LITTLEWOOD-PALEY INEQUALITY FOR A DIFFUSION SATISFYING THE LOGARITHMIC SOBOLEV INEQUALITY AND FOR THE BROWNIAN MOTION ON A RIEMANNIAN MANIFOLD

WITH BOUNDARY

ICHIRO SHIGEKAWA

Abstract. We discuss the Littlewood-Paley inequality for a diffusion process associated with a Dirichlet form of gradient type. We assume that the logarithmic Sobolev inequality holds and the negative part of Γ_{2} is exponentially integrable. Under these and some additional conditions, we showed that ∥∇u∥_{p} ≤ C∥\sqrt{1 − L}u∥_{q} for 1 < p < q. We also discuss the Brownian motion on a Riemannian manifold with boundary.

1. Introduction

In this paper, we discuss the Littlewood-Paley inequality. Typical example is the Brownian motion on the Euclidean space and it leads to the following inequality: for any p > 1 there exist a positive constant C such that

C^{−1}∥∇u∥_{p} ≤ ∥\sqrt{−Δ}u∥_{p} ≤ C∥∇u∥_{p}.  (1.1)

\sqrt{−Δ}, the square root of the minus Laplacian, is called the Cauchy operator. (1.1) is equivalent to the L^{p}-boundedness of the Riesz transformation.

This kind of inequality also holds for the Ornstein-Uhlenbeck process on an abstract Wiener space, which was proved by P. A. Meyer [11] in a probabilistic approach.

In this paper, we attempt to extend this inequality for a diffusion process associated with a Dirichlet form that admits a square field operator. There have been several related works, e.g., Bakry [3, 4], Shigekawa-Yoshida [16]. In these papers, they assumed that Γ_{2} is positive or bounded from below. We replace this boundedness assumption with the exponential integrability of negative part of Γ_{2}. To handle this case, we assume that the logarithmic Sobolev inequality holds. Moreover our square field operator is of the gradient form, i.e., the Dirichlet form E is given as follows;

E(u, v) = ∫_{M}(∇u, ∇v)μ(dx).  (1.2)

We adopt a probabilistic approach which was developed by Meyer and Bakry. We will show the inequality for the Littlewood-Paley G-function. Since the square field operator is given as a gradient, we consider another semigroup that acts on vector valued functions and use the semigroup domination to estimate vector valued functions. Using this method, the estimate for vector valued functions can be reduced to the scalar case. But the unboundedness of Γ_{2} causes some troubles and so we could not prove the exact inequality.

1991 Mathematics Subject Classification. 60J60, 58G32.

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE MINISTRY OF EDUCATION, CULTURE, SPORTS, SCIENCE AND TECHNOLOGY, GRANT-IN-AID FOR SCIENTIFIC RE-SEARCH (B), NO. 11440045, 1999

We only show that the L^{p}-norm is dominated by L^{q}-norm for 1 < p < q (see the precise statement in §2 ).

We also discuss the Brownian motion on a Riemannian manifold with boundary. We impose the Neumann boundary condition on the Brownian motion. In this case, the quantity corresponding to Γ_{2} is singular (i.e., it is not a function but a smooth measure). We deal with it by way of an associated additive functional. The additive functional belongs to the Kato class and we can show the exact inequality (i.e., no loss of exponent).

The organization of the paper is as follows. We give a formulation and a main result in §2 . We define Γ_{2} in our formulation. It is a generalization of Ricci curvature and is based on a square field operator for vector valued functions. In §3 , the maximal ergodic inequality for a semigroup with a potential is given. Here the logarithmic Sobolev inequality is essential. We give a proof of the main theorem in §5 . To do this, we prepare fundamental inequalities for the Littlewood-Paley G-function in §4 . A proof for the Littlewood-Paley inequality is given in §5 . Combining this with the intertwining property of semigroups, we can get the main result. The Brownian motion on a Riemannian manifold with boundary is dealt with in §6 .

2. Symmetric diffusion

Let us introduce a diffusion process that we use in the paper. Let M be a topological space. We assume M to be Souslinian. Suppose we are given a Borel probability measure μ on M and a Dirichlet form E in L^{2}(μ). We assume that there exists a Hunt diffusion process (X_{t}, P_{x})_{x∈M} associated with E. We denotes the generator and the semigroup by L and \{T_{t}\}, respectively. We assume that \sqrt{−1} ∈ Dom(L) and L\sqrt{−1} = 0 where \sqrt{−1} denotes the function that is identically equal to 1. Hence the diffusion (X_{t}) is conservative. We also assume that the Dirichlet form satisfies the following defective logarithmic Sobolev inequality: there exist α > 0 and β ≥ 0 such that

∫_{M} u^{2} log u/∥u∥_{2}μ(dx) ≤ αE(u, u) + β(u, u).  (2.1)

Here ( , ) denotes the inner product in L^{2}.

Further we assume that the square field operator Γ is well-defined. Here Γ: Dom(E) × Dom(E) → L^{1}(μ) is a continuous bilinear map which is characterized as follows:

2(Γ(v, w), u) = E(vw, u) − E(v, wu) − E(vu, w), ∀u, v, w ∈ E ∩ L^{∞}.  (2.2)

A crucial assumption is as follows; there exists a ‘gradient operator’ ∇ such that ∇ is a closed operator from L^{2}(μ) to L^{2}(μ; K) and it satisfies Γ(u, v) = (∇u, ∇v). Here K is a (separable) Hilbert space. L^{2}(μ; K) may be possibly the set of all square integrable section of a vector bundle over M. But we use L^{2}(μ; K) for notational convention. We need another semigroup {T̂_{t}} in L^{2}(μ; K). Let {T̂_{t}} be a contraction symmetric semigroup associated with a bilinear form Ê. We also need a square field operator for {T̂_{t}} and so we assume that (A.1) For θ ∈ Dom(L̂), it holds that |θ|_{K}^{2} ∈ Dom(L_{1}). Here L_{1} is the generator in L^{1}(μ). Under this condition, we define a square field operator Γ̂ as

2Γ̂(θ, η) = L(θ, η)_{K} − (L̂θ, η)_{K} − (θ, L̂η)_{K}.  (2.3)

We assume the following two properties: the positivity and the derivation property.

(A.2) Γ̂(θ, θ) ≥ 0 for θ ∈ Dom(Ê). (A.3) For θ, η ∈ Dom(Ê) ∩ L^{∞} and u ∈ Dom(E) ∩ L^{∞}, it holds that

2uΓ̂(θ, η) = −(∇u, ∇(θ, η)) + Γ̂(θ, uη) + Γ̂(uθ, η).  (2.4)

Then, by the semigroup domination theorem (see [14]), we have

|T̂_{t}θ| ≤ T_{t}|θ|.  (2.5)

Let S_{b}(K) be the space of all self-adjoint operator on K that is bounded from below. Let R be a function on M taking values in S_{b}(K). Define a bilinear form Ê^{R} by

Ê^{R}(θ, η) = Ê(θ, η) + ∫_{M}(R(x)θ(x), η(x))_{K}μ(dx).  (2.6)

The associated semigroup will be denoted by T̂_{t}^{R}. We assume the following intertwining property, which is crucial in the paper.

∇T_{t}u = T̂_{t}^{R}∇u, for u ∈ Dom(∇).  (2.7)

R plays the role of so called Γ_{2}.

We take a scalar function V such that

(R(x)k, k)_{K} ≥ V (x)(k, k)_{K}.  (2.8)

The semigroup generated by L − V is denoted by {T_{t}^{V} }. The generator of T̂_{t}^{R} is L̂ − R. Again by the domination theorem, it holds that

|T̂_{t}^{R}θ| ≤ T_{t}^{V} |θ|.  (2.9)

V can be decomposed as V = V_{+} − V_{−} where V_{+} = V ∨ 0 and V_{−} = (−V) ∨ 0. The last assumption is that (A.4) e^{V_{−}} ∈ L^{∞−} = ⋂_{p≥1} L^{p}.

For scalar functions, we can define two kinds of norms: ∥∇u∥_{p} and ∥\sqrt{1 − L}u∥_{p}. It is a fundamental question whether these norms are equivalent or not. For example, if the generator L is the Ornstein-Uhlenbeck operator on an abstract Wiener space, then the equivalence of two norms are known as the Meyer equivalence.

Under our conditions, we can get the following result.

Theorem 2.1. For any 1 < p < q < ∞, we have

∥∇u∥_{p} ≲ ∥\sqrt{1 − L}u∥_{q},  (2.10)
∥\sqrt{1 − L}u∥_{p} ≲ ∥∇u∥_{q} + ∥u∥_{q}.  (2.11)

In the above theorem, the notation A ≲ B stands for A ≤ kB for a positive constant k. Further, in (2.10) for example, the constant depends only on p but is independent of u. We use this convention in the sequel without mentioning.

To prove the theorem, we use the Littlewood-Paley G-function. We introduce it in §4 and give a proof of the theorem in §5 .

3. Maximal ergodic inequality

In this section, we discuss the maximal ergodic inequality. This inequality is known for a symmetric Markov semigroup (see e.g., Stein [17]). Here we consider a semigroup with a potential. To show the inequality, we adopt a probabilistic method due to Rota [13].

We consider an additive functional A_{t} associated to a smooth signed measure ρ under the Revuz correspondence. We define a Dirichlet form by

E^{ρ}(u, v) = E(u, v) + ∫_{M} ũṽρ(dx)  (3.1)

where ũ denotes the quasi-continuous modification of u. The associated semigroup is denoted by {T_{t}^{ρ}}, which is expressed as

T_{t}^{ρ}u(x) = E_{x}[u(X_{t})e^{−A_{t}}]  (3.2)

where E_{x} denote the expectation under the measure P_{x}.

Theorem 3.1. Assume that for any q ≥ 1, there exist constants c_{q}, β_{q} such that

E_{x}[e^{−qA_{t}}]^{1/q} ≤ c_{q}e^{β_{q}t}  ∀t ≥ 0, q.e.-x.  (3.3)

Here “q.e.” means that it holds except for a set of capacity 0. Then for any p > 1 there exist constants λ, c such that

∥sup_{t≥0} |e^{−λt}T_{t}^{ρ}u|∥_{p} ≤ c∥u∥_{p},  ∀u ∈ L^{p}.  (3.4)

In particular, if ρ is non-negative (i.e., A_{t} is non-negative), we can take λ = 0.

Proof. We note that |T_{t}^{ρ}u| ≤ T_{t}^{−ρ_{−}}|u|, where ρ = ρ_{+} − ρ_{−} is the Hahn decomposition of ρ. Without loss of generality, we may assume that ρ is non-positive.

Set

M_{t} = T_{T −t}^{ρ}u(X_{t})e^{−A_{t}}.  (3.5)

We show first that {M_{t}} is a martingale under P_{μ} := ∫_{M} P_{x}μ(dx). In fact,

E_{μ}[u(X_{T})e^{−A_{T}} |F_{t}] = E_{μ}[u(X_{T −t} ∘ θ_{t})e^{−A_{T −t}∘θ_{t}−A_{t}}|F_{t}]
= e^{−A_{t}}E_{μ}[u(X_{T −t} ∘ θ_{t})e^{−A_{T −t}∘θ_{t}}|F_{t}]
= e^{−A_{t}}E_{X_{t}}[u(X_{T −t})e^{−A_{T −t}}] (Markov property)
= e^{−A_{t}}T_{T −t}^{ρ}u(X_{t}).

We note, by the Markov property,

T_{T −t}^{ρ}u(X_{T}) = E_{X_{T}} [u(X_{T −t})e^{−A_{T −t}}]
= E_{μ}[u(X_{T −t} ∘ θ_{T})e^{−A_{T −t}∘θ_{T}} |X_{T}]
= E_{μ}[u(X_{2T −t})e^{−A_{2T −t}+A_{T}} |X_{T}].

Now, using the reversibility of (X_{t}), i.e., (X_{2T −t})_{0≤t≤2T} has the same law as (X_{t})_{0≤t≤2T}, we have

T_{T −t}^{ρ}u(X_{T}) = E_{μ}[u(X_{t})e^{−A_{T} +A_{t}}|X_{T}].

Hence

T_{2(T −t)}^{ρ}u(X_{T}) = T_{T −t}^{ρ}T_{T −t}^{ρ}u(X_{T})

= E_{μ}[T_{T −t}^{ρ}u(X_{t})e^{−A_{T} +A_{t}}|X_{T}]

= E_{μ}[M_{t}e^{−A_{T} +2A_{t}}|X_{T}].

Noting that we have taken A to be non-positive, we have

sup_{0≤t≤T} |T_{2(T −t)}^{ρ}u(X_{T})| ≤ E_{μ}[ sup_{0≤t≤T} |M_{t}|e^{−A_{T}} |X_{T} ]

≤ E_{μ}[ sup_{0≤t≤T} |M_{t}|^{p}|X_{T} ]^{1/p}E_{μ}[e^{−qA_{T}} |X_{T}]^{1/q} (1/p + 1/q = 1)

≤ c_{q}e^{β_{q}T} E_{μ}[ sup_{0≤t≤T} |M_{t}|^{p}|X_{T} ]^{1/p} (∵ (3.3) .)

Hence, by the Doob inequality

∥ sup_{0≤t≤T}|T_{2(T −t)}^{ρ}u|∥_{p}

≤ c_{q}e^{β_{q}T} E_{μ}[E_{μ}[ sup_{0≤t≤T} |M_{t}|^{p}|X_{T} ]]^{1/p}

= c_{q}e^{β_{q}T} E_{μ}[ sup_{0≤t≤T} |M_{t}|^{p}]^{1/p}

≤ C′c_{q}e^{β_{q}T} E_{μ}[|M_{T} |^{p}]^{1/p} (Doob’s inequality)

= C′c_{q}e^{β_{q}T} E_{μ}[|u(X_{T})|^{p}e^{−pA_{T}}]^{1/p}

= C′c_{q}e^{β_{q}T} E_{μ}[E_{μ}[|u(X_{T})|^{p}e^{−pA_{T}} |X_{T}]]^{1/p}

= C′c_{q}e^{β_{q}T} E_{μ}[|u(X_{T})|^{p}E_{μ}[e^{−pA_{T}} |X_{T}]]^{1/p}

= C′c_{q}e^{β_{q}T} E_{μ}[|u(X_{T})|^{p}E_{μ}[e^{−p(A_{2T} −A_{T})}|X_{T}]]^{1/p} (reversibility)

= C′c_{q}e^{β_{q}T} E_{μ}[|u(X_{T})|^{p}E_{μ}[e^{−pA_{T} ∘θ_{T})}|X_{T}]]^{1/p} (additivity)

= C′c_{q}e^{β_{q}T} E_{μ}[|u(X_{T})|^{p}E_{X_{T}} [e^{−pA_{T}}]]^{1/p}

≤ C′c_{q}e^{β_{q}T} c_{p}e^{β_{p}T} E_{μ}[|u(X_{T})|^{p}]^{1/p}

= C′c_{p}c_{q}e^{(β_{p}+β_{q})T} ∥u∥_{p}.

Thus we can find constants k > 0 and C > 0 which are independent of T and u such that

∥ sup_{0≤t≤2T} |T_{t}^{ρ}u|∥_{p} ≤ Ce^{2kT} ∥u∥_{p}

We take λ > k. Note that for any integer n,

∥ sup_{n≤t≤n+1} e^{−λt}|T_{t}^{ρ}u|∥_{p} ≤ e^{−λn}∥ sup_{0≤t≤n+1} |T_{t}^{ρ}u|∥_{p}
≤ Ce^{−λn}e^{k(n+1)}∥u∥_{r}.

Summing up in n,

∑_{n=0}^{∞}∥ sup_{n≤t≤n+1} e^{−λt}|T_{t}^{ρ}u|∥_{p} ≤ Ce^{k} ∑_{n=0}^{∞} e^{−(λ−k)n}∥u∥_{p}

≤ \frac{C}{e^{−k} − e^{−λ}} ∥u∥_{p}.

Clearly this leads us to

∥ sup_{0≤t<∞} e^{−λt}|T_{t}^{ρ}u|∥_{p} ≤ ∥∑_{n=0}^{∞} sup_{n≤t≤n+1} e^{−λt}|T_{t}^{ρ}u|∥_{p}

≤ ∑_{n=0}^{∞}∥ sup_{n≤t≤n+1} e^{−λt}|T_{t}^{ρ}u|∥_{p}

≤ \frac{C}{e^{−k} − e^{−λ}} ∥u∥_{p}.

This completes the proof.

The assumption (3.3) is rather strong. We replace it with the assumption (A.4). In this case, we set ρ = V m. Hence, the associated additive functional is given by

A_{t} = ∫_{0}^{t} V (X_{s})ds.  (3.6)

Here we denote the semigroup {T_{t}^{ρ}} by {T_{t}^{V} }. Since E satisfies the logarithmic Sobolev inequality (2.1) , we have (see, e.g., [14]),

∥T_{t}^{V} u∥_{p} ≤ ∥e^{−V} ∥_{αp^{2}/4(p−1)}^{t}e^{4βt/α}∥u∥_{p}.  (3.7)

This means that there exists a constant γ_{p} such that

∥T_{t}^{V} ∥_{p→p} ≤ e^{γ_{p}t}.  (3.8)

E.g., set γ_{p} = (4β/α) log ∥e^{−V} ∥_{αp^{2}/4(p−1)}.

In particular, when p = 2,

∥T_{t}^{V} u∥_{2} ≤ ∥e^{−V} ∥_{α}^{t}e^{4βt/α}∥u∥_{2}  (3.9)

In this case, taking u = 1, we have

E_{μ}[e^{−A_{t}}] = ∫_{X} E_{x}[1e^{−A_{t}}]μ(dx)
= ∥T_{t}^{V} 1∥_{1}
≤ ∥T_{t}^{V} 1∥_{2}
≤ ∥e^{−V} ∥_{α}^{t}e^{4βt/α}∥1∥_{2}
≤ ∥e^{−V} ∥_{α}^{t}e^{4βt/α}.

Hence, for any γ > 0, it holds that

E_{μ}[e^{−γA_{t}}] ≤ ∥e^{−γV} ∥_{α}^{t}e^{4βt/α}  (3.10)

Noticing this inequality, we can get the following maximal ergodic inequality.

Theorem 3.2. Take any 1 < p < r < ∞. If we take λ > 0 to be sufficiently large, then there exists a constant c > 0 such that

∥sup_{t≥0} |e^{−λt}T_{t}^{V} u|∥_{p} ≤ c∥u∥_{r}.  (3.11)

Proof. By the same proof as in Theorem 3.1 , we have

sup_{0≤t≤T} |T_{2(T −t)}^{V}u(X_{T})| ≤ E_{μ}[ sup_{0≤t≤T} |M_{t}|e^{−A_{T}} |X_{T} ].

Hence,

∥ sup_{0≤t≤T}|T_{2(T −t)}^{V}u|∥_{p}

≤ E_{μ}[E_{μ}[ sup_{0≤t≤T} |M_{t}|e^{−A_{T}} |X_{T} ]^{p}]^{1/p}

≤ E_{μ}[ sup_{0≤t≤T} |M_{t}|^{p}e^{−pA_{T}} ]^{1/p}

≤ E_{μ}[ sup_{0≤t≤T} |M_{t}|^{pq/p}]^{(p/q)·(1/p)}E_{μ}[e^{−puA_{T}}]^{(q−p)/pq} (\frac{1}{q/p} + \frac{1}{u} = 1)

≤ CE_{μ}[|M_{T} |^{q}]^{1/q}E_{μ}[e^{−puA_{T}}]^{(q−p)/pq} (Doob’s inequality)

= CE_{μ}[|u(X_{T})e^{−A_{T}} |^{q}]^{1/q}E_{μ}[e^{−puA_{T}}]^{(q−p)/pq}

≤ CE_{μ}[|u(X_{T})|^{rq/q}]^{(q/r)·(1/q)}E_{μ}[e^{−qvA_{T}}]^{(r−q)/rq}E_{μ}[e^{−puA_{T}}]^{(q−p)/pq} (\frac{1}{r/q} + \frac{1}{v} = 1)

≤ C∥u∥_{r}∥e^{−qvV} ∥_{α}^{(r−q)T/rq}e^{4β(r−q)T/αrq}∥e^{−puV} ∥_{α}^{(q−p)T/pq}e^{4β(q−p)T/αpq}.

Thus we can find a constant k > 0 which is independent of T and u such that

∥ sup_{0≤t≤2T} |T_{t}^{V} u|∥_{p} ≤ Ce^{2kT} ∥u∥_{r}.

The rest is the same as Theorem 3.1 . This completes the proof.

4. Littlewood-Paley G-functions

Let us introduce the Littlewood-Paley G-functions. To do this, we recall the subordination of a semigroup. Set T_{t}^{λ} = e^{−λt}T_{t} (λ ≥ 0). We take λ to be large enough. For any t ≥ 0, define a measure μ_{t} on [0, ∞) by

μ_{t}(ds) = \frac{t}{2\sqrt{π}} e^{−t^{2}/4s}s^{−3/2}ds.  (4.1)

In terms of the Laplace transform, this measure is characterized as

∫_{0}^{∞} e^{−αs}μ_{t}(ds) = e^{−\sqrt{α}t} for α > 0.

Then the subordination {Q_{t}^{λ}} of {T_{t}^{λ}} is defined by

Q_{t}^{λ} = ∫_{0}^{∞} T_{s}^{λ} μ_{t}(ds).  (4.2)

The generator of \{Q_{t}^{λ}\} in L^{2}(μ) is −\sqrt{λ − L}.

We recall that {T_{t}^{V} } is the semigroup with the potential V. We set T_{t}^{λ+V} = e^{−λt}T_{t}^{V} and we also define the subordination of {T_{t}^{V} } as

Q_{t}^{λ+V} = ∫_{0}^{∞} T_{s}^{λ+V} μ_{t}(ds).  (4.3)

The operator norm of {Q_{t}^{λ+V} } in L^{p} is estimated as

∥Q_{t}^{λ+V} ∥_{p→p} ≤ ∫_{0}^{∞} ∥T_{s}^{λ+V} ∥_{p→p} μ_{t}(ds)
≤ ∫_{0}^{∞} e^{−λs+γ_{p}s} μ_{t}(ds)
= e^{−\sqrt{λ−γ_{p}} t}.

Here γ_{p} is the constant in (3.8) . Moreover, by the semigroup domination |T̂_{t}^{λ+R}θ| ≤ T_{t}^{λ+V} |θ|, we have

∥T̂_{t}^{λ+R}∥_{p→p} ≤ ∥T_{t}^{λ+V} ∥_{p→p} ≤ e^{−(λ−γ_{p})t}.  (4.4)

Similarly we have

∥Q̂_{t}^{λ+R}∥_{p→p} ≤ e^{−\sqrt{λ−γ_{p}} t}.  (4.5)

For any real valued function u, define

g^{→}(x, t) = |∂_{t}Q_{t}^{λ}u(x)|^{2},  (4.6)
g^{↑}(x, t) = |∇Q_{t}^{λ}u(x)|_{K}^{2},  (4.7)
g(x, t) = g^{→}(x, t) + g^{↑}(x, t).  (4.8)

Here ∂_{t} = \frac{∂}{∂t}. Then, the Littlewood-Paley G-function is defined by

G^{→}u(x) = {∫_{0}^{∞} tg^{→}(x, t)dt}^{1/2},  (4.9)

G^{↑}u(x) = {∫_{0}^{∞} tg^{↑}(x, t)dt}^{1/2},  (4.10)

Gu(x) = {∫_{0}^{∞} tg(x, t)dt}^{1/2}.  (4.11)

Moreover, we define the H-functions by

H^{→}u(x) = {∫_{0}^{∞} tQ_{t}g^{→}(x, t)dt}^{1/2},  (4.12)

H^{↑}u(x) = {∫_{0}^{∞} tQ_{t}g^{↑}(x, t)dt}^{1/2},  (4.13)

Hu(x) = {∫_{0}^{∞} tQ_{t}g(x, t)dt}^{1/2}.  (4.14)

For vector valued function θ, we define G-function and H-function, similarly. That is, e.g.,

ĝ^{→}(x, t) = |∂_{t}Q̂_{t}^{λ+R}θ(x)|^{2},  (4.15)

Ĝ^{→}θ(x) = {∫_{0}^{∞} tĝ^{→}(x, t)dt}^{1/2}  (4.16)

Ĥ^{→}θ(x) = {∫_{0}^{∞} tQ_{t}ĝ^{→}(x, t)dt}^{1/2}.  (4.17)

Notice that, in this case, we use the semigroup {Q̂_{t}^{λ+R}} that is the subordination of {T̂_{t}^{λ+R}}. Ĝ^{↑}θ, Ĥ^{↑}θ, Ĝθ, and Ĥθ are defined similarly. For example,

ĝ^{↑}(x, t) = Γ̂(Q̂_{t}^{λ+R}θ, Q̂_{t}^{λ+R}θ)(x)

(see (2.3) for the definition of Γ̂).

The following proposition is easily obtained by the spectral decomposition:

Proposition 4.1. It holds that

∥G^{→}u∥_{2} = \frac{1}{2}∥u∥_{2},  (4.18)

and

∥Ĝ^{→}θ∥_{2} = \frac{1}{2}∥θ∥_{2}.  (4.19)

Later we need the interrelationship between G and H functions and so we first prepare the following.

Lemma 4.2. We have the following estimate:

|T_{t}^{λ+V} u(x)|^{2} ≤ {sup_{s≥0} T_{s}^{2(λ+V)}1(x)}T_{t}|u|^{2}(x),  (4.20)
|Q_{t}^{λ+V} u(x)|^{2} ≤ {sup_{s≥0} T_{s}^{2(λ+V)}1(x)}Q_{t}|u|^{2}(x).  (4.21)

Proof. By the Feynman-Kac formula, we have

|T_{t}^{λ+V} u(x)|^{2} = |E_{x}[exp{−λt − ∫_{0}^{t} V (X_{s})ds}u(X_{t})]|^{2}

≤ E_{x}[exp{−2λ − 2 ∫_{0}^{t} V (X_{s})ds}]E_{x}[|u(X_{t})|^{2}]

= T_{t}^{2(λ+V)}1(x) · T_{t}|u|^{2}(x)

≤ {sup_{s≥0} T_{s}^{2(λ+V)}1(x)} · T_{t}|u|^{2}(x).

Further we have,

|Q^{λ+V} u(x)|^{2} = |∫_{0}^{∞} T_{s}^{λ+V} u(x)λ_{t}(ds)|^{2}

≤ ∫_{0}^{∞} |T_{s}^{λ+V} u(x)|^{2}λ_{t}(ds)

≤ ∫_{0}^{∞}{sup_{r≥0} T_{r}^{2(λ+V)}1(x)} · T_{s}|u|^{2}(x)λ_{t}(ds)

= {sup_{s≥0} T_{s}^{2(λ+V)}1(x)} · Q_{t}|u|^{2}(x).

This completes the proof.

Now we can show the following estimate between G-functions and H-functions. Proposition 4.3. We have that

Ĝ^{→}θ ≤ 2{sup_{s≥0} T_{s}^{2(λ+V)}1(x)}^{1/2} · Ĥ^{→}θ.  (4.22)

For scalar function, we have

G^{→}u ≤ 2H^{→}u,  (4.23)
G^{↑}u ≤ 2{sup_{s≥0} T_{s}^{2(λ+V)}1(x)}^{1/2} · H^{↑}u.  (4.24)

Proof. We have,

|Q̂_{t}^{λ+R}θ(x)| ≤ ∫_{0}^{∞} |T̂_{s}^{λ+R}θ(x)|μ_{t}(ds) ≤ ∫_{0}^{∞} T_{s}^{λ+V} |θ|(x)μ_{t}(ds) = Q_{t}^{λ+V} |θ|(x).

Using Proposition 4.1 , we have

|Q̂_{t}^{λ+R}θ(x)|^{2} ≤ {Q_{t}^{λ+V} |θ|(x)}^{2} ≤ {sup_{s≥0} T_{s}^{2(λ+V)}1(x)} · Q_{t}|θ|^{2}(x).

Therefore

ĝ^{→}(x, 2t) = |∂_{s}Q̂_{s}^{λ+R}θ(x)|^{2}|_{s=2t}

= |\sqrt{λ − L̂ + R} Q̂_{2t}^{λ+R}θ(x)|^{2}

= |Q̂_{t}^{λ+R}\sqrt{λ − L̂ + R} Q̂_{t}^{λ+R}θ(x)|^{2}

≤ \{sup_{s≥0} T_{s}^{2(λ+V)}1(x)\}Q_{t}|\sqrt{λ − L̂ + R} Q̂_{t}^{λ+R}θ|^{2}(x)

= {sup_{s≥0} T_{s}^{2(λ+V)}1(x)}Q_{t}ĝ^{→}(x, t).

From this,

Ĝ^{→}θ(x) = {∫_{0}^{∞} tĝ^{→}(x, t)dt}^{1/2} = {4 ∫_{0}^{∞} tĝ^{→}(x, 2t)dt}^{1/2}

≤ 2{∫_{0}^{∞} t{sup_{s≥0} T_{s}^{2(λ+V)}1(x)}Q_{t}ĝ^{→}(x, t)dt}^{1/2}

≤ 2{sup_{s≥0} T_{s}^{2(λ+V)}1(x)}^{1/2}{∫_{0}^{∞} tQ_{t}ĝ^{→}(x, t)dt}^{1/2}

= 2{sup_{s≥0} T_{s}^{2(λ+V)}1(x)}^{1/2}Ĥ^{→}θ(x).

For the scaler function, it holds that G^{→}u ≤ 2H^{→}u since we have |Q_{t}u(x)|^{2} ≤ Q_{t}|u|^{2}(x). Let us next estimate G^{↑}u.

G^{↑}u(x) = { ∫_{0}^{∞} t|∇Q_{t}^{λ}u(x)|^{2}dt}^{1/2}

= {4 ∫_{0}^{∞} t|∇Q_{2t}^{λ}u(x)|^{2}dt}^{1/2}

= 2{ ∫_{0}^{∞} t|Q̂_{t}^{λ+R}∇Q_{t}^{λ}u(x)|^{2}dt}^{1/2}

≤ 2{ ∫_{0}^{∞} t{Q_{t}^{λ+V} |∇Q_{t}^{λ}u(x)|}^{2}dt}^{1/2}

≤ 2{ ∫_{0}^{∞} t{sup_{s≥0} T_{s}^{2(λ+V)}1(x)}Q_{t}|∇Q_{t}^{λ}u|^{2}(x)dt}^{1/2}

= 2{sup_{s≥0} T_{s}^{2(λ+V)}1(x)}^{1/2}{ ∫_{0}^{∞} tQ_{t}g^{↑}(x, t)dt}^{1/2}

= 2{sup_{s≥0} T_{s}^{2(λ+V)}1(x)}^{1/2}H^{↑}u(x).

Thus we have (4.24) . This completes the proof.

In the next section, we use the diffusion process generated by L + ∂_{a}^{2}. So we will do some calculation on L + ∂_{a}^{2}.

Lemma 4.4. For any θ, set f̂(x, a) = |Q̂_{a}^{λ+R}θ(x)| and for ε > 0, f̂_{ε}(x, a) = \sqrt{f̂(x, a)^{2} + ε}. Then we have

(L + ∂_{a}^{2})f̂^{2} ≥ 2(λ + V)f̂^{2} + 2ĝ.  (4.25)

and for 1 < p ≤ 2, it holds that

(L + ∂_{a}^{2})f̂_{ε}^{p} ≥ p(λ + V)f̂^{2}f̂_{ε}^{p−2} + p(p − 1)f̂_{ε}^{p−2}ĝ.  (4.26)

where ĝ = ĝ(x, a) was defined by

ĝ(x, a) = |∂_{a}Q̂_{a}^{λ+R}θ(x)|^{2} + Γ̂(Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ)(x).

For the scalar case, we define f(x, a) = |Q_{a}^{λ}u(x)|, f_{ε}(x, a) = \sqrt{f(x, a)^{2} + ε}. Then we have

(L + ∂_{a}^{2})f^{2} ≥ 2λf^{2} + 2g  (4.27)

and for 1 < p ≤ 2,

(L + ∂_{a}^{2})f_{ε}^{p} ≥ pλf^{2}f_{ε}^{p−2} + p(p − 1)f_{ε}^{p−2}g.  (4.28)

Proof. We first show (4.25) . To show this, we note that (L̂ − λ − R + ∂_{a}^{2})Q̂_{a}^{λ+R}θ(x) = 0. Moreover, using the identity 2Γ̂(θ, θ) = L|θ|^{2} − 2(L̂θ, θ), it holds that

L|Q̂_{a}^{λ+R}θ|^{2} = 2(L̂Q̂_{a}^{λ+R}θ, θ) + 2Γ̂(Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ).

Hence

(L + ∂_{a}^{2})f̂^{2} = (L + ∂_{a}^{2})|Q̂_{a}^{λ+R}θ|^{2}

= 2(∂_{a}^{2}Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ) + 2(∂_{a}Q̂_{a}^{λ+R}θ, ∂_{a}Q̂_{a}^{λ+R}θ)
+ 2(L̂Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ) + 2Γ̂(Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ)
= −2((L̂ − λ − R)Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ) + 2|∂_{a}Q̂_{a}^{λ+R}θ|^{2}
+ 2(L̂Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ) + 2Γ̂(Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ)
≥ 2(λ + V)|Q̂_{a}^{λ+R}θ|^{2} + 2ĝ(x, a).

Secondly we show (4.12) . To show this we recall the following fundamental relationship between L and ∇: for F (ξ^{1}, ξ^{2}, ⋯, ξ^{n}) ∈ C^{∞}(ℝ^{n}) and f^{1}, f^{2}, ⋯, f^{n} ∈ Dom(L),

LF (f^{1}, f^{2}, ⋯, f^{n}) = ∑_{i=1}^{n} \frac{∂F}{∂ξ^{i}} Lf^{i} + ∑_{i,j=1}^{n} \frac{∂^{2}F}{∂ξ^{i}∂ξ^{j}} (∇f^{i}, ∇f^{j})

(see [5] Lemma 1). Hence we have, for 1 < p ≤ 2,

(L + ∂_{a}^{2})f̂_{ε}^{p} = (L + ∂_{a}^{2})(f̂_{ε}^{2})^{p/2}

= \frac{p}{2}(f̂_{ε}^{2})^{p/2−1}(L + ∂_{a}^{2})f̂_{ε}^{2}

+ \frac{p}{2}(\frac{p}{2} − 1)(f̂_{ε}^{2})^{p/2−2}\{(∂_{a}f̂_{ε}^{2})^{2} + |∇f̂_{ε}^{2}|^{2}\}

= \frac{p}{2}f̂_{ε}^{p−2}(L + ∂_{a}^{2})f̂_{ε}^{2} + \frac{p}{4}(p − 2)f̂_{ε}^{p−4}\{(∂_{a}f̂^{2})^{2} + |∇f̂^{2}|^{2}\}.

Let us recall that (see, e.g., [14, (3.11)])

|∇f̂^{2}|^{2} = |∇(Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ)|^{2} ≤ 4Γ̂(Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ)|Q̂_{a}^{λ+R}θ|^{2}.

Taking this into account, we have

(L + ∂_{a}^{2})f̂_{ε}^{p} ≥ \frac{p}{2}f̂_{ε}^{p−2}\{2(λ + V)|Q̂_{a}^{λ+R}θ|^{2} + 2ĝ\}

+ \frac{p}{4}(p − 2)f̂_{ε}^{p−4}\{4(∂_{a}Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ)^{2} + 4Γ̂(Q̂_{a}^{λ+R}θ, Q̂_{a}^{λ+R}θ)|Q̂_{a}^{λ+R}θ|^{2}\}

≥ \frac{p}{2}f̂_{ε}^{p−2}\{2(λ + V)|Q̂_{a}^{λ+R}θ|^{2} + 2ĝ\} + p(p − 2)f̂_{ε}^{p−4}f̂^{2}ĝ

≥ pf̂_{ε}^{p−2}(λ + V)|Q̂_{a}^{λ+R}θ|^{2} + pf̂_{ε}^{p−2}ĝ + p(p − 2)f̂_{ε}^{p−2}ĝ

≥ p(λ + V)f̂_{ε}^{p−2}f̂^{2} + p(p − 1)f̂_{ε}^{p−2}ĝ.

This completes the proof.

5. Equivalence of L^{p}-norms

In this section, we give estimates of G and H functions by a probabilistic method and then show the domination of norms. The original idea is due to P. A. Meyer [9] but we mainly follow Bakry [4].

Let (X_{t}, P_{x}) be the diffusion process on M associated with E as before. We need an additional 1-dimensional Brownian motion (B_{t})_{t≥0} and we regard M as a vertical space. We write P_{x}^{↑} in place of P_{x}. Let (B_{t}, P\overrightarrow{a}) be a 1-dimensional Brownian motion starting at a ∈ ℝ with the generator \frac{d^{2}}{da^{2}} . Note that this Brownian motion is different from the standard one up to constant. Let τ be the hitting time of (B_{t}) to 0, i.e.,

τ = inf{t ≥ 0 ; B_{t} = 0}.

We consider the following diffusion (Y_{t}, P_{(x,a)}) on the state space M × ℝ;

Y_{t} := (X_{t}, B_{t}), P_{(x,a)} := P_{x}^{↑} ⊗ P\overrightarrow{a}.  (5.1)

So the generator of (Y_{t}) is L + ∂_{a}^{2}. We denote the integration with respect to P_{(x,a)} and ∫_{M} P_{(x,a)}μ(dx) by E_{(x,a)} and E_{μ×δ_{a}}, respectively.

We use the following identities (see Meyer [9] for the proof): Let η: M × ℝ_{+} → [0, ∞) be measurable. Then

E_{μ×δ_{a}}[∫_{0}^{τ} η(X_{t}, B_{t})dt] = ∫_{M} μ(dx) ∫_{0}^{∞}(a ∧ t)η(x, t) dt  (5.2)

and

E_{μ×δ_{a}}[∫_{0}^{τ} η(X_{t}, B_{t})dt|X_{τ} = x] = ∫_{0}^{∞}(a ∧ t)Q_{t}η(x, t) dt.  (5.3)

We need an inequality for submartingales. Let (Z_{t}) be a non-negative continuous submartingale with the following Doob-Meyer decomposition;

Z_{t} = M_{t} + A_{t}

where (M_{t}) is a continuous martingale and (A_{t}) is a continuous increasing process with A_{0} = 0. Then, for p ≥ 1, it holds that

E[A_{∞}^{p}] ≤ C_{p}E[Z_{∞}^{p}].  (5.4)

For the proof, see Lenglart-Lépingle-Pratelli [8].

Before going to estimate G-function we prepare the following;

Proposition 5.1. For any p ≥ 1, we have

sup_{{α≥0 \atop N≥0}} E_{μ×δ_{N}} [\{∫_{0}^{τ} αe^{−\sqrt{α}B_{s}}ds\}^{p}] < ∞.  (5.5)

Proof. By the Itô formula, we have

e^{−\sqrt{α}B_{t}} = e^{−\sqrt{α}B_{0}} − \sqrt{α} ∫_{0}^{t} e^{−\sqrt{α}B_{s}}dB_{s} + ∫_{0}^{t} αe^{−\sqrt{α}B_{s}}ds.

Hence

∫_{0}^{t∧τ} αe^{−\sqrt{α}B_{s}}ds = e^{−\sqrt{α}B_{t∧τ}} − e^{−\sqrt{α}B_{0}} + M_{t}

where (M_{t}) is a martingale defined by

M_{t} = \sqrt{α} ∫_{0}^{t∧τ} e^{−\sqrt{α}B_{s}}dB_{s},

which satisfies

⟨M⟩_{t} = 2α ∫_{0}^{t∧τ} e^{−2\sqrt{α}B_{s}}ds.

Now, by the Burkholder inequality

E_{μ×δ_{N}} [\{∫_{0}^{τ} αe^{−\sqrt{α}B_{s}}ds\}^{p}]
≤ C_{p}E_{μ×δ_{N}} [(e^{−\sqrt{α}B_{τ}} − e^{−\sqrt{α}B_{0}})^{p}] + C_{p}E_{μ×δ_{N}} [⟨M⟩_{τ}^{p/2}]

≤ C_{p} + C_{p}E_{μ×δ_{N}} [\{∫_{0}^{τ} 4αe^{−\sqrt{4α}B_{s}}ds\}^{p/2}].

Thus it is enough to show (5.5) when p = 1.

E_{μ×δ_{N}} [∫_{0}^{τ} αe^{−\sqrt{α}B_{s}}ds] = ∫_{0}^{∞}(N ∧ a)αe^{−\sqrt{α}a}da ≤ ∫_{0}^{∞} aαe^{−\sqrt{α}a}da = 1.

This completes the proof.

G-functions are now estimated as follows. Proposition 5.2. For any 1 < p < q < 2, we have

∥Ĝθ∥_{p} ≲ ∥θ∥_{q}  (5.6)

and

∥θ∥_{q′} ≲ ∥G^{→}θ∥_{p′}  (5.7)

where p′ and q′ are the conjugate exponent of p and q, respectively.

For scalar functions, we have

∥Gu∥_{p} ≲ ∥u∥_{p}.  (5.8)

Proof. Set f̂(x, a) = |Q̂_{a}^{λ+R}θ(x)| and for ε > 0, f̂_{ε}(x, a) = \sqrt{f̂(x, a)^{2} + ε}. Define

Z_{t}^{(ε)} = f̂_{ε}(X_{t∧τ}, B_{t∧τ})^{p}

and

Z_{t} = f̂(X_{t∧τ}, B_{t∧τ})^{p}.

Then

M_{t}^{(ε)} = Z_{t}^{(ε)} − ∫_{0}^{t∧τ}(L + ∂_{a}^{2})f̂_{ε}(X_{s}, B_{s})^{p}ds

is a martingale. Therefore,

E_{μ×δ_{N}} [Z_{∞}^{(ε)} − Z_{0}^{(ε)}] = E_{μ×δ_{N}} [∫_{0}^{τ}(L + ∂_{a}^{2})f̂_{ε}(X_{s}, B_{s})^{p}ds].  (5.9)

According to Lemma 4.4 , we have

(L + ∂_{a}^{2})f̂_{ε}^{p} ≥ p(λ + V)f̂^{2}f̂_{ε}^{p−2} + p(p − 1)ĝf̂_{ε}^{p−2}
≥ −pV_{−}f̂^{2}f̂_{ε}^{p−2} + p(p − 1)ĝf̂_{ε}^{p−2}.  (5.10)

By letting ε → 0, we have

lim inf_{ε→0}(L + ∂_{a}^{2})f̂_{ε}^{p} ≥ −pV_{−}f̂^{p} + p(p − 1)ĝf̂^{p−2}.

Now, taking limit in (5.9) , we have

E_{μ×δ_{N}} [Z_{∞} − Z_{0}] ≥ E_{μ×δ_{N}} [∫_{0}^{τ} lim inf_{ε→0}(L + ∂_{a}^{2})f̂_{ε}(X_{s}, B_{s})^{p}ds]

≥ E_{μ×δ_{N}} [∫_{0}^{τ} −pV_{−}(X_{s})f̂(X_{s}, B_{s})^{p} + p(p − 1)ĝ(X_{s}, B_{s})f̂^{p−2}ds]. (5.11)

This limit is justified because E_{μ×δ_{N}} [∫_{0}^{τ} pV_{−}(X_{s})f̂(X_{s}, B_{s})^{p}ds] < ∞. From (5.10) , we have,

ĝ ≤ \frac{1}{p(p − 1)} lim inf_{ε→0}(L + ∂_{a}^{2})f̂_{ε}^{p} · f̂^{2−p} + \frac{1}{p − 1}V_{−}f̂^{2}.  (5.12)

Now we can estimate Ĝθ.

∥Ĝθ∥_{p}^{p} =∥{ ∫_{0}^{∞} aĝ(x, a)da}^{p/2}∥_{1}

≲∥{ ∫_{0}^{∞} a lim inf_{ε→0}(L + ∂_{a}^{2})f̂_{ε}^{p} · f̂^{2−p} + aV_{−}f̂^{2}da}^{p/2}∥_{1}

≤∥{ ∫_{0}^{∞} a lim inf_{ε→0}(L + ∂_{a}^{2})f̂_{ε}^{p} · f̂^{2−p}da}^{p/2}∥_{1}+∥{ ∫_{0}^{∞} aV_{−}f̂^{2}da}^{p/2}∥_{1}

=: I_{1} + I_{2}.

We recall that

f̂(x, a) = |Q̂_{a}^{λ+R}θ(x)| ≤ sup_{t≥0} T_{t}^{λ+V} |θ|(x).  (5.13)

I_{1} is estimated as

I_{1} ≤∥{sup_{t≥0} T_{t}^{λ+V} |θ|}^{p(2−p)/2}{ ∫_{0}^{∞} a lim inf_{ε→0}(L + ∂_{a}^{2})f̂_{ε}^{p}da}^{p/2}∥_{1}

≲∥{sup_{t≥0} T_{t}^{λ+V} |θ|}^{p}∥_{1}^{(2−p)/2}∥∫_{0}^{∞} a lim inf_{ε→0}(L + ∂_{a}^{2})f̂_{ε}^{p}da∥_{1}^{p/2}.

(∵ the Hölder inequality for the exponents 2/(2 − p) and 2/p)

Further we have

∥∫_{0}^{∞}a lim inf_{ε→0}(L + ∂_{a}^{2})f̂_{ε}^{p}da∥_{1}

= lim_{N→∞} ∫_{M} μ(dx) ∫_{0}^{∞}(N ∧ a) lim inf_{ε→0}(L + ∂_{a}^{2})f̂_{ε}^{p}da

= lim_{N→∞} E_{μ×δ_{N}} [∫_{0}^{τ} lim inf_{ε→0}(L + ∂_{a}^{2})f̂_{ε}^{p}(X_{t}, B_{t})dt] (∵ (5.2) )

≤ lim_{N→∞} E_{μ×δ_{N}} [Z_{∞} − Z_{0}] (∵ (5.11) )

≤ lim_{N→∞} E_{μ×δ_{N}} [|θ(X_{τ})|^{p}]

= ∥θ∥_{p}^{p}.

On the other hand, by virtue of Theorem 3.2 , we have

∥{sup_{t≥0} T_{t}^{λ+V} |θ|}^{p}∥_{1}^{(2−p)/2} ≤∥ sup_{t≥0} T_{t}^{λ+V} |θ|∥_{p}^{p(2−p)/2} ≲ ∥θ∥_{q}^{p(2−p)/2}.  (5.14)

Thus we have I_{1} ≲ ∥θ∥_{q}^{p}.

As for I_{2},

∥{ ∫_{0}^{∞} aV_{−}f̂^{2}da}^{p/2}∥_{1} ≤∥{ ∫_{0}^{∞} f̂^{2−p}aV_{−}f̂^{p}da}^{p/2}∥_{1}

≤∥{sup_{t≥0} T_{t}^{λ+V} |θ|}^{p(2−p)/2}{ ∫_{0}^{∞} aV_{−}f̂^{p}da}^{p/2}∥_{1}

≤∥{sup_{t≥0} T_{t}^{λ+V} |θ|}^{p}∥_{1}^{(2−p)/2}∥ ∫_{0}^{∞} aV_{−}f̂^{p}da∥_{1}^{p/2}

(∵ the Hölder inequality for the exponents 2/(2 − p) and 2/p)

Further we have

∥ ∫_{0}^{∞} aV_{−}f̂^{p}da∥_{1} ≤ ∫_{0}^{∞} a∥V_{−}f̂^{p}∥_{1}da

≤ ∫_{0}^{∞} a∥V_{−}∥_{r}∥f̂^{p}∥_{q/p}da (\frac{1}{q/p} + \frac{1}{r} = 1)

≤ ∥V_{−}∥_{r} ∫_{0}^{∞} a∥Q̂_{a}^{λ+R}θ∥_{q}^{p}da

≤ ∥V_{−}∥_{r} ∫_{0}^{∞} a∥Q̂_{a}^{λ+R}∥_{q→q}^{p}∥θ∥_{q}^{p}da

≤ ∥V_{−}∥_{r} ∫_{0}^{∞} ae^{−\sqrt{λ−γ_{q}}ap}∥θ∥_{q}^{p}da (∵ (4.5) )

= \frac{∥V_{−}∥_{r}∥θ∥_{q}^{p}}{(λ − γ_{q})p^{2}} .

This combined with (5.14) implies I_{2} ≲ ∥θ∥_{q}^{p} and we have proved (5.6) .

(5.7) is obtained by the duality argument. In fact, using Proposition 4.1 , we have

∫_{M}(θ(x), η(x))_{K}μ(dx) = 4 ∫_{M} μ(dx) ∫_{0}^{∞} a(∂_{a}Q̂_{a}^{λ+R}θ(x), ∂_{a}Q̂_{a}^{λ+R}η(x))_{K}da
≤ 4 ∫_{M} Ĝ^{→}θ(x)Ĝ^{→}η(x)μ(dx)
≤ 4∥Ĝ^{→}θ∥_{p}∥Ĝ^{→}η∥_{p′}
≲ ∥θ∥_{q}∥Ĝ^{→}η∥_{p′}.

Now (5.7) follows easily.

(5.8) for scalar functions can be shown much easily.

When p ≥ 2, we estimate Ĥu and Hu. Proposition 5.3. For any 2 < p < r, we have

∥Ĥθ∥_{p} ≲ ∥θ∥_{r}.  (5.15)

For scalar functions, we have

∥Hu∥_{p} ≲ ∥u∥_{p}.  (5.16)

Proof. We set f(x, a) = |Q̂_{a}^{λ+R}θ(x)| and define

Z_{t} = f̂(X_{t∧τ}, B_{t∧τ})^{2}.

Then,

M_{t} = Z_{t} − ∫_{0}^{t∧τ}(L + ∂_{a}^{2})f̂^{2}(X_{s}, B_{s})ds

is a martingale. By Lemma 4.4 , we have

2ĝ ≤ (L + ∂_{a}^{2})f̂^{2} + 2V_{−}f̂^{2}.  (5.17)

Then, setting

A_{t} = ∫_{0}^{t∧τ}{(L + ∂_{a}^{2})f̂^{2}(X_{s}, B_{s}) + 2V_{−}f̂^{2}(X_{s}, B_{s})}ds,  (5.18)

we can see that (A_{t}) is an increasing process and have that

Z_{t} + ∫_{0}^{t∧τ} 2V_{−}f̂^{2}(X_{s}, B_{s})ds = M_{t} + A_{t}.  (5.19)

Hence Z_{t} + ∫_{0}^{t∧τ} 2V_{−}f^{2}ds is a non-negative submartingale and its increasing part (A_{t}) satisfies

A_{t} ≥ ∫_{0}^{t∧τ} 2ĝ(X_{s}, B_{s})ds.  (5.20)

Therefore, by (5.4) , the following inequality hold.

E_{μ×δ_{N}} [{∫_{0}^{τ} 2ĝ(X_{s}, B_{s})ds}^{p/2}] ≤ E_{μ×δ_{N}} [A_{∞}^{p/2}]

≲ E_{μ×δ_{N}} [{Z_{∞} + ∫_{0}^{τ} 2V_{−}f̂^{2}ds}^{p/2}]

≲ E_{μ×δ_{N}} [Z_{∞}^{p/2}] + E_{μ×δ_{N}} [{∫_{0}^{τ} V_{−}f̂^{2}ds}^{p/2}]

= ∥θ∥_{p}^{p} + E_{μ×δ_{N}} [{∫_{0}^{τ} V_{−}f̂^{2}ds}^{p/2}].

The second term can be estimated as follows. We take any p < q < r.

E_{μ×δ_{N}} [{∫_{0}^{τ} V_{−}f̂^{2}ds}^{p/2}]

= E_{μ×δ_{n}}[{∫_{0}^{τ} e^{−ηB_{s}}e^{ηB_{s}}V_{−}f̂^{2}ds}^{p/2}]

≤ E_{μ×δ_{N}} [{∫_{0}^{τ} e^{−ηB_{s}q/(q−2)}ds}^{(q−2)p/2q}{∫_{0}^{τ} e^{ηB_{s}q/2}V_{−}^{q/2}f̂^{q}ds}^{2p/2q}]

(∵ the Hölder inequality for the exponents q/(q − 2) and q/2)

≤ E_{μ×δ_{N}} [{∫_{0}^{τ} e^{−ηB_{s}q/(q−2)}ds}^{u(q−2)p/2q}]^{1/u}E_{μ×δ_{N}} [∫_{0}^{τ} e^{ηB_{s}q/2}V_{−}^{q/2}f̂^{q}ds]^{p/q}

(∵ the Hölder inequality for the exponents u and q/p where \frac{1}{u} + \frac{1}{q/p} = 1)

≲ E_{μ×δ_{N}} [∫_{0}^{τ} e^{ηqB_{s}/2}V_{−}^{q/2}f̂^{q}ds]^{p/q}

= {∫_{M} μ(dx) ∫_{0}^{∞}(N ∧ a)e^{ηqa/2}V_{−}^{q/2}(x)|Q̂_{a}^{λ+R}θ(x)|^{q}da}^{p/q}.

To estimate the integral above, we recall that ∥Q̂_{t}^{λ+R}θ∥_{r} ≤ e^{−\sqrt{λ−γ_{r}}t}∥θ∥_{r}. Therefore,

∫_{M} μ(dx) ∫_{0}^{∞}(N ∧ a)e^{ηqa/2}V_{−}^{q/2}(x)|Q̂_{a}^{λ+R}θ(x)|^{q}da

≤ ∫_{0}^{∞} ae^{ηqa/2}da ∫_{M} V_{−}^{q/2}(x)|Q̂_{a}^{λ+R}θ(x)|^{q}μ(dx)

≤ ∫_{0}^{∞} ae^{ηqa/2}da{∫_{M} V_{−}^{vq/2}(x)μ(dx)}^{1/v}{∫_{M} |Q̂_{a}^{λ+R}θ(x)|^{qr/q}μ(dx)}^{q/r}

(\frac{1}{v} + \frac{1}{r/q} = 1)

≲ ∫_{0}^{∞} ae^{ηqa/2}∥Q̂_{a}^{λ+R}θ(x)∥_{r}^{q}da

≲ ∫_{0}^{∞} ae^{ηqa/2}e^{−\sqrt{λ−γ_{r}}qa}∥θ∥_{r}^{q}da

≲ ∥θ∥_{r}^{q} (∵ \sqrt{λ − γ_{r}}q > ηq/2).

Thus we have obtained

E_{μ×δ_{N}} [{∫_{0}^{τ} ĝ(X_{s}, B_{s})ds}^{p/2}] ≲ ∥θ∥_{r}^{p}.  (5.21)

Now we can estimate Ĥθ.

∥Ĥθ∥_{p}^{p} =∥{ ∫_{0}^{∞} aQ_{a}ĝ(x, a)da}^{p/2}∥_{1}

= lim_{N→∞} ∫_{M} μ(dx){ ∫_{0}^{∞}(a ∧ N)Q_{a}ĝ(x, a)da}^{p/2}

= lim_{N→∞} ∫_{M} μ(dx)E_{μ×δ_{N}} [ ∫_{0}^{τ} ĝ(X_{s}, B_{s})ds|X_{τ} = x]^{p/2}

≤ lim_{N→∞} ∫_{M} μ(dx)E_{μ×δ_{N}} [{∫_{0}^{τ} ĝ(X_{s}, B_{s})ds}^{p/2}|X_{τ} = x]

= lim_{N→∞} E_{μ×δ_{N}} [{∫_{0}^{τ} ĝ(X_{s}, B_{s})ds}^{p/2}]

≲ ∥θ∥_{r}^{p}. (∵ (5.21) )

The scalar case is easier. Combining Propositions 4.3 5.3, we can get Proposition 5.4. For any 2 ≤ p < q < ∞, we have

∥Ĝ^{→}θ∥_{p} ≲ ∥θ∥_{q}  (5.22)

and

∥θ∥_{q′} ≲ ∥Ĝ^{→}θ∥_{p′}  (5.23)

where p′ and q′ are the conjugate exponents of p and q, respectively.

For scalar functions, we have

∥Gu∥_{p} ≲ ∥u∥_{p}.  (5.24)

We are now ready to prove Theorem 2.1 .

Proof of Theorem 2.1

We take λ to be large enough. Recall that {Q_{t}^{λ}} is the subordination of {T_{t}^{λ}}. Then, by the intertwining property (2.7) , we have

∇Q_{t}^{λ} = Q̂_{t}^{λ+R}∇.

Now take any 1 < p < q < ∞. Then we have

∥∇u∥_{p} ≲ ∥Ĝ^{→}∇u∥_{q} = ∥{∫_{0}^{∞} a|∂_{a}Q̂_{a}^{λ+R}∇u(x)|^{2}da}^{1/2}∥_{q}

= ∥\{∫_{0}^{∞} a|∇Q_{a}^{λ}\sqrt{λ − L}u|^{2}da\}^{1/2}∥_{q}
= ∥G^{↑}\sqrt{λ − L}u∥_{q}
≲ ∥\sqrt{λ − L}u∥_{q}

which proves (2.10) .

The reversed inequality (2.11) is obtained by the duality argument. This completes the proof.

6. Riemannian manifold with boundary

In this section, we discuss the reflected Brownian motion on a Riemannian manifold with boundary. Let M be a compact Riemannian manifold with boundary ∂M. Let (X_{t}, P_{x})_{x∈M} be the Brownian motion on M with the Neumann boundary condition. We denote the Riemannian volume by m. In this section, the semigroup {T_{t}} is generated by L = Δ with the Neumann boundary condition. {T_{t}} is a symmetric and strongly continuous contraction semigroup in L^{2}(m). Further {T̂_{t}} is the semigroup generated by the Hodge-Kodaira Laplacian L̂ = −dd^{∗} − d^{∗}d with absolute boundary condition. The associated bilinear forms with L and L̂ are denoted by E and Ê. Moreover {T_{t}} and {T̂_{t}} satisfy the following intertwining property:

∇T_{t} = T̂_{t}∇.  (6.1)

As in §5 , we use an additional 1-dimensional Brownian motion (B_{t}, P\overrightarrow{a}) generated by \frac{d^{2}}{da^{2}} . Let τ be the hitting time of (B_{t}) to 0, and (Y_{t}, P_{(x,a)}) be the product diffusion process on the state space M × ℝ.

Y_{t} := (X_{t}, B_{t}), P_{(x,a)} := P_{x} ⊗ P\overrightarrow{a}.  (6.2)

So the generator of (Y_{t}) is L + ∂_{a}^{2}.

We use the notation E_{m×δ_{a}} = ∫_{M} P_{(x,a)}m(dx) in the same way as in §5 . For any f ∈ C^{∞}(M), we have

f(X_{t}, B_{t}) − f(X_{0}, B_{0}) = a martingale + ∫_{0}^{t}(L + ∂_{a}^{2})f(X_{s}, B_{s})ds + ∫_{0}^{t} ∇_{N}f(X_{s}, B_{s})dl_{s}.

Here {l_{t}} is an additive functional corresponding to the smooth measure σ (σ is the surface measure of ∂M), N is the inner normal vector and ∇ denotes the covariant differentiation. In particular we take 1-form θ with absolute boundary condition and set f(x, a) = |Q_{a}θ(x)|^{2}. Then,

f(X_{t∧τ}, B_{t∧τ}) − f(X_{0}, B_{0})

= M_{t} + ∫_{0}^{t∧τ}(L + ∂_{a}^{2})f(X_{s}, B_{s})ds + ∫_{0}^{t∧τ} \frac{∂f}{∂N} (X_{s}, B_{s})dl_{s}

= M_{t} + ∫_{0}^{t∧τ}(L + ∂_{a}^{2})f(X_{s}, B_{s})ds + ∫_{0}^{t∧τ} α(Q_{B_{s}}θ(X_{s}), Q_{B_{s}}θ(X_{s}))dl_{s}.

Here α is the second fundamental form of ∂M (see [15] for this identity.) The quadratic variation of (M_{t}) is given by

⟨M⟩_{t} = 2 ∫_{0}^{t∧τ} |∇Q_{B_{s}}θ(X_{s})|^{2} + |∂_{a}Q_{B_{s}}θ(X_{s})|^{2}ds.

Hence we can do the same argument as in the previous section. But we have to tackle the additional term ∫_{0}^{t∧τ} α(Q_{B_{s}}θ(X_{s}), Q_{B_{s}}θ(X_{s}))dl_{s}.

Next we see the semigroup domination. We note that for 1-forms θ, η and f ∈ C^{∞}(M),

−E((θ, η), f) + Ê(fθ, η) + Ê(θ, fη)
= 2 ∫_{M}(∇θ, ∇η)fm(dx) + 2 ∫_{M} Ric(θ, η)fm(dx) + 2 ∫_{∂M} α(θ, η)fσ(dx) (6.3)

where Ric is the Ricci curvature (refer to [15] for this identity.)

We take γ ≥ 0 and β ≥ 0 so that Ric(θ, θ) ≥ −γ|θ|^{2} and α(θ, θ) ≥ −β|θ|^{2}. Then α(θ, θ)σ ≥ −β|θ|^{2}σ as measures. It is easy to see that σ is a smooth measure. We also note that

L(θ, η) − (L̂θ, η) − (θ, L̂η) = 2(∇θ, ∇η) − 2 Ric(θ, η).  (6.4)

By (6.3) and (6.4) , the semigroup domination theorem implies (see [14, 15])

|T̂_{t}θ| ≤ T_{t}^{−γ−βσ}|θ|.  (6.5)

Here T_{t}^{−γ−βσ} is the semigroup which has −γ − βσ as a potential. It can be represented as

T_{t}^{−γ−βσ}u(x) = E_{x}[u(X_{t})e^{γt+βl_{t}}]  (6.6)

We can also show that (−l_{t}) satisfies the assumption of Theorem 3.1 . In fact, it is enough to see that there exists a function h ∈ C^{∞}(M) such that ∇_{N}h = 1 on ∂M. Then

h(X_{t}) − h(X_{0}) = M_{t} + ∫_{0}^{t} Δh(X_{s})ds + l_{t}

where (M_{t}) is a martingale with d⟨M⟩_{t} ≤ Cdt for a constant C > 0. Hence

E_{x}[e^{ql_{t}}] = E_{x}[exp{qh(X_{t}) − qh(X_{0}) − qM_{t} − q ∫_{0}^{t} Δh(X_{s})ds}].

The right hand side is bounded in x because h, Δh and ⟨M⟩ is bounded (this implies that σ is a Kato class potential; for Kato class potentials, see Albeverio-Ma [2]). Therefore, there exist constant c_{q} > 0 and β_{q} > 0 such that, for q.e.-x,

E_{x}[e^{ql_{t}}]^{1/q} ≤ c_{q}e^{β_{q}t}, ∀t ≥ 0.  (6.7)

Now we can apply Theorem 3.1 to T_{t}^{λ−γ−βσ}. For simplicity, we introduce the following notation:

M^{λ−γ−βσ}u(x) = sup_{t≥0}|T_{t}^{λ−γ−βσ}u(x)|.

When λ − γ = 0 and β = 0, we simply denote Mu in place of M^{λ−γ−βσ}u. Then, if λ is large enough, we have for any p > 1,

∥M^{λ−γ−βσ}u∥_{p} ≲ ∥u∥_{p}.  (6.8)

We can also obtain an estimate for the subordination. Let {Q_{t}^{λ−γ−βσ}} be the subordination of {T_{t}^{λ−γ−βσ}}. Then

|Q_{t}^{λ−γ−βσ}u| = |∫_{0}^{∞} T_{s}^{λ−γ−βσ}uμ_{t}(ds)|

≤ ∫_{0}^{∞} e^{−α^{2}s} sup_{r≥0}|T_{r}^{λ−α^{2}−γ−βσ}u|μ_{t}(ds)

= e^{−αt}M^{λ−α^{2}−γ−βσ}u.

Thus we have

sup_{t≥0}{e^{αt}|Q_{t}^{λ−γ−βσ}u|} ≤ M^{λ−α^{2}−γ−βσ}u.  (6.9)

We also note that {T̂_{t}} is a bounded operator in L^{p} by virtue of (6.5) and there exist constants c_{p} > 0 and γ_{p} > 0 so that

∥T̂_{t}∥_{p→p} ≤ c_{p}e^{γ_{p}t}.  (6.10)

Let {Q̂_{t}^{λ}} be the subordination of {T̂_{t}^{λ} = e^{−λt}T̂_{t}}. (6.5) implies |Q̂_{t}^{λ}θ| ≤ Q_{t}^{λ−γ−βσ}|θ|. We define Ĝ and Ĥ in terms of {Q̂_{t}^{λ}}. Now we can easily see that Proposition 4.3 holds in this case. We have more. In fact, by virtue of (6.7) , we can and do take λ large enough so that sup_{t≥0} T_{t}^{λ−γ−βσ}1(x) is bounded in q.e.-x and thereby we have

Ĝ^{→}θ ≲ Ĥ^{→}θ.  (6.11)

Similar estimate holds for G^{↑}u and H^{↑}u.

Lastly we note that, by combining the domination and (6.9) ,

sup_{t≥0}{e^{αt}|Q̂_{t}^{λ}θ|} ≤ M^{λ−α^{2}−γ−βσ}|θ|.  (6.12)

Next we extend (5.2) to additive functionals. Take any smooth measure ρ and let A_{t} be the additive functional associated with ρ. Then we have the following identity.

Proposition 6.1. For any non-negative function f on M × [0, ∞) and k on M, the following identity holds:

E_{m×δ_{a}}[∫_{0}^{τ} f(X_{t}, B_{t})dA_{t}] = ∫_{M} ρ(dx) ∫_{0}^{∞}(a ∧ t)f(x, t) dt  (6.13)
E_{m×δ_{a}}[k(X_{τ}) ∫_{0}^{τ} f(X_{t}, B_{t})dA_{t}] = ∫_{M} ρ(dx) ∫_{0}^{∞}(a ∧ t)Q_{t}k(x)f(x, t) dt (6.14)

Proof. Let us first recall the resolvent kernel for the absorbing Brownian motion on (0, ∞). Here, the generator is \frac{d^{2}}{da^{2}} . For α > 0, set

g_{α}(x, y) = \{ \begin{matrix}\frac{1}{2\sqrt{α}}(e^{\sqrt{α}a} − e^{−\sqrt{α}a})e^{−\sqrt{α}b}, & a ≤ b, \\ \frac{1}{2\sqrt{α}}e^{−\sqrt{α}a}(e^{\sqrt{α}b} − e^{−\sqrt{α}b}), & a ≥ b. \end{matrix}  (6.15)

Then the resolvent G_{α} = (α − \frac{d}{da^{2}} )^{−1} is given by

G_{α}h(a) = ∫_{0}^{∞} g_{α}(a, b)h(b)db.  (6.16)

Moreover we note that lim_{α→0} g_{α}(a, b) = a ∧ b.

By the Revuz correspondence, (see [6, the equation (5.1.14)]) we have

∫_{0}^{∞}h(a)daE_{m×δ_{a}}[∫_{0}^{τ} e^{−αs}f(X_{s}, B_{s})dA_{s}]
= ∫_{0}^{∞} G_{α}h(a)da ∫_{M} f(x, a)ρ(dx)
= ∫_{0}^{∞} h(a)da ∫_{M} ρ(dx) ∫_{0}^{∞} g_{α}(a, b)f(x, b)db (∵ g_{α} is symmetric)

Hence we have, for a.e.-a,

E_{m×δ_{a}}[∫_{0}^{τ} e^{−αs}f(X_{s}, B_{s})dA_{s}] = ∫_{M} ρ(dx) ∫_{0}^{∞} g_{α}(a, b)f(x, b)db.

But both hands are quasi-continuous in a and one point has positive capacity, the above identity holds for all a ≥ 0. By letting α → 0, we can get (6.13) .

To show (6.14) , set

H_{t} = ∫_{0}^{t∧τ} f(X_{s}, B_{s})dA_{s}.

Then H_{t} is a process of bounded variation. Hence, by the Itô formula,

Q_{B_{t∧τ}} k(X_{t∧τ})H_{t} = a martingale + ∫_{0}^{t∧τ}(L + ∂_{a}^{2})Q_{B_{s}}k(X_{s})H_{s}ds + ∫_{0}^{t∧τ} Q_{B_{s}}k(X_{s})dH_{s}
= a martingale + ∫_{0}^{t∧τ} Q_{B_{s}}k(X_{s})f(X_{s}, B_{s})dA_{s}.

Here we used that ∇_{N}Q_{a}k = 0 on ∂M because Q_{a}k belongs to the domain of the Neumann Laplacian. By taking expectation and letting t → ∞, we have

E_{m×δ_{a}}[Q_{B_{τ}} k(X_{τ}) ∫_{0}^{τ} f(X_{s}, B_{s})dA_{s}]

= E_{m×δ_{a}}[∫_{0}^{τ} Q_{B_{s}}k(X_{s})f(X_{s}, B_{s})dA_{s}]
= ∫_{M} ρ(dx) ∫_{0}^{∞}(a ∧ t)Q_{t}k(x)f(x, t) dt. (∵ (6.13) )

This completes the proof.

Recall that {Q_{t}^{λ}}, {Q̂_{t}^{λ}} are subordinations of {T_{t}^{λ}}, {T̂_{t}^{λ}}, respectively and G and H functions are defined in terms of {Q̂_{t}^{λ}}. Then we have the following estimate. Proposition 6.2. For 1 < p ≤ 2, we have

∥Ĝθ∥_{p} ≲ ∥θ∥_{p}  (6.17)

and

∥θ∥_{p′} ≲ ∥G^{→}θ∥_{p′}  (6.18)

where p′ is the conjugate exponent of p.

For scalar functions, we have

∥Gu∥_{p} ≲ ∥u∥_{p}.  (6.19)

Proof. We only show the 1-form case. We set f̂(x, a) = |Q̂_{a}^{λ}θ(x)| and f̂_{ε} = \sqrt{f̂^{2} + ε} (ε > 0). Define

Z_{t}^{(ε)} = f̂_{ε}(X_{t∧τ}, B_{t∧τ})^{p}

and

Z_{t} = f̂(X_{t∧τ}, B_{t∧τ})^{p}.

Then,

M_{t}^{(ε)} = Z_{t}^{(ε)} − ∫_{0}^{t∧τ}(L + ∂_{a}^{2})f̂_{ε}(X_{s}, B_{s})^{p}ds − ∫_{0}^{t∧τ} ∇_{N} f̂_{ε}(X_{s}, B_{s})^{p}dl_{s}.

is a martingale. Note that

∇_{N} f̂_{ε}^{p} = ∇_{N}(f̂^{2} + ε)^{p/2} = \frac{p}{2}(f̂^{2} + ε)^{(p−2)/2}α(Q̂_{a}^{λ}θ, Q̂_{a}^{λ}θ).

Therefore,

E_{m×δ_{N}} [ ∫_{0}^{τ}(L + ∂_{a}^{2})f̂_{ε}(X_{s}, B_{s})^{p}ds]

= E_{m×δ_{N}} [Z_{∞}^{(ε)} − Z_{0}^{(ε)}] − \frac{p}{2}E_{m×δ_{N}} [∫_{0}^{τ} f̂_{ε}^{p−2}α(Q̂_{B_{s}}^{λ}θ, Q̂_{B_{s}}^{λ}θ)dl_{s}]

≤ E_{m×δ_{N}} [Z_{∞}^{(ε)} − Z_{0}^{(ε)}] + \frac{pβ}{2} E_{m×δ_{N}} [∫_{0}^{τ} f̂_{ε}^{p−2}|Q̂_{B_{s}}^{λ}θ|^{2}dl_{s}].

By taking limit, we have

E_{m×δ_{N}} [ lim inf_{ε→0} ∫_{0}^{τ}(L + ∂_{a}^{2})f̂_{ε}(X_{s}, B_{s})^{p}ds]

≤ lim inf_{ε→0} E_{m×δ_{N}} [Z_{∞}^{(ε)} − Z_{0}^{(ε)}] + \frac{pβ}{2} lim inf_{ε→0} E_{m×δ_{N}} [∫_{0}^{τ} f̂_{ε}^{p−2}|Q̂_{B_{s}}^{λ}θ(X_{s})|^{2}dl_{s}]

≤ ∥u∥_{p}^{p} + \frac{pβ}{2} lim inf_{ε→0} ∫_{0}^{∞}(N ∧ a)da ∫_{M} f̂_{ε}^{p−2}|Q̂_{a}^{λ}θ|^{2}σ(dx)

≤ ∥u∥_{p}^{p} + \frac{pβ}{2} lim inf_{ε→0} ∫_{0}^{∞}(N ∧ a)da ∫_{M} f̂_{ε}^{p}σ(dx).

We estimate the second term. We use the interpolation space. Taking ξ = 1 − (1/p), we introduce the interpolation norm ∥ · ∥_{ξ,p} of ∥ · ∥_{0,p} and ∥ · ∥_{1,p}. Here ∥ · ∥_{0,p} is the L^{p} norm in L^{p}(M, dx) and ∥ · ∥_{1,p} is the Sobolev norm:

∥u∥_{1,p}^{p} = ∫_{M} |u|^{p}m(dx) + ∫_{M} |∇u|^{p}m(dx).

Then the following inequality holds (see e.g., [1, Chapter VII]):

∫_{∂M} |u|^{p}dσ(x) ≲ ∥u∥_{ξ,p}^{p}.

Moreover, the general theory of interpolation implies (see [1, LEMMA 7.16])

∥u∥_{ξ,p}^{p} ≲ ∥u∥_{0,p}^{(1−ξ)p}∥u∥_{1,p}^{ξp}.

Thus we have

∫_{∂M} |u|^{p}σ(dx) ≲ ∥u∥_{0,p}^{(1−ξ)p}∥u∥_{1,p}^{ξp}.  (6.20)

On the other hand

|∇f̂_{ε}| = |∇\sqrt{f̂^{2} + ε}| = \frac{1}{2}(f̂^{2} + ε)^{−1/2}|∇f̂^{2}| ≤ \frac{1}{2}(f̂^{2} + ε)^{−1/2}2f̂|∇Q̂_{a}^{λ}θ| ≤ |∇Q̂_{a}^{λ}θ|.

Using these inequalities, we have

∫_{0}^{∞}(N ∧ a)da ∫_{M} f̂_{ε}^{p}σ(dx) ≲ ∫_{0}^{∞}(N ∧ a)∥f̂_{ε}∥_{0,p}^{(1−ξ)p}∥f̂_{ε}∥_{1,p}^{ξp}da
≲ ∫_{0}^{∞}(N ∧ a){∥f̂_{ε}∥_{p}^{p} + ∥f̂_{ε}∥_{p}^{(1−ξ)p}∥∇f̂_{ε}∥_{p}^{ξp}}da
≲ ∫_{0}^{∞}(N ∧ a){∥f̂_{ε}∥_{p}^{p} + ∥f̂_{ε}∥_{p}^{(1−ξ)p}∥∇Q̂_{a}^{λ}θ∥_{p}^{ξp}}da.

By taking limit, we have

lim_{ε→0} ∫_{0}^{∞}(N ∧ a)da ∫_{M} f̂_{ε}^{p}σ(dx)

≲ ∫_{0}^{∞}(N ∧ a){∥f̂∥_{p}^{p} + ∥f̂∥_{p}^{(1−ξ)p}∥∇Q̂_{a}^{λ}θ∥_{p}^{ξp}}da

≲ ∫_{0}^{∞} ae^{−\sqrt{λ−γ_{p}}pa}∥θ∥_{p}^{p}da + ∫_{0}^{∞} ae^{−\sqrt{λ−γ_{p}}(1−ξ)pa}∥θ∥_{p}^{(1−ξ)p}∥∇Q̂_{a}^{λ}θ∥_{p}^{ξp}da

≲ ∥θ∥_{p}^{p} + ∥θ∥_{p}^{(1−ξ)p} ∫_{0}^{∞} ae^{−\sqrt{λ−γ_{p}}(1−ξ)pa}da\{∫_{M} |∇Q̂_{a}^{λ}θ(x)|^{p}m(dx)\}^{ξ}

≤ ∥θ∥_{p}^{p} + ∥θ∥_{p}^{(1−ξ)p}\{∫_{0}^{∞} ae^{−\sqrt{λ−γ_{p}}(1−ξ)pa}1^{1/(1−ξ)}da\}^{1−ξ}

× \{∫_{0}^{∞} ae^{−\sqrt{λ−γ_{p}}(1−ξ)pa}da ∫_{M} |∇Q̂_{a}^{λ}θ(x)|^{p}m(dx)\}^{ξ} (\frac{1}{1/(1 − ξ)} + \frac{1}{1/ξ} = 1)

≲ ∥θ∥_{p}^{p} + ∥θ∥_{p}^{(1−ξ)p}\{∫_{M} m(dx) ∫_{0}^{∞} ae^{−\sqrt{λ−γ_{p}}(1−ξ)pa}|∇Q̂_{a}^{λ}θ(x)|^{p}da\}^{ξ}

≲ ∥θ∥_{p}^{p} + ∥θ∥_{p}^{(1−ξ)p}[∫_{M} m(dx)\{∫_{0}^{∞} ae^{−\sqrt{λ−γ_{p}}(1−ξ)pνa}da\}^{1/ν}

× \{∫_{0}^{∞} a|∇Q̂_{a}^{λ}θ(x)|^{p·2/p}da\}^{p/2}]^{ξ} (\frac{1}{ν} + \frac{1}{2/p} = 1)

≲ ∥θ∥_{p}^{p} + ∥θ∥_{p}^{(1−ξ)p}[∫_{M}{∫_{0}^{∞} a|∇Q̂_{a}^{λ}θ(x)|^{2}da}^{p/2}m(dx)]^{ξ}

≲ ∥θ∥_{p}^{p} + ∥θ∥_{p}^{(1−ξ)p}∥Ĝ^{↑}θ∥_{p}^{ξp}. Further, as in the proof of Proposition 5.2 , we can show that

∥Ĝθ∥_{p}^{p} ≲ ∥θ∥_{p}^{p(2−p)/2}∥∫_{0}^{∞} a lim inf_{ε→0}(L + ∂_{a}^{2})f̂_{ε}^{p}da∥^{p/2}.

Combining these inequalities, we have

∥Ĝθ∥_{p}^{p} ≲ ∥θ∥_{p}^{p(2−p)/2}{∥θ∥_{p}^{p·p/2} + ∥θ∥_{p}^{(1−ξ)p·p/2}∥Ĝ^{↑}θ∥_{p}^{ξp·p/2}}

≤ ∥θ∥_{p}^{p} + ∥θ∥_{p}^{(2−ξp)p/2}∥Ĝ^{↑}θ∥_{p}^{ξp^{2}/2}

= ∥θ∥_{p}^{p} + ∥θ∥_{p}^{(3−p)p/2}∥Ĝ^{↑}θ∥_{p}^{(p−1)p/2} (ξ = 1 − \frac{1}{p})

≤ ∥θ∥_{p}^{p} + \frac{3 − p}{2}δ^{−(p−1)/(3−p)}∥θ∥_{p}^{p} + \frac{p − 1}{2}δ∥Ĝ^{↑}θ∥_{p}^{p} (\frac{3 − p}{2} + \frac{p − 1}{2} = 1)

≲ ∥θ∥_{p}^{p} + δ^{−(p−1)/(3−p)}∥θ∥_{p}^{p} + δ∥Ĝθ∥_{p}^{p}.

Since δ is arbitrary, we can get

∥Ĝθ∥_{p}^{p} ≲ ∥θ∥_{p}^{p}

which is (6.17) . Now the rest is easy.

When p ≥ 2, we estimate Ĥθ and Hu. Proposition 6.3. We further assume that the second fundamental form α is non-negative definite. Then, for any p ≥ 2, we have

∥Ĥθ∥_{p} ≲ ∥θ∥_{p}.  (6.21)

For scalar functions, we have

∥Hu∥_{p} ≲ ∥u∥_{p}.  (6.22)

Proof. Set f̂(x, a) = |Q̂_{a}^{λ}θ(x)| for 1-form θ and define

Z_{t} = f̂(X_{t∧τ}, B_{t∧τ})^{2}.

Then

Z_{t} = Z_{0} + M_{t} − ∫_{0}^{t∧τ}(L + ∂_{a}^{2})f̂^{2}(X_{s}, B_{s})ds − ∫_{0}^{t∧τ} α(Q̂_{B_{s}}^{λ}θ(X_{s}), Q̂_{B_{s}}^{λ}θ(X_{s}))dl_{s} where (M_{t}) is a martingale with the quadratic variation

⟨M⟩_{t} = 2 ∫_{0}^{t∧τ}{|∇f̂^{2}(X_{s}, B_{s})|^{2} + |∂_{a}f̂^{2}(X_{s}, B_{s})|^{2}}ds.

By the assumption that α is non-negative definite, (Z_{t}) is a submartingale and the increasing part is given as

A_{t} := ∫_{0}^{t∧τ}(L + ∂_{a}^{2})f̂^{2}(X_{s}, B_{s})ds + ∫_{0}^{t∧τ} α(Q̂_{B_{s}}^{λ}θ(X_{s}), Q̂_{B_{s}}^{λ}θ(X_{s}))dl_{s}.

Now, recalling that (see Lemma 4.4 )

(L + ∂_{a}^{2})f̂^{2} ≥ 2ĝ,

we have

A_{t} ≥ ∫_{0}^{t∧τ} 2ĝ(X_{s}, B_{s})ds.

By virtue of the submartingale inequality (5.4) , we obtain

E_{m×δ_{N}} [{∫_{0}^{τ} 2ĝ(X_{s}, B_{s})ds}^{p/2}] ≤ E_{m×δ_{N}} [A_{∞}^{p/2}]
≲ E_{m×δ_{N}} [Z_{∞}^{p/2}]
= ∥θ∥_{p}^{p}.

Thus we have

∥Ĥθ∥_{p}^{p} =∥{ ∫_{0}^{∞} aQ_{a}ĝ(x, a)da}^{p/2}∥_{1}

= lim_{N→∞} ∫_{M} μ(dx){ ∫_{0}^{∞}(a ∧ N)Q_{a}ĝ(x, a)da}^{p/2}

= lim_{N→∞} ∫_{M} μ(dx)E_{m×δ_{N}} [ ∫_{0}^{τ} ĝ(X_{s}, B_{s})ds|X_{τ} = x]^{p/2}

≤ lim_{N→∞} ∫_{M} μ(dx)E_{m×δ_{N}} [{∫_{0}^{τ} ĝ(X_{s}, B_{s})ds}^{p/2}|X_{τ} = x]

= lim_{N→∞} E_{m×δ_{N}} [{∫_{0}^{τ} ĝ(X_{s}, B_{s})ds}^{p/2}]

≲ ∥θ∥_{p}^{p}.

Scalar case is easier.

By combining Propositions 4.3 and 6.3, we easily obtain the following estimates for G-functions:

Proposition 6.4. Assume that α is non-negative definite. Then, for any p ≥ 2, we have

∥Ĝ^{→}θ∥_{p} ≲ ∥θ∥_{p}  (6.23)

and

∥θ∥_{p′} ≲ ∥Ĝ^{→}θ∥_{p′}  (6.24)

where p′ is the conjugate exponent of p.

For scalar functions, we have

∥Gu∥_{p} ≲ ∥u∥_{p}.  (6.25)

Now the following theorem can be proved in the same way as Theorem 2.1 .

Theorem 6.5. For any 1 < p ≤ 2, it holds that

∥u∥_{p} + ∥∇u∥_{p} ≲ ∥\sqrt{1 − Δ}u∥_{p},  (6.26)
∥\sqrt{1 − Δ}u∥_{p′} ≲ ∥u∥_{p′} + ∥∇u∥_{p′}.  (6.27)

where p′ is the conjugate exponent of p.

If we further assume that the second fundamental form α is non-negative definite, then the inequalities above hold for p ≥ 2.

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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/