|,

(2) |G⃗_{α+λ}u| ≤ G_{α}|u|, α > max{0, −λ}, (3) A|u| ≥ ℜ((A⃗ − λ)u| sgn u)_{K}, u ∈ Dom(A⃗) where

sgn u = \{ \begin{matrix}\frac{u}{|u|}, & u ≠ 0, \\ 0, & u = 0. \end{matrix}  (2.8)

Proof. We give here a proof for reader’s convenience. The equivalence (1) and (2) can be easily seen by noting

G_{α} = ∫_{0}^{∞} e^{−αt}T_{t}dt, G⃗_{α+λ} = ∫_{0}^{∞} e^{−(α+λ)t}T⃗_{t}dt

and

T_{t} = lim_{n→∞}(\frac{n}{t})^{n}G_{n/t}^{n}, T⃗_{t} = e^{λt} lim_{n→∞}(\frac{n}{t})^{n}G⃗_{(n/t)+λ}^{n}.

Next we see the implication (1) ⇒ (3). For u ∈ Dom(A_{2}), take any f ∈ Dom(A_{2}) and f ≥ 0. Then we have,

(T_{t}|u| | f) ≥ (ℜ(T⃗_{t}u| sgn u)_{K}|f)

and hence

(|u| | T_{t}^{∗}f) ≥ ℜ(T⃗_{t}u|f sgn u).

In the case of t = 0, the following identity holds evidently:

(|u| | f) = (u|f sgn u).

Thus we have

(|u||\frac{T_{t}^{∗}f − f}{t} ) ≥ ℜ(\frac{T⃗_{t}u − u}{t}|f sgn u).

Now letting t → 0, we have

(|u| | A^{∗}f) ≥ ℜ(A⃗u|f sgn u) = (ℜ(A⃗u|u)_{K}|f),

which implies (3).

We prove the implication (3) ⇒ (1). For any g ∈ Dom(A_{2}^{∗})_{+}, we have

(|u| | (A^{∗} − α)g) ≥ (ℜ(A⃗ − λ − α)u| sgn u)_{K}|g) ≥ −(|(A⃗ − λ − α)u| | g).

Now we set u = G⃗_{α+λ}v and g = G_{α}ψ, ψ ≥ 0. Then we have

(|G⃗_{α+λ}v| | (A^{∗} − α)G_{α}ψ) ≥ −(|(A⃗ − λ − α)G⃗_{α+λ}v| | G_{α}^{∗}ψ).

Hence

(|G⃗_{α+λ}v| | ψ) ≤ (G_{α}|v| | ψ).

Thus we have |G⃗_{α+λ}v| ≤ G_{α}|v|.

So far, we have not assumed the symmetry of T_{t} and T⃗_{t}. If they are symmetric, we can think of the associated quadratic forms. In this case, we denote the quadratic forms associated with {T_{t}} and {T⃗_{t}} by E and E⃗, respectively.

Theorem 2.5. Assume that T_{t} and T⃗_{t} are symmetric and one of the conditions in Theorem 2.4 is satisfied. Then we have that if u ∈ Dom(E⃗), then |u| ∈ Dom(E) and

E(|u|, |u|) ≤ E⃗(u, u) + λ∥u∥_{2}^{2}.  (2.9)

Moreover if g ≥ 0 and u ∈ Dom(E⃗) satisfy gu ∈ Dom(E⃗), then g|u| ∈ Dom(E) and

E(|u|, g|u|) ≤ E⃗(u, gu) + λ⟨g, |u|^{2}⟩.  (2.10)

Proof. Assuming (2) in Theorem 2.4, we show (2.9). We note that, for u ∈ Dom(E⃗),

((α − α^{2}G_{α})|u| | |u|)_{L^{2}} ≤ ((α − α^{2}G⃗_{α+λ})u|u)_{L^{2}}.

Letting α → ∞,

lim_{α→∞}((α − α^{2}G_{α})|u| | |u|)_{L^{2}} ≤ lim_{α→∞}((α − α^{2}G⃗_{α+λ})u|u)_{L^{2}} = E⃗(u, u) + λ∥u∥_{2}^{2}.

We have therefore |u| ∈ Dom(E) and

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

Next we show (2.10). Assume that gu ∈ Dom(E⃗). Then, by the above result, we have g|u| ∈ Dom(E). Now we notice the following inequality, which follows from (2),

((α − α^{2}G_{α})|u| | g|u|)_{L^{2}} ≤ ((α − α^{2}G⃗_{α+λ})u|gu)_{L^{2}}

Letting α → ∞, we get the desired result (2.10).

The above theorem can be used in the following situation. We suppose all conditions in Theorem 2.5. We suppose further that the following defective logarithmic Sobolev inequality holds for E: there exists a α, β > 0 such that

∫_{X} f^{2} log(f^{2}/∥f∥_{2}^{2})dm ≤ αE(f, f) + β∥f∥_{2}^{2} for f ∈ Dom(E).  (2.11)

Then we have a similar inequality for E⃗:

∫_{X} |u|^{2} log(|u|^{2}/∥u∥_{2}^{2})dm ≤ αE⃗(u, u) + (β + λ)∥u∥_{2}^{2} for u ∈ Dom(E⃗). (2.12)

To show this, set f = |u|. By virtue of (2.9), we have

∫_{X} |u|^{2} log(|u|^{2}/∥u∥_{2}^{2})dm ≤ αE(|u|, |u|) + β∥|u|∥_{2}^{2} ≤ αE⃗(u, u) + (β + λ)∥u∥_{2}^{2}.

The inequality (2.12) plays an important role in dealing with the Laplacian acting on tensor fields.

Proposition 2.6. Suppose that one of the conditions in Theorem 2.4 is fulfilled. Then {T⃗_{t}} (resp. {T⃗_{t}^{∗}}) is a strongly continuous semigroup in L^{p} for p ∈ [1, ∞).

Proof. First we prove that lim_{t→0} ∥T⃗_{t}u−u∥_{1} = 0 for u ∈ L^{1} ∩L^{2}. Otherwise there exist ε > 0 and a sequence {t_{n}} such that t_{n} → 0 and

∫_{X} |T⃗_{t_{n}}u − u|_{K}dm ≥ 5ε.  (2.13)

We may assume that T⃗_{t_{n}}u → u m-a.e. by taking a subsequence if necessary. By the Fatou lemma, we have

∫_{X} |u|dm ≤ \underline{lim}_{n→∞} ∫_{X} |T⃗_{t_{n}}u|dm ≤ \overline{lim}_{n→∞} ∫_{X} |T⃗_{t_{n}}u|dm ≤ \overline{lim}_{n→∞} ∫_{X} T_{t_{n}}|u|dm ≤ ∫_{X} |u|dm

which implies

lim_{n→∞} ∫_{X} |T⃗_{t_{n}}u|dm = ∫_{X} |u|dm.

We have therefore |T_{t_{n}}u| → |u| in L^{1} and further, for any E ⊂ X,

lim_{n→∞} ∫_{E} |T⃗_{t_{n}}u|dm = ∫_{E} |u|dm

(see, [21, Theorem (13.47)]). Now we take E ⊆ X so that m(E) < ∞ and

∫_{X\E} |u|dm ≤ ε.

We set v_{n} = 1_{E}T_{t_{n}}u and v = 1_{E}u. Since {|v_{n}|} is uniformly integrable, we can take N > 0 so that

∫ |v_{n}|1_{{|v_{n}|>N}}dm < ε ∀n.

We may assume that the above inequality holds for v. For such N > 0, we define φ_{N}: K → K by

φ_{N}(k) = \{ \begin{matrix}k & if |k| ≤ N, \\ Nk/|k| & if |k| > N. \end{matrix}

Set v_{n}^{(N)} = φ_{N}(v_{n}), v^{(N)} = φ_{N}(v). Then we have

∫_{X} |T_{t_{n}}u − u|dm ≤ ∫_{X} 1_{E}|T_{t_{n}}u − u|dm + ∫_{X\E} |T_{t_{n}}u − u|dm
≤ ∫_{X}{|v_{n} − v_{n}^{(N)}| + |v_{n}^{(N)} − v^{(N)}| + |v^{(N)} − v|}dm
+ ∫_{X\E}{|T_{t_{n}}u| + |u|}dm
≤ ∫_{X} |v_{n}|1_{{|v_{n}|>N}}dm + ∫_{X} |v_{n}^{(N)} − v^{(N)}|dm + ∫_{X} |v|1_{{|v|>N}}dm
+ ∫_{X\E} |T_{t_{n}}u|dm + ∫_{X} |u|dm.

Now by the dominated convergence theorem, we have

\overline{lim}_{n→∞} ∫_{X} |T_{t_{n}}u − u|dm ≤ ε + lim_{n→∞} ∫_{X} |v_{n}^{(N)} − v^{(N)}|dm + ε + lim_{n→∞} ∫_{X\E} |T_{t_{n}}u|dm + ε
= 4ε

which contradicts to (2.13). Now we have proved the strong continuity of {T_{t}} in L^{1}.

For general p ∈ [1, ∞), it is enough to notice the same inequality as (2.7). The proof is completed.

3. 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. From now on, we assume that {T_{t}} is a symmetric semigroup. Following Bouleau-Hirsch [13], we assume the following condition: (Γ) For f, g ∈ Dom(A_{2}), we have f · ḡ ∈ Dom(A_{1}).

Under the above assumption, we set

Γ(f, g) = \frac{1}{2}\{A_{1}(f · ḡ) − A_{2}f · ḡ − f · A_{2}ḡ\}.  (3.1)

The reader can see an extensive discussion of Γ in [13, Chapter 1, §4] (but he should note that our definition of Γ is different from that of [13] up to a constant). We list some fundamental properties of Γ for later use. Γ can be uniquely extended to a continuous sesquilinear form from Dom(E) × Dom(E) into L^{1} and has the following properties. For f, g ∈ Dom(E),

Γ(f, f) ≥ 0,  (3.2)
|Γ(f, g)|^{2} ≤ Γ(f, f)Γ(g, g),  (3.3)
∫_{X} Γ(f, f)dm ≤ E(f, f) ≤ 2 ∫_{X} Γ(f, f)dm.  (3.4)

(but see also (3.17)) below). Further for f, g ∈ Dom(A_{2})

E(f, g) = ∫_{X} Γ(f, g)dm − \frac{1}{2} ∫_{X} A_{1}(f · ḡ)dm.  (3.5)

Dom(A_{1}) ∩ L^{∞} is an algebra and for f, g ∈ Dom(A_{1}) ∩ L^{∞}, we have

A_{1}(f · ḡ) = A_{1}f · ḡ + f · A_{1}ḡ + 2Γ(f, g).  (3.6)

We remark that Dom(A_{1}) ∩ L^{∞} ⊆ Dom(E). We can see this by noting

E(f, f) = lim_{t→0}(\frac{f − T_{t}f}{t} |f) = −(A_{1}f|f) ≤ ∥A_{1}f∥_{1}∥f∥_{∞}.

In [13], the reader can find several conditions that are equivalent to (Γ). Among them, the following condition is rather practical: (Γ′) There exists a subspace C ⊆ Dom(A_{2}) such that C is dense in Dom(E), closed under complex conjugation, and |f|^{2} ∈ Dom(A_{1}) for all f ∈ C.

Let us consider the semigroup {T⃗_{t}}. We also assume that {T⃗_{t}} is symmetric. To define the square field operator for {T⃗_{t}}, we impose the following condition: (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.  (3.7)

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}\}.  (3.8)

Γ⃗ satisfies

Γ⃗(u, u) + λ|u|^{2} ≥ 0 for u ∈ Dom(A_{2}).  (3.9)

Now recall that the positivity of Γ corresponds to

|T_{t}f|^{2} ≤ T_{t}|f|^{2}, ∀f ∈ L^{2}.

The following theorem is a natural generalization of the above fact.

Theorem 3.1. Under the assumptions (Γ) and (Γ⃗_{λ}), we have, for u ∈ L^{2},

|T⃗_{t}u|^{2} ≤ e^{2λt}T_{t}|u|^{2}  (3.10)

and {T⃗_{t}} is a strongly continuous contraction semigroup in L^{p} for p ∈ [1, ∞). Moreover, Γ⃗(u, v) is well-defined for u, v ∈ Dom(E⃗) and we have

∫_{X} Γ⃗(u, u)dm ≤ E⃗(u, u), u ∈ Dom(E⃗).  (3.11)

Proof. By considering A⃗ − λ in place of A⃗, we may assume that λ = 0.

Take any t > 0 and fix it. For u ∈ Dom(A⃗_{2}) and g ∈ L∞_{+}, define Φ(s), s ∈ [0, t] by

Φ(s) = (T_{t−s}|T⃗_{s}u|^{2} | g).

Noting that |T⃗_{t−s}u|^{2} ∈ Dom(A_{1}) and T⃗_{s}u ∈ Dom(A⃗_{2}) and using (Γ⃗), we have

Φ′(s) = (2T_{t−s}(A⃗_{2}T⃗_{s}u|T⃗_{s}u)_{K} | g) − (A_{1}T_{t−s}|T⃗_{s}u|^{2} | g)
= (2(A⃗_{2}T⃗_{s}u|T⃗_{s}u)_{K} − A_{1}|T⃗_{s}u|^{2} | T_{t−s}g)
≤ 0.

Thus we have Φ(0) ≥ Φ(t) which implies (3.10) for u ∈ Dom(A⃗_{2}). The general case easily follows from this.

For p ∈ [2, ∞], the contraction property of {e^{−λt}T⃗_{t}} on L^{p} is easily obtained from (2.3):

e^{−λt}|T⃗_{t}u| ≤ {T_{t}|u|^{2}}^{1/2} ≤ {T_{t}|u|^{p}}^{1/p}.

(Recall that when p = ∞, T⃗_{t} is defined on \overline{L^{2} ∩ L^{∞}}^{∥·∥_{∞}}.) By duality, \{e^{−λt}T⃗_{t}\} is also a contraction semigroup on L^{p} for p ∈ [1, 2].

To see the strong continuity of {T⃗_{t}}, we note that

∫_{X} |u|dm ≤ \underline{lim}_{t→0} ∫_{X} |T⃗_{t}u|dm ≤ \overline{lim}_{t→0} ∫_{X} |T⃗_{t}u|dm ≤ ∫_{X} |u|dm.

Then the rest is the same as the proof of Proposition 2.6.

To show (3.11), we repeat the argument in [6]. From the definition of Γ⃗, we see that

∫_{X} Γ⃗(u, v)dm = E⃗(u, v) + \frac{1}{2} ∫_{X} A_{1}(u|v)_{K}dm.  (3.12)

Further, by the contraction property of {T_{t}}, we have

∫_{X} T_{t}|u|^{2}dm ≤ ∫_{X} |u|^{2}dm

and thereby

∫_{X} A_{1}|u|^{2}dm = lim_{t→0} ∫_{X} \frac{T_{t}|u|^{2} − |u|^{2}}{t} dm ≤ 0.

Thus we have, for u ∈ Dom(A⃗_{2}),

∫_{X} Γ⃗(u, u)dm ≤ E⃗(u, u).  (3.13)

On the other hand, by the positivity of (3.9), we have the following Schwarz inequality:

|Γ⃗(u, v) + λ(u|v)_{K}|^{2} ≤ {Γ⃗(u, u) + λ|u|^{2}}{Γ⃗(v, v) + λ|v|^{2}}.  (3.14)

Therefore we have ∥E⃗(u, v)∥_{1} ≤ {E⃗(u, u) + λ∥u∥_{2}^{2}}^{1/2}{E⃗(v, v) + λ∥v∥_{2}^{2}}^{1/2} + |λ|∥u∥_{2}∥v∥_{2}

Now it is easy to see that Γ⃗ can be extended to continuous bilinear map from Dom(E⃗) × Dom(E⃗) into L^{1}. This completes the proof.

We give a sufficient condition for the assumption of the above theorem. Proposition 3.2. Assume that there exists a subspace D ⊆ Dom(A⃗_{2}) such that D is a core for A⃗_{2} and (u|v)_{K} ∈ Dom(A_{1}) for u, v ∈ D. Further we suppose that there exists λ ∈ ℝ such that

Γ⃗(u, u) + λ|u|^{2} ≥ 0 for u ∈ D.  (3.15)

Then the condition (Γ⃗_{λ}) holds. Proof. By the same proof as in Theorem 3.1, we have

∫_{X} Γ⃗(u, u)dm ≤ E⃗(u, u), u ∈ Dom(E⃗).

From this, it is easy to see that Γ⃗ is well-defined on Dom(E⃗)×Dom(E⃗) and Γ⃗ is a bounded sesquilinear operator from Dom(E⃗) × Dom(E⃗) into L^{1}.

Recall that for u, v ∈ D, A_{1}(u|v)_{K} = 2Γ⃗(u, v) + (A⃗_{2}u|v)_{K} + (u|A⃗_{2}v)_{K}.

Since the right hand side is continuous from Dom(A⃗_{2}) × Dom(A⃗_{2}) to L^{1} and D is a core for Dom(A⃗_{2}), we can deduce that (u|v)_{k} ∈ Dom(A_{1}) for u, v ∈ Dom(A⃗_{2}). The rest is easy.

From now on we suppose that E has the local property in the following sense (see [13, 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,  (3.16)

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).

Under the condition (L), the following identity holds:

E(f, g) = ∫_{X} Γ(f, g)dm.  (3.17)

Moreover the following derivation property of Γ follows (for the proof, see, e.g., [13, Chapter I, §5]):

(1) Let F: ℝ^{n} → ℝ be a C^{1} function with F (0) = 0. Then for real valued functions f_{1}, …, f_{n} ∈ Dom(E) ∩ L^{∞}, we have F (f_{1}, …, f_{n}) ∈ Dom(E) and

Γ(F (f_{1}, …, f_{n}), g) = ∑_{j} ∂_{j}F (f_{1}, …, f_{n})Γ(f_{j}, g), ∀g ∈ Dom(E).  (3.18)

If, in addition, ∂_{j}F are bounded, then the above identity holds for f_{n}, …, f_{n} ∈ Dom(E).

(2) Let F: ℝ^{n} → ℝ be a C^{2} function with F (0) = 0. Then for real valued functions f_{1}, …, f_{n} ∈ Dom(A_{1}) ∩ L^{∞}, we have F (f_{1}, …, f_{n}) ∈ Dom(A_{1}) ∩ L^{∞} and

A_{1}F (f_{1}, …, f_{n}) = ∑_{i} ∂_{i}F (f_{1}, …, f_{n})A_{1}f_{i} + ∑_{i,j} ∂_{i}∂_{j}F (f_{1}, …, f_{n})Γ(f_{i}, f_{j}). (3.19)

Moreover if ∂_{i}F, ∂_{i}∂_{j}F, i, j = 1, …, n are all bounded and F (0) = 0, then F (f_{1}, …, f_{n}) ∈ Dom(A_{1}) ∩ Dom(E) for f_{1}, …, f_{n} ∈ Dom(A_{1}) ∩ Dom(E) and the above identity holds.

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).  (3.20)

Now we can prove the main theorem in this section:

Theorem 3.3. Assume conditions (Γ) and (L). Moreover we suppose the following:

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

E(|u|, |u|) ≤ E⃗(u, u) + λ∥u∥_{2}^{2}.  (3.21)

(2) If u ∈ Dom(E⃗) ∩ L^{∞} and f ∈ Dom(E) ∩ L^{∞}, then fu ∈ Dom(E⃗) ∩ L^{∞}.

(3) For u ∈ Dom(E⃗) ∩ L^{∞} and g ∈ Dom(E) ∩ L∞_{+}, it holds that

E(|u|, g|u|) ≤ E⃗(u, gu) + λ⟨g, |u|^{2}⟩.  (3.22)

Then A_{2}|u| ≥ ℜ((A⃗ − λ)u, sgn u)_{K} in the sense of distributions for u ∈ Dom(A⃗_{2}) and therefore the assumption of Theorem 2.4 is satisfied.

Proof. We may assume λ = 0. Take any u ∈ Dom(A⃗_{2}) ∩ L^{∞}. For ε > 0, set |u|_{ε} = \sqrt{|u|^{2} + ε^{2}}. Then 1/|u|_{ε} ∈ Dom(E). For g ∈ Dom(A_{2}) ∩ L∞_{+}, Taking f = g/|u|_{ε} in (3.22), we have

(A_{2}u|g \frac{u}{|u|_{ε}}) = −E⃗(u, g \frac{u}{|u|_{ε}})

≤ −E(|u|, g \frac{|u|}{|u|_{ε}})

= − ∫_{X} Γ(|u|, g \frac{|u|}{|u|_{ε}})dm (∵ (3.17))

= − ∫_{X}\{g|u|(−\frac{|u|}{|u|_{ε}^{3}} Γ(|u|, |u|) + \frac{g}{|u|_{ε}} Γ(|u|, |u|) + \frac{|u|}{|u|_{ε}} Γ(|u|, g)\}dm

= − ∫_{X}\{g \frac{ε^{2}}{|u|_{ε}^{3}} Γ(|u|, |u|) + \frac{|u|}{|u|_{ε}} Γ(|u|, g)\}dm

≤ − ∫_{X} \frac{|u|}{|u|_{ε}} Γ(|u|, g)dm.

Now letting ε → 0, we have

∫_{X} \frac{|u|}{|u|_{ε}} Γ(|u|, g)dm → ∫_{X} 1_{\{u≠0\}}Γ(|u|, g)dm.

On the other hand, by the derivation property of Γ,

Γ(|u|^{1+ε}, |u|^{1+ε}) = (1 + ε)^{2}|u|^{2ε}Γ(|u|, |u|).

Hence 1_{{u=0}}Γ(|u|^{1+ε}, |u|^{1+ε}) = 0. Letting ε → 0, we have

1_{{u=0}}Γ(|u|, |u|) = 0.

Using the Schwarz inequality |Γ(|u|, g)| ≤ Γ(g, g)^{1/2}Γ(|u|, |u|)^{1/2}, we obtain

1_{{u=0}}Γ(|u|, g) = 0.

Now we eventualy have

(A⃗_{2}u|g sgn u) ≤ − ∫_{X} Γ(|u|, g)dm = −E(|u|, g) = (|u| | A_{2}g).

To see that the above inequality holds for g ∈ Dom(A_{2})∩L2_{+}, we note that αG_{α}(g∧n) ∈ Dom(A_{2}) ∩ L∞_{+} for n ∈ ℕ. Letting n → ∞ and then α → ∞, we get the desired result. This completes the proof.

Now we introduce the following condition on Γ⃗ and E⃗. It is related to the derivation property of Γ⃗. (D⃗) For u, v ∈ Dom(E⃗) ∩ L^{∞}, f ∈ Dom(E) ∩ L^{∞}, it holds that (u|v)_{K} ∈ Dom(E), fu ∈ Dom(E⃗) and

2fΓ⃗(u, v) = −Γ(f, (v|u)_{K}) + Γ⃗(u, f̄v) + Γ⃗(fu, v),  (3.23)
Γ⃗(f̄u, u) = \overline{Γ⃗(fu, u)},  (3.24)
E⃗(f̄u, u) = \overline{E⃗(fu, u)}.  (3.25)

On the other hand, from the definition of Γ⃗, for u, v ∈ Dom(A⃗_{2})∩L^{∞} and f ∈ Dom(E)∩ L^{∞},

2fΓ⃗(u, v) = fA_{1}(u|v)_{K} − f(A_{2}u|v)_{K} − f(u|A_{2}v)_{K}.

Integrating both sides over X, we have

∫_{X} 2fΓ⃗(u, v) = −E(f, (v|u)_{K}) + E⃗(u, f̄v) + E⃗(fu, v).  (3.26)

The above identity holds for u, v ∈ Dom(E⃗) ∩ L^{∞} and f ∈ Dom(E) ∩ L^{∞}.

Now we return to (3.23) when u = v:

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

= −Γ(f, |u|^{2}) + \overline{Γ⃗(f̄u, u)} + Γ⃗(fu, u)
= −Γ(f, |u|^{2}) + 2Γ⃗(fu, u).

We integrate both sides over X:

∫_{X} 2fΓ⃗(u, u) = −E(f, |u|^{2}) + 2 ∫_{X} Γ⃗(fu, u).

Comparing this with (3.26) and (3.25), we have

∫_{X} Γ⃗(fu, u) = E⃗(fu, u).

By polarization, we eventually obtain

∫_{X} Γ⃗(fu, v) = E⃗(fu, v).  (3.27)

We can give a sufficient condition for Theorem 3.3. Theorem 3.4. 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}.  (3.28)

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}.  (3.29)

Furthermore the conditions of Theorem 3.3 are satisfied. Proof. 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 (3.23):

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\{2f̄Γ⃗(u, u) + Γ(f̄, |u|^{2})\} + Γ(f, f|u|^{2})

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

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

= |f|^{2}Γ⃗(u, u) + \frac{1}{2}fΓ(f̄, |u|^{2}) + \frac{1}{2}f̄Γ(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}).  (3.30)

On the other hand, substituting v = u and f = |u|^{2} in (3.23), 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) (∵ (3.30))
= 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).  (3.31)

Now for ε > 0, set φ_{ε}(t) = \sqrt{t + ε^{2}} − ε. Then by the derivation property and (3.31),

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).

Letting ε → 0, we easily obtain that |u| ∈ Dom(E) and

Γ(|u|, |u|) ≤ Γ⃗(u, u).

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(f̄, |u|^{2}) + \frac{1}{2}f̄Γ(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 (3.29).

Next we check the conditions of Theorem 3.3. (1) is clear from (3.28). (2) is already assumed. We show (3). Take g ∈ Dom(E) ∩ L∞_{+} and u ∈ Dom(E⃗) ∩ L^{∞}. Then, by (D⃗) and (3.28),

E⃗(u, gu) = ∫_{X} Γ⃗(u, gu)dm (∵ (3.27))

= ∫_{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|).

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 3.5. 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 for f ∈ C and u, v ∈ D, we have (u|v)_{K} ∈ Dom(A_{1}) ∩ L^{∞}, fu ∈ D and

Γ⃗(fu, v) + Γ⃗(u, fv) = 2fΓ⃗(u, v) + Γ(f, (u|v)_{K})  (3.32)
Γ⃗(f̄u, u) = \overline{Γ⃗(fu, u)},  (3.33)
E⃗(f̄u, u) = \overline{E⃗(fu, u)}.  (3.34)
Γ⃗(u, u) + λ|u|^{2} ≥ 0.  (3.35)

Finally, we suppose that C is closed under complex conjugation and a core for E and D is a core for E⃗. Then (Γ⃗_{λ}) and (D⃗) are satisfied.

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 3.4, we have

Γ⃗(fu, fu) = f^{2}Γ⃗(u, u) + \frac{1}{2}fΓ(f̄, |u|^{2}) + \frac{1}{2}f̄Γ(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) ≤ 2 ∫_{X} Γ⃗(fu, fu)dm
≤ 2∥f∥_{∞}^{2} ∫_{X} Γ⃗(u, u)dm + 2∥f∥_{∞} ∫_{X} Γ(f, f)^{1/2}Γ(|u|^{2}, |u|^{2})^{1/2}dm
+ 2∥u∥_{∞}^{2} ∫_{X} Γ(f, f)dm

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

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

We claim that for f ∈ Dom(E) ∩ L^{∞}, u ∈ D, we have fu ∈ Dom(E⃗) and (3.32), (3.36) 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)  (3.36)

E⃗(F (f)u, F (f)u) ≤ 2∥F (f)∥_{∞}^{2}E⃗(u, u) + 2∥F (f)∥_{∞}E(F (f)f, F (f)f)^{1/2}E(|u|^{2}, |u|^{2})^{1/2}

+ 2∥u∥_{∞}^{2}E(F (f)f, F (f)).  (3.37)

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(ℜf_{n}) + \sqrt{−1}φ(ℑf_{n}) → f weakly in Dom(E) and \{g_{n}\} is uniformly bounded. Moreover, by (3.36) 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 (3.32) hold.

Since (3.32) 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 3.4, 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}  (3.38)
Γ(|u|, |u|) ≤ Γ⃗(u, u)  (3.39)

Since D is dense in Dom(E⃗), it is easy to see that (3.39) holds for u ∈ Dom(E⃗).

Now we can prove (3.32) for f ∈ Dom(E)∩L^{∞} and u ∈ Dom(E⃗)∩L^{∞}. Take any u ∈ D. For ε > 0, set

ψ_{e}(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), we have

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 (3.32) holds for u, v ∈ Dom(E⃗) ∩ L^{∞} and f ∈ Dom(E). 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, f̄v) − 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. Using this fact, we shall show (Γ⃗).

Take u, v ∈ D and f ∈ Dom(A_{2}) ∩ L^{∞}. Then

(A_{2}f|(u|v)_{K}) = (f|A_{1}(u|v)_{K})
= (f, (A⃗_{2}u|v)_{K} + (A⃗_{2}v|f̄u) + (f|2Γ⃗(u, v))
= −E⃗(fv, u) + (A⃗_{2}v|f̄u) + (f|2Γ⃗(u, v)).

Now, for u ∈ Dom(A⃗_{2})∩L^{∞}, we take a sequence {u_{n}} ⊆ D that converges to u in Dom(E⃗). Then, for v ∈ D and f ∈ Dom(A_{2}) ∩ L^{∞},

(A_{2}f|(u|v)_{K}) = lim_{n→∞}(A_{2}f|(u_{n}|v)_{K})

= − lim_{n→∞} E⃗(fv, u_{n}) + lim_{n→∞}(A⃗_{2}v|f̄u_{n}) + lim_{n→∞}(f|2Γ⃗(u_{n}, v))

= −E⃗(fv, u) + (A⃗_{2}v|f̄u) + (f|2Γ⃗(u, v))

= (fv|A⃗_{2}u) − E⃗(v, f̄u) + (f|2Γ⃗(u, v)).

Secondly, for v ∈ Dom(A⃗_{2}) ∩ L^{∞}, we take a sequence {v_{n}} ⊆ D that converges to v in Dom(E⃗). Then, for u ∈ Dom(A⃗_{2}) ∩ L^{∞} and f ∈ Dom(A_{2}) ∩ L^{∞},

(A_{2}f|(u|v)_{K}) = lim_{n→∞}(A_{2}f, (u|v_{n})_{K})

= lim_{n→∞}(A⃗_{2}u|fv_{n}) − lim_{n→∞} E⃗(v_{n}|f̄u) + lim_{n→∞}(f, 2Γ⃗(u, v_{n}))

= (A⃗_{2}u|fv) − E⃗(v, f̄u) + (f|2Γ⃗(u, v))

= (A_{2}u, fv) + (A⃗_{2}v|f̄u) + (f|2Γ⃗(u, v))

= (f|(A⃗_{2}u|v)_{K} + (u|A⃗_{2}v)_{K} + 2Γ⃗(u, v)).

Hence we have, for u, v ∈ Dom(A⃗_{2}) ∩ L^{∞},

A_{1}(u|v)_{K} = (A⃗_{2}u|v) + (u|A⃗_{2}v) + 2Γ⃗(u, v).

Noticing that Dom(A⃗_{2}) ∩ L^{∞} is dense in Dom(A⃗_{2}), we get the desired result.

This completes the proof.

4. Perturbation of a contraction semigroup

So far, our basic semigroup is a contraction Markovian semigroup. In this section, we discuss perturbed semigroups. Let {T_{t}} and {T⃗_{t}} be two semigroups as in the previous section. Our basic assumption is that {T_{t}} satisfies the hyperboundedness. Under this assumption, the perturbation theory was developed by Segal, Høegh-Krohn, Simon (see [32, 20, 36]). We shall deal with the semigroup {T⃗_{t}} acting on vector valued functions. We mainly follow the Segal method and make use of the comparison between semigroups.

Suppose that a real valued function V and an SL(K)-valued function R are given. Here, SL(K) is a set of all bounded symmetric operators on K and the norm in SL(K) is the operator norm which we denote by ∥ · ∥_{op}. We always assume that V and R are measurable.

We consider the following quadratic form:

E^{V} (f, g) = E(f, g) + ∫_{X} V fḡ dm  (4.1)

and

E⃗^{R}(u, v) = E⃗(u, v) + ∫_{X}(Ru|v)_{K}dm  (4.2)

where E and E⃗ are the quadratic forms associated with semigroups {T_{t}} and {T⃗_{t}}, respectively.

First we consider the case that V and R are bounded. In this case, E^{V} and E⃗^{R} are closed quadratic forms with the same domains as E and E⃗, respectively.

Proposition 4.1. Assume that V and R are bounded and

V (x)|k|^{2} ≤ (R(x)k|k)_{K}, ∀k ∈ K a.e. x.  (4.3)

Then, T_{t}^{V} is a positivity preserving semigroup and

|T⃗_{t}^{R}u| ≤ T_{t}^{V} |u|.  (4.4)

Proof. We can easily see that E(|f|, |f|) ≤ E(f, f) and hence T_{t}^{V} is positivity preserving (see, e.g. [29, Theorem XIII.50]).

Next we see (4.4). By the Trotter product formula, we have

T_{t}^{V} f = lim_{n→∞}(e^{−tV/n}T_{t/n})^{n}f

and

T⃗_{t}^{R}u = lim_{n→∞}(e^{−tR/n}T⃗_{t/n})^{n}u.

By noting

|T⃗_{t}u| ≤ T_{t}|u|,

and

∥e^{−tR(x)}∥_{op} ≤ e^{−tV (x)} a.e. x,

we have

|T⃗_{t}^{R}u| = lim_{n→∞} |(e^{−tR/n}T⃗_{t/n})^{n}u| ≤ lim_{n→∞}(e^{−tV/n}T_{t/n})^{n}|u| = T_{t}^{V} |u|.

This completes the proof.

We further assume the following condition (DLS) (the defective logarithmic Sobolev inequality): (DLS) There exist α, β > 0 such that

∫_{X} f^{2} log(f^{2}/∥f∥_{2}^{2})dm ≤ αE(f, f) + β∥f∥_{2}^{2} for f ∈ Dom(E).  (4.5)

Then, as we have mentioned after Theorem 2.5, we have a similar inequality for E⃗:

∫_{X} |u|^{2} log(|u|^{2}/∥u∥_{2}^{2})dm ≤ αE⃗(u, u) + β∥u∥_{2}^{2} for u ∈ Dom(E⃗).  (4.6)

Under the above assumption, we have, for t > 0 and 1 < p < q < ∞ with \frac{q−1}{p−1} ≤ e^{4t/α},

∥T_{t}∥_{p→q} ≤ exp\{β(\frac{1}{p} − \frac{1}{q})\}  (4.7)

(see [14, Theorem 6.1.14] and [18, Lemma 5.5]). Proposition 4.2. Assume that (DLS) holds. Then for t > 0 and 1 < p < q < ∞ such

that \frac{q − 1}{p − 1} ≤ e^{4t/α′} with α′ > α, we have

∥T_{t}^{V} ∥_{p→q} ≤ exp\{β(\frac{1}{p} − \frac{1}{q})\}∥e^{−V} ∥_{r}^{t}e^{St}  (4.8)

where

r = \frac{1}{4}\{\frac{1}{α} − \frac{1}{α′} \}^{−1}\{\frac{p^{2}}{p − 1} ∨ \frac{q^{2}}{q − 1}\}, S = \frac{4β}{α} .

where ∨ denotes the maximum. In particular, for any t > 0,

∥T_{t}^{V} ∥_{p→p} ≤ ∥e^{−V} ∥_{αp^{2}/4(p−1)}^{t}e^{4βt/α}.  (4.9)

Proof. Define α′′ so that \frac{q − 1}{p − 1} = e^{4t/α′′}. For n ∈ ℕ, we define a sequence \{p_{0}, p_{1}, …, p_{n}\} inductively as follows:

p_{0} = p,
e^{4t/nα′′} = \frac{p_{k+1} − 1}{p_{k} − 1}, k = 0, …, n − 1.

Clearly p_{n} = q. Moreover define q_{k} and r_{k} as

e^{4t/nα} = \frac{q_{k} − 1}{p_{k} − 1}

and

\frac{1}{p_{k+1}} = \frac{1}{r_{k}} + \frac{1}{q_{k}} .

Then,

∥e^{−tV/n}T_{t/n}f∥_{p_{k+1}} ≤ ∥e^{−tV/n}∥_{r_{k}}∥T_{t/n}f∥_{q_{k}}

≤ ∥e^{−tV/n}∥_{r_{k}}∥T_{t/n}∥_{p_{k}→q_{k}}∥f∥_{p_{k}}

≤ ∥e^{−tV/n}∥_{r_{k}} exp\{4β(\frac{1}{p_{k}} − \frac{1}{q_{k}} )\}∥f∥_{p_{k}}.

On the other hand,

\frac{1}{r_{k}} = \frac{1}{p_{k+1}} − \frac{1}{q_{k}}

= \frac{1}{e^{4t/nα′′}(p_{k} − 1) + 1} − \frac{1}{e^{4t/nα}(p_{k} − 1) + 1}

= \frac{(e^{4t/nα} − e^{4t/nα′′})(p_{k} − 1)}{(e^{4t/nα′′}(p_{k} − 1) + 1)(e^{4t/nα}(p_{k} − 1) + 1)}.

In general, for 0 < a < b, set

f(x) = \frac{1}{ax + 1} − \frac{1}{bx + 1} x ≥ 0.

Then,

f′(x) = −\frac{a}{(ax + 1)^{2}} + \frac{b}{(bx + 1)^{2}} = \frac{(b − a)(1 − abx^{2})}{(ax + 1)^{2}(bx + 1)^{2}} .

f takes its maximum at x = (ab)^{−1/2}. Note that

f((ab)^{−1/2}) = \frac{\sqrt{b}}{\sqrt{a} + \sqrt{b}} − \frac{\sqrt{a}}{\sqrt{a} + \sqrt{b}} = \frac{\sqrt{b} − \sqrt{a}}{\sqrt{a} + \sqrt{b}}.

Hence we have

\frac{(e^{4t/nα} − e^{4t/nα′′})(p − 1)}{(e^{4t/nα′′}(p − 1) + 1)(e^{4t/nα}(p − 1) + 1)} ∧ \frac{(e^{4t/nα} − e^{4t/nα′′})(q − 1)}{(e^{4t/nα′′}(q − 1) + 1)(e^{4t/nα}(q − 1) + 1)}

≤ \frac{1}{r_{k}} ≤ \frac{e^{2t/nα} − e^{2t/nα′′}}{e^{2t/nα′′} + e^{2t/nα}} .

and therefore

P_{n} := \frac{(e^{4t/nα′′}(p − 1) + 1)(e^{4t/nα}(p − 1) + 1)}{(e^{4t/nα} − e^{4t/nα′′})(p − 1)} ∨ \frac{(e^{4t/nα′′}(q − 1) + 1)(e^{4t/nα}(q − 1) + 1)}{(e^{4t/nα} − e^{4t/nα′′})(q − 1)}
≥ r_{k}
Q_{n} := \frac{e^{2t/nα} − e^{2t/nα′′}}{e^{2t/nα′′} + e^{2t/nα}} ≥ \frac{1}{r_{k}} .

Further

∑_{k=0}^{n−1}(\frac{1}{p_{k}} − \frac{1}{q_{k}} ) = ∑_{k=0}^{n−1}(\frac{1}{p_{k}} − \frac{1}{p_{k+1}} + \frac{1}{r_{k}} ) = \frac{1}{p} − \frac{1}{q} + ∑_{k=0}^{n−1} \frac{1}{r_{k}} ≤ \frac{1}{p} − \frac{1}{q} + nQ_{n}.

Now

∥(e^{−tV/n}T_{t/n})^{n}∥_{p→q} ≤ ∏_{k=0}^{n−1} ∥e^{−tV/n}T_{t/n}∥_{p_{k}→p_{k+1}}

≤ ∏_{k=0}^{n−1} ∥e^{−tV/n}∥_{r_{k}} exp\{4β(\frac{1}{p_{k}} − \frac{1}{q_{k}} )\}

≤ ∥e^{−tV/n}∥_{P_{n}}^{n} exp\{4β(\frac{1}{p} − \frac{1}{q} + nQ_{n})\}

≤ exp\{4β(\frac{1}{p} − \frac{1}{q})\}\{ ∫_{X} e^{−tP_{n}V/n}dm\}^{n/P_{n}}e^{4βnQ_{n}}.

Letting n → ∞, we have

\frac{P_{n}}{n} = \frac{(e^{4t/nα′′}(p − 1) + 1)(e^{4t/nα}(p − 1) + 1)}{n(e^{4t/nα} − e^{4t/nα′′})(p − 1)} ∨ \frac{(e^{4t/nα′′}(q − 1) + 1)(e^{4t/nα}(q − 1) + 1)}{n(e^{4t/nα} − e^{4t/nα′′})(q − 1)}

→ \frac{1}{4t}\{\frac{1}{α} − \frac{1}{α′′} \}^{−1}\{\frac{p^{2}}{p − 1} ∨ \frac{q^{2}}{q − 1}\} ≤ U/t

and

4βnQ_{n} = \frac{4βn(e^{2t/nα} − e^{2t/nα′′})}{e^{2t/nα′′} + e^{2t/nα}} → 4βt\{\frac{1}{α} − \frac{1}{α′′} \} ≤ St.

Thus we have

∥T_{t}^{V} ∥_{p→q} ≤ exp\{4β(\frac{1}{p} − \frac{1}{q})\}∥e^{−V} ∥_{r}^{t}e^{St}.

Lastly, we note that when q tends to p, α′′ tends to ∞ and hence we have (4.9). Proposition 4.3. Suppose that there exists α̃ > α such that exp{α̃∥R∥_{op}} ∈ L^{1}. Then

E⃗^{R}(u, v) := E(u, v) + ∫_{X}(Ru|v)_{K}dm

is well-defined for u, v ∈ Dom(E⃗) as a lower semibounded closed quadratic form in L^{2}(m). Moreover if D ⊆ Dom(E⃗) is dense in Dom(E⃗), then so is in Dom(E⃗^{R}). Proof. By the Hausdorff-Young inequality st ≤ s log s − s + e^{t}, s > 0, t ∈ ℝ, we have for u ∈ Dom(E⃗) with ∥u∥_{2} = 1,

∫_{X} |(Ru|u)_{K}|dm ≤ ∫_{X} ∥R∥_{op}|u|^{2}dm

= \frac{1}{α̃} ∫_{X} |u|^{2}(α̃∥R∥_{op})dm

= \frac{1}{α̃} ∫_{X}\{|u|^{2} log |u|^{2} − |u|^{2} + exp(α̃∥R∥_{op})\}dm

= \frac{1}{α̃}\{αE⃗(u, u) + β − 1 + ∥ exp\{α̃∥R∥_{op}\}∥_{1}\}.

For general u, we have

∫_{X} |(Ru|u)_{K}|dm = \frac{α}{α̃}E⃗(u, u) + \frac{1}{α̃}(β − 1 + ∥ exp\{α̃∥R∥_{op}\}∥_{1})∥u∥_{2}^{2}.

Now, by the KLMN theorem, the assertion follows.

We denote the semigroup associated with E⃗^{R} and E^{V} by {T⃗_{t}^{R}} and {T_{t}^{V} }, respectively. We have the following comparison theorem. Theorem 4.4. We have

|T⃗_{t}^{R}u| ≤ T_{t}^{V} |u|  (4.10)

To prove this theorem, we need an approximation argument. We set

X_{n} = ∥x ∈ X; ∥R(x)∥_{op} ≤ n, |V (x)| ≤ n}

and define

R_{n} = 1_{X_{n}}R, V_{n} = 1_{X_{n}}V.

We shall show the convergence of E⃗^{R_{n}}. To do this, we use the convergence of quadratic forms. We say that a sequence of quadratic forms E^{n} converges to E in the sense of Mosco (see, e.g., [27]) if (M.1) For any sequence {u_{n}} that converges to u weakly, it holds that

E(u, u) ≤ \underline{lim}_{n→∞} E^{n}(u_{n}, u_{n})  (4.11)

(M.2) For any u ∈ H, there exists a sequence {u_{n}} converging to u strongly, such that

E(u, u) ≥ \overline{lim}_{n→∞} E^{n}(u_{n}, u_{n})  (4.12)

Proposition 4.5. Suppose that there exists α′ > α such that exp{2α′∥R∥_{op}} ∈ L^{1}. Then, E⃗^{R_{n}} converges to E⃗^{R} in the sense of Mosco.

Proof. We first show (M.1). Let {u_{n}} be a sequence that converges to u weakly. We may assume that {E^{R_{n}}(u_{n}, u_{n})} is a converging sequence. Since there exist constants a, b > 0 (independent of n) such that

E⃗^{R}(u, u) ≤ aE⃗^{R_{n}}(u, u) + b∥u∥_{2}^{2}.

Hence {u_{n}} is a bounced sequence in Dom(E⃗^{R}) and u ∈ Dom(E⃗). By the lower semicontinuity,

E⃗(u, u) + ∫_{X}(Ru|u)_{K}dm − ∫_{X} 1_{X\X_{N}} ∥R∥_{op}|u|^{2}dm

≤ \underline{lim}_{n→∞}\{E⃗(u_{n}, u_{n}) + ∫_{X}(Ru_{n}|u_{n})_{K}dm − ∫_{X} 1_{X\X_{N}} ∥R∥_{op}|u_{n}|^{2}dm\}

≤ lim_{n→∞}\{E⃗(u_{n}, u_{n}) + ∫_{X}(Ru_{n}|u_{n})_{K}dm\} − \overline{lim}_{n→∞} ∫_{X} 1_{X\X_{n}}∥R∥_{op}|u_{n}|^{2}dm.

Hence

\overline{lim}_{n→∞} ∫_{X} ∥R∥_{op}1_{X\X_{n}}|u_{n}|^{2}dm ≤ E⃗(u, u) + ∫_{X}(Ru|u)_{K}dm − ∫_{X} 1_{X\X_{N}} ∥R∥_{op}|u|^{2}dm

− lim_{n→∞}{E⃗(u_{n}, u_{n}) + ∫_{X}(Ru_{n}|u_{n})_{K}dm}.

Since the left hand side is independent of N, by letting N → ∞, we have

\overline{lim}_{n→∞} ∫_{X} 1_{X\X_{n}}∥R∥_{op}|u_{n}|^{2}dm ≤ E⃗(u, u) + ∫_{X}(Ru|u)_{K}dm

− lim_{n→∞}{E⃗(u_{n}, u_{n}) + ∫_{X}(Ru_{n}|u_{n})_{K}dm}.

We now turn to E⃗^{R_{n}}(u_{n}, u_{n}).

\underline{lim}_{n→∞} E⃗^{R_{n}}(u_{n}, u_{n}) = \underline{lim}_{n→∞}\{E⃗(u_{n}, u_{n}) + ∫_{X}(R_{n}u_{n}|u_{n})_{K}dm\}

= \underline{lim}_{n→∞}\{E⃗(u_{n}, u_{n}) + ∫_{X}(Ru_{n}|u_{n})_{K}dm + ∫_{X}(1_{X\X_{n}}Ru_{n}|u_{n})dm\}

≥ \underline{lim}_{n→∞}\{E⃗(u_{n}, u_{n}) + ∫_{X}(Ru_{n}|u_{n})_{K}dm − ∫_{X} 1_{X\X_{n}}∥R∥_{op}|u_{n}|^{2}dm\}

= lim_{n→∞}\{E⃗(u_{n}, u_{n}) + ∫_{X}(Ru_{n}|u_{n})_{K}dm\} − \overline{lim}_{n→∞} ∫_{X} 1_{X\X_{n}}∥R∥_{op}|u_{n}|^{2}dm\}

≥ lim_{n→∞}{E⃗(u_{n}, u_{n}) + ∫_{X}(Ru_{n}|u_{n})_{K}dm} + E⃗(u, u) + ∫_{X}(Ru|u)_{K}dm

− lim_{n→∞}{E⃗(u_{n}, u_{n}) + ∫_{X}(Ru_{n}|u_{n})_{K}dm}

= E⃗(u, u) + ∫_{X}(Ru|u)_{K}dm.

Next we see (M.2). For any u ∈ Dom(E⃗) = Dom(E⃗^{R}), we take u_{n} = u. Then, by the dominated convergence theorem, we have

E⃗^{R_{n}}(u, u) = E⃗(u, u) + ∫_{X}(1_{X_{n}}Ru, u)dm → E⃗(u, u) + ∫_{X}(Ru, u)dm = E⃗^{R}(u, u).

This completes the proof.

Proof of Theorem 4.4. By Proposition 4.1, we have

|T⃗_{t}^{R_{n}}u| ≤ T_{t}^{V_{n}}|u|.

By letting n → ∞, we can get the desired result.

We have proved the strong convergence of {T⃗_{t}^{R_{n}}}. But we can even obtain convergence in norm.

Proposition 4.6. Suppose that exp{∥R∥_{op}} ∈ L^{∞−}. Then, {T⃗_{t}^{R}} is a strongly continuous semigroup in L^{p} for any p > 1.

Moreover {T⃗_{t}^{R_{n}}} converges to {T⃗_{t}^{R}} in norm sense as an operator in L^{p}.

Proof. We follow the Segal method. We take V = −∥R∥_{op} and let R_{n} be the same as before. We first note the following Duhamel formula:

T⃗_{t}^{R_{n}}u = T⃗_{t}^{R_{m}}u + ∫_{0}^{t} T⃗_{t−s}^{R_{n}}(R_{m} − R_{n})T⃗_{s}^{R_{m}}uds.

We choose q′ < p < q so that \frac{q−1}{p−1} < e^{2t/α}, \frac{p−1}{q′−1} < e^{2t/α}. By Proposition 4.2, there exist constants C > 0 and M > 0 such that for s ≥ t/2,

∥T⃗_{s}^{R_{n}}∥_{p→q} ≤ CM^{s},
∥T⃗_{s}^{R_{n}}∥_{q′→p} ≤ CM^{s},

and for all s > 0,

∥T⃗_{s}^{R_{n}}∥_{p→p} ≤ M^{s}.

We choose r > 1 so that

\frac{1}{p} ≥ \frac{1}{q} + \frac{1}{r},
\frac{1}{q′} ≥ \frac{1}{p} + \frac{1}{r}.

Then, by the Hölder inequality, we have,

∥T⃗_{t}^{R_{n}}u − T⃗_{t}^{R_{m}}u∥_{p} ≤ ∫_{0}^{t/2} ∥T⃗_{t−s}^{R_{n}}∥_{q′→p}∥R_{m} − R_{n}∥_{r}∥T⃗_{s}^{R_{m}}∥_{p→p}∥u∥_{p}ds
+ ∫_{t/2}^{t} ∥T⃗_{t−s}^{R_{n}}∥_{p→p}∥R_{m} − R_{n}∥_{r}∥T⃗_{s}^{R_{m}}u∥_{p→q}∥u∥_{p}ds.

Therefore

∥T⃗_{t}^{R_{n}} − T⃗_{t}^{R_{m}}∥_{p→p} ≤ CM^{t}∥R_{m} − R_{n}∥_{r}.

Now it is easy to see that T⃗_{t}^{R_{n}} is a Cauchy sequence with respect to the operator norm in L^{p}. But T⃗_{t}^{R_{n}} converges to T⃗_{t}^{R} strongly in L^{2}. Hence the limit of T⃗_{t}^{R_{n}} is T⃗_{t}^{R}. This completes the proof.

Now we give a sufficient condition of essential self-adjointness of A⃗^{R}.

Theorem 4.7. For any p ∈ (1, ∞), a subspace D is dense in Dom(A⃗_{p}^{R}) if it is dense in Dom(A⃗_{q}) for some q > p.

Proof. We mimic the method of L. M. Wu [38, Theorem 2.5]. Let q′ be the conjugate exponent of q: \frac{1}{q} + \frac{1}{q′} = 1. We first show that for any r < q′, Dom(A⃗_{q′}) ⊆ Dom(A⃗_{r}^{R}) and for u ∈ Dom(A⃗_{q′})

A⃗_{r}^{R}u = A⃗_{q′}u − Ru.  (4.13)

To show this, we first note

G⃗_{α}^{R_{n}}(A⃗_{q′}u − Ru) = G⃗_{α}^{R_{n}}(A⃗_{q′}u − R_{n}u + R_{n}u − Ru)
= −u + αG⃗_{α}^{R_{n}}u + G⃗_{α}^{R_{n}}(R_{n}u − Ru).

By the Hölder inequality

∥G⃗_{α}^{R_{n}}(R_{n}u − Ru)∥_{r} ≤ ∥G⃗_{α}^{R_{n}}∥_{p→p}∥R_{n} − R∥_{s}∥u∥_{q′} → 0.

Here we take r so that \frac{1}{r} = \frac{1}{q′} + \frac{1}{s}. We recall that G⃗_{α}^{R_{n}} converges to G⃗_{α}^{R} in norm sense by Proposition 4.6. Hence, by letting n → ∞, we have

G⃗_{α}^{R}(A⃗_{q′}u − Ru) = −u + αG⃗_{α}^{R}u

which means (4.13).

We show that (α − A_{p}^{R})(D) is dense in L^{p}. To see this, let p′ and q′ be the conjugate exponents of p and q, respectively: \frac{1}{p} + \frac{1}{p′} = 1, \frac{1}{q} + \frac{1}{q′} = 1. We take v ∈ L^{p′} that is perpendicular to (α − A_{p}^{R})(D). for any w ∈ D,

(v + G⃗_{α}(Rv)|(α − A⃗_{q})w) = (v|(α − A⃗_{p})w) + (Rv|G⃗_{α}(α − A⃗_{q})w)
= (v|(α − A⃗_{p} + R)w)
= 0.

From the assumption, (α − A⃗_{q})(D) is dense in L^{q} and thereby

v = −G⃗_{α}(Rv).

Since Rv ∈ L^{q′}, G⃗_{α}^{R}(Rv) ∈ Dom(A⃗_{q′}), we have, v ∈ Dom(A⃗_{q′}) and

(α − A⃗_{q′})v = −Rv.

Hence, by (4.13), we have

(α − A⃗_{r}^{R})v = 0.

Since α − A⃗_{r}^{R} is injective, we have v = 0. This completes the proof.

5. Examples

In this section, we give examples.

Our main interest is in infinite dimensional spaces and so we first take an abstract Wiener space. Let (B, H, μ) be an abstract Wiener space, i.e., B is a real separable Banach space and μ is a Gaussian measure with a reproducing kernel Hilbert space H. Let F = (F^{1}, …, F^{d}) be a smooth function in the sense of Malliavin. We further assume that F is non-degenerate in the following sense: setting σ = (σ^{i,j}), σ^{i,j} = (DF^{i}, DF^{j}), we have that (det σ)^{−1} ∈ L^{∞−}. Take any point a ∈ ℝ^{d} and set S = {x ∈ B; F (x) = a}. We regard S as a submanifold of B. Since F is non-degenerate, we can define a measure on S in the following manner:

m(dx) := \sqrt{det(σ)}δ_{a}(F).  (5.1)

Here, in the right hand side, δ_{a} denotes the Dirac measure at a and δ_{a}(F) stands for the composite of a distribution δ_{a} and a Wiener function F in the sense of Watanabe (see, e.g. [37]). We regard m as a reference measure. Let W^{∞,∞−} be the set of all real-valued smooth functions in the sense of Malliavin. We identify two functions f, g ∈ W^{∞,∞−} if they coincide m-a.e. To be precise, denoting the above equivalence relation by ∼, we set F(S) := W^{∞,∞−}/ ∼ and we regard an element of F(S) as a smooth function on S. Clearly, F(S) is an algebra.

Next let us define tensor bundles. Set T (S)_{x} = {h ∈ H; ⟨DF^{i}(x), h⟩ = 0, i = 1, …, d}. Then T (S) = ⋃_{x∈S} T (S)_{x} is the tangent bundle of S. The cotangent bundle and tensor bundle can be defined similarly. We denote the set of all smooth tensor fields of type (l, m) by Γ(T_{m}^{l}(S)). The restriction of the inner product of H defines a metric on T (M). It can be regarded as a Riemannian metric, denoted by (·|·). We can also define a metric on the tensor bundle.

We introduce a Dirichlet form on L^{2}(m). The gradient operator on S is denoted by ∇. Then, ∇ defines a Dirichlet form as follows:

E(f, g) = ∫_{S}(∇f|∇g)dm.  (5.2)

Let {T_{t}} be the associated semigroup. The generator A is given by A = −∇^{∗}∇. In this case, Γ is expressed as

Γ(f, g) = (∇f|∇g)  (5.3)

and we have

E(f, g) = ∫_{S} Γ(f, g)dm.  (5.4)

It is easy to see that Γ has the derivation property and so the conditions (Γ) and (D) in Section 3 are fulfilled.

Aida [3] considered the operator A when F is given as the solution of a stochastic differential equation on a compact Riemannian manifold. He showed that A is essentially self-adjoint on W^{∞,∞−}. Moreover, in the L^{p} setting, he obtained the denseness of W^{∞,∞−} in Dom(A) and therefore we can apply our perturbation method discussed in Section 4.

We can also define a covariant derivative ∇ acting on tensor fields on S (see, e.g. [22]). Then the quadratic form

E⃗(u, v) = ∫_{S}(∇u|∇v)dm u, v ∈ Γ(T_{m}^{l}(S))  (5.5)

induces a contraction semigroup {T⃗_{t}} in L^{2}(Γ(T_{m}^{l}(S)) where L^{2}(Γ(T_{m}^{l}(S)) is the set of all measurable sections u with

∥u∥_{2} = {∫_{S} |u|^{2}dm}^{1/2} < ∞.

The generator is given by A⃗ = ∇^{∗}∇ and Γ⃗ is given by

Γ⃗(u, v) = (∇u|∇v).  (5.6)

Now the positivity of Γ⃗ is clear.

Next let us check that the assumptions in Proposition 3.5 are all satisfied. We first recall that for f ∈ F(S) and u ∈ Γ(T_{m}^{l}(S)), we have fu ∈ Γ(T_{m}^{l}(S)) and

∇(fu) = ∇f ⊗ u + f∇u.

Hence we have

(∇(fu)|∇v) + (u|∇(fv)) = (∇f ⊗ u + f∇u|∇v) + (∇u|∇f ⊗ v + f∇v)
= (∇f|(u|∇v) + (∇u|v)) + 2f(∇u|∇v)
= (∇f|∇(u|v)) + 2f(∇u|∇v).

Here, in the last line, we used

∇(u|v) = (∇u|v) + (u|∇v).

Thus we have that Γ and Γ⃗ satisfy (3.32). Thus, we can apply Theorem 3.4 and obtain

|T⃗_{t}u| ≤ T_{t}|u|.

We now turn to the finite dimensional case. Let M be a compact Riemannian manifold. We denote the Laplacian by Δ and the Levi-Civita covariant derivative by ∇. Let us consider the Hodge-Kodaira Laplacian □ = dd^{∗} + d^{∗}d where d is the exterior differential and d^{∗} is its dual. On the space of 1-forms, the Weitzenböck formula says that

□u = ∇^{∗}∇u + R(·, u).

Here R denotes the Ricci curvature. We consider two semigroups {T_{t}} and {T⃗_{t}}: {T_{t}} is generated by Δ and {T⃗_{t}} is generated by −□. Then, the Γ⃗ associated with □ is given by

2Γ⃗(u, v) := Δ(u|v) + (□u|v) + (u| □ v)
= (∇u|∇v) + 2R(u, v).

Since M is compact, there exists λ ∈ ℝ such that

R(u, u) ≥ −λ|u|^{2}

which implies

Γ⃗(u, u) + λ|u|^{2} ≥ 0.

Moreover we can easily check that (3.23) holds for C^{∞} sections. Then, by Proposition 3.5 and Theorem 3.4, we have

|T⃗_{t}u| ≤ e^{λt}T_{t}|u|.

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Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-01, Japan

E-mail address: ichiro@kusm.kyoto-u.ac.jp