A PROBABILISTIC CONSTRUCTION OF THE HEAT KERNEL

FOR THE ¯∂-NEUMANN PROBLEM ON A STRONGLY PSEUDOCONVEX SIEGEL DOMAIN

Naomasa Ueki (received September 11, 1990)

1. Introduction

In this paper, we consider the heat equation for the ¯∂-Neumann problem. This is an initial boundary value problem whose boundary condition includes an imaginary directional differentiation. In [7], Malliavin constructed the solution of the heat equation on a domain by using a method of singular perturbations and pointed out that a method related to the Fourier transform can be applied to this problem. On the other hand, a strongly pseudoconvex Siegel domain is well known as one of the most fundamental complex manifolds with boundary. This domain D can be regarded as the product of a Heisenberg group H_{n} and ℝ^{+}: D = H_{n} × ℝ^{+}. In [4], Gaveau constructed explicitly the heat kernel for Kohn’s Laplacian on H_{n}, by combining a probabilistic method and the Fourier transform. In this paper, by referring their works, we construct the heat kernel for the ¯∂-Neumann problem on the Siegel domain D explicitly in terms of the theory of generalized Wiener functionals by Watanabe [13]. For the heat kernel on this domain, Stanton gave an explicit formula in the (0, q)-form case (q > 0), by using methods of the partial differential equations [10], [11]. We here consider the general (p, q)-form case. The

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main part of our discussion is the proof of well-definedness of the heat kernel. In [12], our main results (Theorems 2.1 and 2,2 below) were announced.

We briefly explain our methods. The equation we consider is the following:

(1.1)  \{ \begin{matrix}\frac{∂}{∂t}F (t, X) & = −□F (t, X), t > 0, X ∈ D, \\ lim_{t→0} F (t, X) & = f(X) ∈ Λ_{0}^{p,q}(D), uniformly on D̄, \\ P F (t, X) & = Q(\frac{∂}{∂r} − i\frac{∂}{∂u})F (t, X) = 0 on bD, \end{matrix}

where

□ = −\frac{1}{4} ∑_{j=1}^{2n} X_{−}_{j}^{2} − \frac{1}{2} (\frac{∂^{2}}{∂u^{2}} + \frac{∂^{2}}{∂r^{2}} ) + i(n − 2C)\frac{∂}{∂u} − ∑_{j=1}^{2n} \frac{A_{j}}{\sqrt{2}}X_{−}_{j} − B.

For the notations, see Section 2 below. Except for the term i∂/∂u in the boundary condition, this equation has the same form as the d-Neumann boundary condition case that Airault [1], Ikeda and Watanabe [6] (see also [9]) considered. Hence, as in the d-Neumann case, we may expect the heat kernel of (1.1) can be expressed as

(1.2)  h_{t}(X, X′) = E[m(t)M(t)K(t)δ_{X′}(X(t), U(t) − iϕ(t), R(t))]

where (X(t), U(t), R(t)) is the diffusion process generated by 1/4 ∑_{j=1}^{2n} X_{j}^{2} + 1/2 (∂^{2}/∂u^{2}+∂^{2}/∂r^{2}), ϕ(t) is the local time of the diffusion {R(t), t ≧ 0} at 0 and δ_{X′} is the Dirac δ function on D. m(t) and M(t) are operator valued functionals for −i(n− 2C)∂/∂u and ∑_{j=1}^{2n} A_{j}/\sqrt{2}X_{−}_{j} + B, respectively (see (2.12) and (3.4)). K(t) = I − P I_{{min_{0≦s≦t} R(s)=0}} is the functional for the boundary condition. However the process (X(t), U(t) − iϕ(t), R(t)) does not lie on D. Hence we do not know how to give a mathematical sense to the right hand side of (1.2).

Now we note that the metric, the differential structure and the complex structure of our case are constant in the direction that we must consider the imaginary directional differentiation. By using these facts, we can use the independence of the above processes and Fourier transform effectively. Then we can formally rewrite the expression (1.2) as

(1.3)  \begin{matrix}h_{t}((x, u, r), (x′, u′, r′)) \\ = \frac{1}{2π} ∫_{−∞}^{∞} dλe^{iλ(u′−u)}E[e^{−iλu(t)}m(t)] \\ × θ_{t}^{λ}(x, x′)E[e^{−λϕ(t)}K(t)|R(t) = r′]r_{t}(r, r′) \end{matrix}

where

u(t) = U(t) − u − S_{x}(t),
θ_{t}^{λ}(x, x′) = E[e^{−iλS_{x}(t)}M(t)δ_{x′}(X(t))],
S_{x}(t) = ∑_{j=1}^{n} ∫_{0}^{t}(X^{n+j}(s) ∘ dX^{j}(s) − X^{j}(s) ∘ dX^{n+j}(s))

and r_{t}(r, r′) is the transition probability function of R(t). We prove that this is the heat kernel for (1.1). In (0, q)-form case, M(t) = I and θ_{t}^{λ}(x, x′) is rewritten more explicitly ((2.18) below). However, in (p, q)-form case, we can not rewrite θ_{t}^{λ}(x, x′) explicitely as in (0, q)-form case. Then it is difficult to show the exponential decay of θ_{t}^{λ}(x, x′) in λ, which is necessary to show the well-definedness of the right hand side of (1.3). Now we note that θ_{t}^{λ}(x, x′) is, as a function of (t, x, x′), the heat kernel for the operator □_{λ}^{0} defined by (5.26) below. By using the semigroup property of this heat kernel, we can prove the exponential decay of θ_{t}^{λ}(x, x′) which is sufficient to prove the convergence of the right hand side of (1.3). This decay of θ_{t}^{λ}(x, x′) is an interesting property of the oscillatory integral over the Wiener space. Furthermore, by using the Malliavin calculus, we can prove the smoothness of (1.3). We note that we need not the partial Malliavin calculus used for the boundary value problem (e.g., [9]), because of the independence of the processes.

The organization of this paper is as follows. In Section 2, we first formulate the problem by following Stanton [10] and then state our main theorem. In Section 3, we give a probabilistic explanation of our formula (2.15) of the heat kernel; we give a few lemmas by which we expect the formula (2.15). In Section 4, we prove Theorem 2.2 below. This theorem plays a crucial role in this paper. In Section 5, we prove our main theorem, Theorem 2.1. In Section 6, we consider the short time asymptotic behavior of the heat kernel on the diagonal. Finally, in Section 7, we generalize the above results to certain domains with nondegenerate indefinite Levi forms.

2. Preliminaries and Main theorems

In this section, first of all, following Stanton [10], we review the formulation of the heat equation on a strongly pseudoconvex Siegel domain. The strongly pseudoconvex Siegel domain is defined by

D = {(z, w) ∈ ℂ^{n} × ℂ, Im w > |z|^{2}}.

We consider the Hermitian metric for which {Z_{1}, Z_{2}, ⋯, Z_{n+1}} is an orthonormal basis of T^{(1,0)}(D), where

(2.1) Z_{j} = \frac{∂}{∂z^{j}} + 2iz̄^{j} \frac{∂}{∂w} , j = 1, 2, ⋯, n, Z_{n+1} = i\sqrt{2}\frac{∂}{∂w} .

We note that the volume form for the metric is the restriction of 2^{n} times the standard Euclidean volume element on ℂ^{n+1}. The differential forms given by

(2.2) ω^{j} = dz^{j}, ω^{n+1} = \frac{1}{i\sqrt{2}}(dw − 2i ∑ z̄^{j}dz^{j})

form the dual basis for T_{(1,0)}^{∗}(D).

We now introduce several spaces of differential forms and several operators. Let Λ_{0}^{p,q}(D) be the space of C^{∞} (p, q) forms with compact supported in D. Let S^{p,q}(D̄) be the space of (p, q) forms on D whose coefficients relative to ω^{I} ∧ ω̄^{J} can be extended to rapidly decreasing functions on ℂ^{n+1} and L_{2}^{p,q}(D) that of square integrable (p, q) forms on D. Then ¯∂ maps S^{p,q}(D̄) to S^{p,q+1}(D̄) and has a smallest closed extention to L_{2}^{p,q}(D). For simplicity we also denote by ¯∂ its closed extension. Let ¯∂^{∗} be the adjoint of ¯∂ on L_{2}^{(p,q)}(D). We define the ¯∂-Laplacian □ by

(2.3)

□ = ¯∂¯∂^{∗} + ¯∂^{∗} ¯∂,
Dom □ = {f ∈ L_{2}^{p,q}(D); f ∈ Dom ¯∂ ∩ Dom ¯∂^{∗},
¯∂f ∈ Dom ¯∂^{∗}, ¯∂^{∗}f ∈ Dom ¯∂}.

Then the operator □ is a positive selfadjoint operator on L_{2}^{p,q}(D) (Folland-Kohn [3] Proposition 1.3.8).

To express the above operators explicitly, we prepare several notations. For ω ∈ T^{∗}D ⊗ ℂ, let ext(ω) be the exterior multiplication, i.e.,

ext(ω)η = ω ∧ η for η ∈ Λ(T^{∗}D ⊗ ℂ),

and int(ω̄) be interior multiplication, i.e., the dual operator of ext(ω). This int(ω) is complex linear in ω. We note that in some literature, e.g., Folland-Kohn [3], int(ω) is conjugate linear in ω. The condition for f ∈ S^{p,q}(D̄) to be in Dom ¯∂^{∗} is

int(ω^{n+1})f ↾_{bD}= 0,

where bD is the boundary of the domain D. Thus the condition for f ∈ S^{p,q}(D̄) to be in Dom □ are

(2.4) int(ω̄^{n+1})f ↾_{bD}= 0, int(ω̄^{n+1})¯∂f ↾_{bD}= 0.

These mean the ¯∂-Neumann boundary conditions. For a differential operator A, we define A_{−} to be the operator acting only on the coefficients of ω^{I} ∧ ω̄^{J}, i.e.,

A_{−}(∑ f_{IJ̄}ω^{I} ∧ ω̄^{J}) = ∑(Af_{IJ̄})ω^{I} ∧ ω̄^{J}.

Since ¯∂ω^{j} = ¯∂ω̄^{j} = ¯∂ω̄^{n+1} = 0 for j = 1, 2, ⋯, n, and

¯∂ω^{n+1} = \sqrt{2} ∑_{j=1}^{n} ω^{j} ∧ ω̄^{j},

in terms of the above notations, ¯∂ and ¯∂^{∗} can be represented as follows:

(2.5)

¯∂ = ∑_{j=1}^{n+1} ext(ω̄^{j})Z̄_{−}_{j} + \sqrt{2} ∑_{j=1}^{n} ext(ω^{j})ext(ω̄^{j})int(ω̄^{n+1}),
¯∂^{∗} = − ∑_{j=1}^{n+1} int(ω^{j})Z_{−}_{j} + \sqrt{2} ∑_{j=1}^{n} ext(ω^{n+1})int(ω^{j})int(ω̄^{j}),

on S^{p,q}(D̄).

We use as coordinates on D (z, u, r) where u = Re w and r = Im w − |z|^{2} for w ∈ ℂ. Then we can regard D as the product of the boundary bD and ℝ^{+}. We identify bD with the Heisenberg group H_{n}. The group law on H_{n} is

(2.6)  \begin{aligned}(z′, u′)(z, u) & = (z′ + z, u′ + u + 2 ∑_{j=1}^{n}(x^{′n+j}x^{j} − x^{′j}x^{n+j})) \\ & for (z′, u′), (z, u) ∈ H_{n}, \end{aligned}

where z^{j} = x^{j} + ix^{n+j}. The metric we gave is the product metric of an invariant metric on H_{n} and the standard metric on ℝ^{+}. In terms of these coordinates,

(2.7)  Z_{j} = \frac{∂}{∂z^{j}} + iz̄^{j} \frac{∂}{∂u}, j = 1, 2, ⋯, n,  Z_{n+1} = \frac{1}{\sqrt{2}} (\frac{∂}{∂r} + i\frac{∂}{∂u}) .

Then we have

Z_{j} = 1/2 (X_{j} − iX_{n+j})

where

(2.8)  X_{j} = \frac{∂}{∂x^{j}} + 2x^{n+j} \frac{∂}{∂u},  X_{n+j} = \frac{∂}{∂x^{n+j}} − 2x^{j} \frac{∂}{∂u}.

We also define the projections P and Q by

P = ext(ω̄^{n+1})int(ω^{n+1}), Q = int(ω^{n+1})ext(ω̄^{n+1}).

Then the ¯∂-Neumann boundary condition (2.6) is rewritten as follows:

(2.9)  P f ↾_{bD}= Q (\frac{∂}{∂r} − i\frac{∂}{∂u}) f ↾_{bD}= 0

for f ∈ S^{p,q}(D̄). Now we see that this condition is similar to the absolute boundary condition except the term i∂/∂u (cf. Ikeda-Watanabe [5]).

The operator □ is expressed on S^{p,q}(D̄) as follows:

(2.10)  \begin{aligned}□ & = −\frac{1}{4} ∑_{j=1}^{2n} X_{−}_{j}^{2} − \frac{1}{2} (\frac{∂^{2}}{∂u^{2}} + \frac{∂^{2}}{∂r^{2}} ) + i(n − 2C)\frac{∂}{∂u} \\ & − ∑_{j=1}^{2n} \frac{A_{j}}{\sqrt{2}}X_{−}_{j} − B \end{aligned}

where

C = qQ + (q − 1)P,
A_{j} = ext(ω^{n+1})int(ω̄^{j}) − ext(ω^{j})int(ω̄^{n+1}), j = 1, 2, ⋯, n,
A_{n+j} = i(ext(ω^{n+1})int(ω̄^{j}) + ext(ω^{j})int(ω̄^{n+1})), j = 1, 2, ⋯, n,
B = −2 ∑_{j,k=1}^{n} ext(ω^{j})ext(ω̄^{j})int(ω^{k})int(ω̄^{k})
− 2 ∑_{j=1}^{n}(int(ω^{j})ext(ω̄^{j}) − ext(ω^{j})int(ω̄^{j}))ext(ω^{n+1})int(ω̄^{n+1}).

Remark 2.1. If p = 0, A_{j} = A_{n+j} = B = 0 and □ acts diagonally on (0, q) forms. If p ≠ 0, □ does not act on diagonally but A_{j}, A_{n+j} and B commutes with P and Q. Thus □ preserves the orthogonal decomposition Λ^{p,q}(T^{∗}D) = Ran P ⊕ Ran Q.

A fundamental solution of the heat equation is a one parameter family of bounded operators H_{t}, t > 0, on L_{2}^{p,q}(D) such that for f ∈ Λ_{0}^{p,q}(D),

(2.11)

(i) for t ∈ [0, T], ∥H_{t}f∥ ≦ C where C is independent of

t but may depend on T and f;

(ii) H_{t}f is differentiable in t;
(iii) H_{t}f ∈ Dom □;
(iv) (∂/∂t + □)H_{t}f = 0;
(v) H_{t}f ⟶ f in L_{2}^{p,q}(D) as t ⟶ 0;
(iv) □H_{t}f = H_{t}□f.

Then we obtain the following results (Stanton [10]):

Proposition 2.1. There is a unique fundamental solution H_{t}. Furthermore, H_{t} is the semigroup generated by −□.

Stanton proved this proposition in (0, q)-form case. Her proof is applicable to general (p, q)-form case.

Before we state our main theorem, we prepare a few notations. Let (W_{0}^{2n}, P) be a 2n-dimensional Wiener space, i.e.,

W_{0}^{2n} = {(x^{1}(·), x^{2}(·), ⋯, x^{2n}(·)) : [0, ∞) ⟶ ℝ^{2n} : continuous,
x(0) = 0}

and P is the Wiener measure. For any ϵ > 0 and m ∈ ℕ, we define the End(Λ^{p,q}(T^{∗}D))-valued process M_{m}^{ϵ}(t) by the solution of the SDE

(2.12) \{ \begin{matrix}dM_{m}^{ϵ}(t) & = M_{m}^{ϵ}(t)dΞ_{m}^{ϵ}(t), \\ M_{m}^{ϵ}(0) & = I \end{matrix}

where

Ξ_{m}^{ϵ}(t) = ∑_{j=1}^{2n} \frac{ϵ}{\sqrt{m}}A_{j}x^{j}(t) + \frac{ϵ^{2}}{m}Bt.

When m = 1, we omit the subscript m: Ξ_{1}^{ϵ}(t) =: Ξ^{ϵ}(t), M_{1}^{ϵ}(t) =: M^{ϵ}(t). When m = ϵ = 1, we also omit the superscript ϵ: Ξ_{1}^{1}(t) =: Ξ(t), M^{1}(t) =: M(t). For any α ∈ ℝ, λ ∈ ℝ and (x, u) ∈ H_{n},

(2.13)  f_{t}^{α}(x, u, λ) = \frac{1}{2π} exp(iλu − tλα)E [e^{−iλS_{0}(t)}M(t)δ_{x}(\frac{x(t)}{\sqrt{2}})]

where

(2.14)  S_{0}(t) = ∑_{j=1}^{n} ∫_{0}^{t}(x^{n+j}(s) ∘ dx^{j}(s) − x^{j}(s) ∘ dx^{n+j}(s)).

Here, ∘dx^{j}(t) means the Stratonovich differential. The right hand side of (2.13) is well defined as a generalized expectation of a generalized Wiener functional [13]. Then our main theorem is stated as follows:

Theorem 2.1. The unique fundamental solution H_{t} of the heat equation for the ¯∂-Neumann problem on (p, q)-forms on D has the following smooth kernel:

(2.15)  \begin{matrix}h_{t}((x, u, r), (x′, u′, r′)) \\ = p_{t}^{n−2q+2}((x, u)^{−1}(x′, u′))e_{t}^{−}(r, r′)P \\ + p_{t}^{n−2q}((x, u)^{−1}(x′, u′))e_{t}^{+}(r, r′)Q \\ + q_{t}^{n−2q}((x, u)^{−1}(x′, u′), r + r′)Q \end{matrix}

where

(2.16)  \begin{matrix}p_{t}^{α}(x, u) = & ∫_{−∞}^{∞} dλ f_{t}^{α}(x, u, λ) exp (−\frac{t}{2}λ^{2}) , \\ q_{t}^{α}((x, u), r) = & − \frac{2}{\sqrt{π}} ∫_{−∞}^{∞} dλ · λf_{t}^{α}(x, u, λ)e^{rλ} ∫_{\sqrt{t/2}(r/t+λ)}^{∞} dμ e^{−μ^{2}} \end{matrix}

and

e_{t}^{±}(r, r′) = \frac{1}{\sqrt{2πt}} \{exp ( − \frac{(r − r′)^{2}}{2t} ) ± exp ( − \frac{(r + r′)^{2}}{2t} )\} .

Remark 2.2. In the case of p = 0, M(t) = I and so

(2.17)  \begin{aligned}f_{t}^{α}(x, u, λ) & = \frac{1}{2π} exp(iλu − tλα)E [e^{−iλS_{0}(t)}δ_{x}(\frac{x(t)}{\sqrt{2}})] I \\ & = f_{t}^{0,α}(x, u, λ)I \end{aligned}

where

(2.18)  f_{t}^{0,α}(x, u, λ) = \frac{1}{2π^{n+1}} (\frac{λ}{sinh(tλ)})^{n} exp(iλu − tλα − |x|^{2}λ coth(tλ))

([5]). Now we see that the h_{t} in (2.15) coincides with the heat kernel obtained by Stanton [10].

In our proof of Theorem 2.1, the most essential part is to show the convergence of the λ-integration in the right hand side of (2.16). For this purpose, we show Theorem 2.2 below. Before we state the theorem, we prepare a few notations. We use the multi-index notation: α = (α_{1}, α_{2}, ⋯, α_{2n}) ∈ ℤ_{+}^{2n}, |α| = α_{1} + α_{2} + ⋯ + α_{2n}, a^{α} = a_{1}^{α_{1}}a_{2}^{α_{2}} ⋯ a_{2n}^{α_{2n}} for a = (a_{1}, a_{2}, ⋯, a_{2n}) ∈ ℝ^{2n} and ∂_{x}^{α} = (\frac{∂}{∂x^{1}} )^{α_{1}} (\frac{∂}{∂x^{2}} )^{α_{2}} ⋯ (\frac{∂}{∂x^{2n}} )^{α_{2n}}. For A = ∑_{I,I′∈I(p),J,J′∈I(q)} A_{IJI′J′}ext(ω^{I} ∧ ω̄^{J})int(ω̄^{I′} ∧ ω^{J′}) ∈ End(Λ^{p,q}(T^{∗}D)), we set

∥A∥ := ( ∑_{I,I′∈I(p),J,J′∈I(q)} |A_{IJI′J′}|^{2})^{\frac{1}{2}}

where I(p) = {I = (i_{1}, i_{2}, ⋯, i_{p}); 1 ≦ i_{1} < i_{2} < ⋯ < i_{p} ≦ 2(n + 1)}, i.e., ∥A∥ denotes the Hilbert-Schmidt norm. Then we obtain the following: Theorem 2.2. (i) For any ϵ_{0} > 0, α, β ∈ ℤ_{+}^{2n} and k ∈ ℤ_{+},

(2.19)

\overline{lim}_{λ→±∞} \frac{1}{|λ|} log sup_{\begin{matrix}x∈ℝ^{2n} \\ 0<ϵ≦ϵ_{0} \end{matrix}} ∥x^{β}∂_{x}^{α}E[S_{0}(1)^{k}e^{−iλS_{0}(1)}

× M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥

≦ −n.

(ii) For any ϵ_{0} > 0, α, β ∈ ℤ_{+}^{2n} and k ∈ ℤ_{+},

(2.20)

\overline{lim}_{λ→±∞} \frac{1}{|λ|} log sup_{\begin{matrix}x∈ℝ^{2n} \\ 0<ϵ≦ϵ_{0} \end{matrix}} ∥x^{β}∂_{x}^{α}E[S_{0}(1)^{k}e^{−iλS_{0}(1)}

× (M^{ϵ}(1) − I)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]/ϵ∥

≦ −n.

By using scaling property of Brownian motions, we can easily prove the following: Corollary. For any α, β ∈ ℤ_{+}^{2n} and k ∈ ℤ_{+},

(2.21)

\overline{lim}_{λ→±∞} sup_{t_{0}≦t≦t_{1}} \frac{1}{t|λ|} log sup_{x∈ℝ^{2n}} ∥x^{β}∂_{x}^{α}E[S_{0}(t)^{k}e^{−iλS_{0}(t)}

× M(t)δ_{x} (\frac{x(t)}{\sqrt{2}}) ]∥

≦ −n.

3. A formal construction of the heat kernel

The purpose of this section is to show that h_{t} in (2.15) is a candidate for the integral kernel of H_{t}. We first consider the initial value problem (1.1). Let (W_{0}^{2(n+1)}, P) be a 2(n + 1)-dimensional Wiener space, i.e.,

W_{0}^{2(n+1)} = {(x^{1}(·), x^{2}(·), ⋯, x^{2n}(·), u(·), r(·)) : [0, ∞) ⟶ ℝ^{2(n+1)} : continuous,
(x(0), u(0), r(0)) = 0}

and P is the Wiener measure. Then the diffusion process Z(t) = (X^{1}(t), X^{2}(t), ⋯, X^{2n}(t), U′(t), R′(t)) defined by

(3.1)  \begin{aligned}X^{j}(t) & = x^{j} + \frac{x^{j}(t)}{\sqrt{2}},  j = 1, 2, ⋯, 2n, \\ U′(t) & = u + u(t) + S_{x}(t), \\ R′(t) & = r + r(t), \\ S_{x}(t) & = ∑_{j=1}^{n} ∫_{0}^{t} \sqrt{2}(X^{n+j}(s) ∘ dx^{j}(s) − X^{j}(s) ∘ dx^{n+j}(s)) \end{aligned}

is a solution of the following SDE on H_{n} × ℝ:

(3.2) \{ \begin{matrix}dZ(t) & = \frac{1}{\sqrt{2}} ∑_{j=1}^{2n} X_{j}(Z(t)) ∘ dx^{j}(t) + \frac{∂}{∂u}(Z(t)) ∘ du(t) + \frac{∂}{∂r} (Z(t)) ∘ dr(t), \\ Z(0) & = (x^{1}, x^{2}, ⋯, x^{2n}, u, r). \end{matrix}

R(t) := |r + r(t)| has the following decomposition by Skolohod:

(3.3)  R(t) = r + B(t) + ϕ(t)

where

B(t) = ∫_{0}^{t} sgn(r + r(s))dr(s)

and

ϕ(t) = lim_{η↘0} \frac{1}{2η} ∫_{0}^{t} I_{(−η,η)}(r + r(s))ds

is the local time of R(t) at 0. We define the End(Λ^{p,q}(T^{∗}D))-valued process m(t) by the solutions of the SDE

(3.4)  \{ \begin{matrix}dm(t) & = m(t)\{−i(n − 2C)\}du(t), \\ m(0) & = I \end{matrix}

and define the End(Λ^{p,q}(T^{∗}D)) -valued process K(t) by

(3.5)  K(t) = I_{{σ≦t}}Q + I_{{σ>t}}

where σ = inf{s : R(s) = 0}. Then we have the following lemma:

Lemma 3.1. Let G(t, X) be a smooth form on [0, ∞) × D̄ such that G(t, ·) ∈ S^{p,q}(D̄) for each t ≧ 0, uniformly with respect to t ∈ [0, T] for any T > 0, i.e.,any seminorms of G(t, ·) are bounded in t ∈ [0, ∞] for any T > 0. Then it holds that

(3.6) m(t)M(t)K(t)e^{−iλU(t)} ∫_{−∞}^{∞} e^{iλζ}G(s, X(t), ζ, R(t))dζ − K(0)e^{−iλu} ∫_{−∞}^{∞} e^{iλζ}G(0, x, ζ, r)dζ = martingale

+ ∫_{0}^{t} m(s)M(s)K(s)e^{−iλU(s)} ∫_{−∞}^{∞} e^{iλζ}

× \{\frac{∂G}{∂s} − □G\} (s, X(s), ζ, R(s))dζds

+ ∫_{0}^{t} m(s)M(s)K(s)e^{−iλU(s)} ∫_{−∞}^{∞} e^{iλζ}

× \{\frac{∂G}{∂r} − i\frac{∂G}{∂u}\} (s, X(s), ζ, R(s))dζdϕ(s),

where X(t) = (X^{1}(t), X^{2}(t), ⋯, X^{2n}(t)) and U(t) = U′(t) − iϕ(t).

Proof. By using Itô’s formula and noting that m(t), M(t) and K(t) are commutative, we have

(3.7)  \begin{matrix}m(t)M(t)K(t)e^{−iλU(t)}e^{iλζ}G(t, X(t), ζ, R(t)) \\ − m(0)M(0)K(0)e^{−iλU(0)}e^{iλζ}G(0, X(0), ζ, R(0)) \\ = ∫_{0}^{t} m(s)M(s)K(s)e^{−iλU(s)}e^{iλζ} \\ × \{ − i(n − 2c + λ)G(s, X(s), ζ, R(s))du(s) \\ + ∑_{j=1}^{2n} (A_{j}G + \frac{1}{\sqrt{2}} \frac{∂G}{∂x^{j}} ) (s, X(s), ζ, R(s))dx^{j}(s) \\ − iλG(s, X(s), ζ, R(s))dS_{x}(s) \\ + \frac{∂G}{∂r}(s, X(s), ζ, R(s))dB(s)\} \\ + ∫_{0}^{t} m(s)M(s)K(s)e^{−iλU(s)}e^{iλζ} \\ × \{\frac{∂G}{∂t} + \frac{1}{4} ∑_{j=1}^{n} (\frac{∂}{∂x^{j}} − 2iλX^{n+j}(s))^{2} G \\ + \frac{1}{4} ∑_{j=1}^{n} (\frac{∂}{∂x^{n+j}} + 2iλX^{j}(s))^{2} G − \frac{λ^{2}}{2} G \\ + \frac{1}{2} \frac{∂^{2}G}{∂r^{2}} − λ(n − 2C)G \\ + ∑_{j=1}^{n} \frac{A_{j}}{\sqrt{2}} (\frac{∂}{∂x^{j}} − 2iλX^{n+j}(s)) G \\ + ∑_{j=1}^{n} \frac{A_{n+j}}{\sqrt{2}} (\frac{∂}{∂x^{n+j}} + 2iλX^{j}(s)) G \\ + BG\}(s, X(s), ζ, R(s))ds \\ + ∫_{0}^{t} m(s)M(s)K(s)e^{−iλU(s)}e^{iλζ} \\ × \{\frac{∂G}{∂r} − λG\} (s, X(s), ζ, R(s))dϕ(s). \end{matrix}

Since G is assumed to be smooth and rapidly decreasing in X, by integraring in ζ variables, exchanging the order of integration and using the integration by parts, we obtain (3.6) from (3.7). □

If the solution F (t, X) of (1.1) is smooth in (t, X) and rapidly decreasing in X uniformly with respect to t ∈ [δ, T] for any 0 < δ < T, by Lemma 3.1, we have

e^{−iλu} ∫_{−∞}^{∞} dζe^{iλζ}F (T, x, ζ, r)

= E [m(T)M(T)K(T)e^{−iλU(T)} ∫_{−∞}^{∞} dζe^{iλζ}f(X(T), ζ, R(T))]

and so

(3.8)  \begin{matrix}F (t, x, u, r) \\ = \frac{1}{2π} ∫_{−∞}^{∞} dλE[m(t)M(t)K(t)e^{−iλU(t)} ∫_{−∞}^{∞} dζe^{iλζ}f(X(t), ζ, R(t))]. \end{matrix}

Now we set

(3.9)  \begin{matrix}H_{t,λ}((x, u, r), (x′, u′, r′)) \\ := \frac{1}{2π} E[m(t)M(t)K(t)e^{−iλU(t)} ∫_{−∞}^{∞} dζe^{iλζ}δ_{(x′,u′,r′)}(X(t), ζ, R(t))] \\ = \frac{1}{2π} e^{iλ(u′−u)}E[e^{−iλu(t)}m(t)]E[e^{−iλS_{x}(t)}M(t)δ_{x′}(X(t))] \\ × E[e^{−λϕ(t)}K(t)|R(t) = r′]r_{t}(r, r′), \end{matrix}

where r_{t}(r, r′) is the probability density function of R(t). We can rewrite this H_{t,λ}((x, u, r), (x′, u′, r′)) as follows: Lemma 3.2. H_{t,λ}((x, u, r), (x′, u′, r′)) is expressed as follows:

(3.10)

H_{t,λ}((x, u, r), (x′, u′, r′))

= f_{t}^{n−2q+2}((x, u)^{−1}(x′, u′), λ) exp (−\frac{t}{2}λ^{2}) e_{t}^{−}(r, r′)P

+ f_{t}^{n−2q}((x, u)^{−1}(x′, u′), λ) exp (−\frac{t}{2}λ^{2}) e_{t}^{+}(r, r′)Q

− \frac{2}{\sqrt{π}} λf_{t}^{n−2q}((x, u)^{−1}(x′, u′), λ)e^{rλ} ∫_{\sqrt{t/2}(r/t+λ)}^{∞} dμ exp(−μ^{2})Q

where e_{t}^{±}(r, r′) and f_{t}^{α}(x, u, λ) are defined in (2.13) and (2.16), respectively. Proof. Since

(3.11)  \{ \begin{matrix}m(t)Q & = exp (−i(n − 2q)u(t) + (n − 2q)^{2} \frac{t}{2}) Q, \\ m(t)P & = exp (−i(n − 2q + 2)u(t) + (n − 2q + 2)^{2} \frac{t}{2}) P, \end{matrix}

we have

(3.12) \{ \begin{matrix}E[e^{−iλu(t)}m(t)]Q & = exp (−\frac{t^{2}}{2} λ^{2} − tλ(n − 2q)) Q, \\ E[e^{−iλu(t)}m(t)]P & = exp (−\frac{t^{2}}{2} λ^{2} − tλ(n − 2q + 2)) P. \end{matrix}

We also have

(3.13)  \begin{matrix}E [e^{−iλS_{x}(t)}M(t)δ_{x′}(X(t))] \\ = exp ( − iλ ∑_{j=1}^{n} 2(x^{n+j}x^{′j} − x^{j}x^{′n+j})) \\ × E [e^{−iλS_{0}(t)}M(t)δ_{x′−x}(\frac{x(t)}{\sqrt{2}})] . \end{matrix}

By using Lemma 3.3 below, we obtain (3.14)

E[e^{−λϕ(t)}K(t)|R(t) = r′]r_{t}(r, r′)

= \frac{1}{\sqrt{2πt}} \{exp ( − \frac{(r − r′)^{2}}{2t} ) − exp ( − \frac{(r + r′)^{2}}{2t} )\}

+ ∫_{0}^{∞} db e^{−λb} \frac{2(r + r′ + b)}{\sqrt{2πt^{3}}} exp (−\frac{(r + r′ + b)^{2}}{2t} ) Q

= \frac{1}{\sqrt{2πt}} \{exp ( − \frac{(r − r′)^{2}}{2t} ) − exp ( − \frac{(r + r′)^{2}}{2t} )\} P

+ \frac{1}{\sqrt{2πt}} \{exp ( − \frac{(r − r′)^{2}}{2t} ) + exp ( − \frac{(r + r′)^{2}}{2t} )\} Q

− ∫_{0}^{∞} db λe^{−λb} \frac{2}{\sqrt{2πt}} exp (−\frac{(r + r′ + b)^{2}}{2t} ) Q.

By (3.12),(3.13) and (3.14), we conclude (3.10). □ Lemma 3.3. (cf. Itô-McKean [6] p.45) Let r(·) be a 1-dimensional Brownian motion. For r ≧ 0, we set

\begin{aligned}R(t, r) & = |r + r(t)|, \\ ϕ(t, r) & = lim_{η↘0} \frac{1}{2η} ∫_{0}^{t} I_{(−η,η)}(r + r(s))ds, \\ σ(r) & = inf\{t ≧ 0, R(t, r) = 0\}. \end{aligned}

Then the joint distribution of (R(t, r), ϕ(t, r)) is (3.15)

P (R(t, r) ∈ da, ϕ(t, r) ∈ db, σ(r) ≦ t)

= \frac{2(r + a + b)}{\sqrt{2πt^{3}}} exp (−\frac{(r + a + b)^{2}}{2t} ) dadb,

P (R(t, r) ∈ da, ϕ(t, r) ∈ db, σ(r) > t)

= \frac{1}{\sqrt{2πt}} \{exp ( − \frac{(r − a)^{2}}{2t} ) − exp ( − \frac{(r + a)^{2}}{2t} )\} δ_{0}(b)dadb,

a, b ≧ 0.

By (3.8) and Lemma 3.2 we can expect that (2.15) holds.

4. Proof of Theorem 2.2

In this section, we prove Theorem 2.2. First we prepare a few lemmas. The main idea of the proof of Theorem 2.2 is to use semigroup property of a heat kernel as follows:

Lemma 4.1. For any m ∈ ℕ and ϵ > 0,

(4.1)  \begin{matrix}E[S_{0}(1)^{k}e^{imλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ] \\ = ∑_{k_{1}+···+k_{m}=k} \frac{k!}{k_{1}! ⋯ k_{m}!} \\ × ∫_{ℝ^{2n}} dx_{1}E[ (\frac{S_{0}(1)}{m})^{k_{1}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x_{1}} (\frac{x(1)}{\sqrt{2m}}) ] \\ × ∫_{ℝ^{2n}} dx_{2}E[ (\frac{S_{x_{1}}(1)}{m})^{k_{2}} e^{iλS_{x_{1}} (1)}M_{m}^{ϵ}(1)δ_{x_{2}} (x_{1} + \frac{x(1)}{\sqrt{2m}}) ] \\ × ⋯ × ∫_{ℝ^{2n}} dx_{m−1}E[ (\frac{S_{x_{m−2}}(1)}{m})^{k_{m−1}} e^{iλS_{x_{m−2}} (1)}M_{m}^{ϵ}(1) \\ × δ_{x_{m−1}} (x_{m−2} + \frac{x(1)}{\sqrt{2m}}) ] \\ × E[ (\frac{S_{x_{m−1}}(1)}{m})^{k_{m}} e^{iλS_{x_{m−1}} (1)}M_{m}^{ϵ}(1)δ_{x} (x_{m−1} + \frac{x(1)}{\sqrt{2m}}) ]. \end{matrix}

Proof. By using scaling property and Markovian property of the Brownian motion, for any ζ ∈ ℝ, we have

(4.2)  \begin{matrix}∑_{l=0}^{∞} \frac{(iζ)^{l}}{l!} E[S_{0}(1)^{l}e^{imλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ] \\ = E[e^{i(mλ+ζ)S_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ] \\ = E[e^{i(λ+ζ/m)S_{0}(m)}M_{m}^{ϵ}(m)δ_{x} (\frac{x(m)}{\sqrt{2m}}) ] \\ = ∫_{ℝ^{2n}} dx_{1}E[e^{i(λ+ζ/m)S_{0}(1)}M_{m}^{ϵ}(1)δ_{x_{1}} (\frac{x(1)}{\sqrt{2m}}) ] \\ × ∫_{ℝ^{2n}} dx_{2}E[e^{i(λ+ζ/m)S_{x_{1}} (1)}M_{m}^{ϵ}(1)δ_{x_{2}} (x_{1} + \frac{x(1)}{\sqrt{2m}}) ] \end{matrix}

× ⋯ × ∫_{ℝ^{2n}} dx_{m−1}E[e^{i(λ+ζ/m)S_{x_{m−2}} (1)}M_{m}^{ϵ}(1)

× δ_{x_{m−1}} (x_{m−2} + \frac{x(1)}{\sqrt{2m}}) ]

× E[e^{i(λ+ζ/m)S_{x_{m−1}} (1)}M_{m}^{ϵ}(1)δ_{x} (x_{m−1} + \frac{x(1)}{\sqrt{2m}}) ]

= ∑_{l_{1},···l_{m}=0}^{∞} \frac{1}{l_{1}! ⋯ l_{m}!}(iζ)^{l_{1}+···+l_{m}}

× ∫_{ℝ^{2n}} dx_{1}E[ (\frac{S_{0}(1)}{m})^{l_{1}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x_{1}} (\frac{x(1)}{\sqrt{2m}}) ]

× ∫_{ℝ^{2n}} dx_{2}E[ (\frac{S_{x_{1}}(1)}{m})^{l_{2}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x_{2}} (x_{1} + \frac{x(1)}{\sqrt{2m}}) ]

× ⋯ × ∫_{ℝ^{2n}} dx_{m−1}E[ (\frac{S_{x_{m−2}}(1)}{m})^{l_{m−1}} e^{iλS_{x_{m−2}} (1)}M_{m}^{ϵ}(1)

× δ_{x_{m−1}} (x_{m−2} + \frac{x(1)}{\sqrt{2m}}) ]

× E[ (\frac{S_{x_{m−1}}(1)}{m})^{l_{m}} e^{iλS_{x_{m−1}} (1)}M_{m}^{ϵ}(1)δ_{x} (x_{m−1} + \frac{x(1)}{\sqrt{2m}}) ].

Since ζ is arbitrary, we obtain (4.1).□

In the followings, we write C(ϵ_{1}, ϵ_{2}, ⋯, ϵ_{r}) a constant which depends only on the parameter ϵ_{1}, ϵ_{2}, ⋯, ϵ_{r} but may vary at each hand of the equations. We suppress the dependense on n. The next two lemmas give the necessary estimates in Malliavin calculus: Lemma 4.2. For any p > 1 and s ∈ ℤ_{+},

(4.3)  ∥M_{m}^{ϵ}(1)∥_{p,s} ≦ C(p, s) exp \{ (\frac{ϵ}{\sqrt{m}})^{p} C(p, s)\}

and

(4.4)  ∥M_{m}^{ϵ}(1) − I∥_{p,s} ≦ C(p, s)\frac{ϵ}{\sqrt{m}} exp \{ (\frac{ϵ}{\sqrt{m}})^{p} C(p, s)\}

where ∥ · ∥_{p,s} is the Sobolev norm on the Wiener space (see e.g. [13]).

Proof. For any continuous adapted Hilbert space valued process N(·) and 0 ≦ t ≦ 1, by using the inequality of Burkholder (Ikeda-Watanabe [5] Theorem III-3.1), we have

(4.5)  \begin{matrix}E[∥ ∫_{0}^{t} N(s)dΞ_{m}^{ϵ}(s)∥^{p}] \\ ≦ ∑_{j=1}^{2n} E[∥ ∫_{0}^{t} N(s)A_{j} \frac{ϵ}{\sqrt{m}}dx^{j}(s)∥^{p}](2n + 1)^{p−1} \\ + E[∥ ∫_{0}^{t} N(s)B \frac{ϵ^{2}}{m}ds∥^{p}](2n + 1)^{p−1} \\ ≦ C(p)\{ ∑_{j=1}^{2n} E[( ∫_{0}^{t} ∥N(s)∥^{2}∥A_{j}∥^{2} \frac{ϵ^{2}}{m}ds)^{\frac{p}{2}} ] \\ + E[ ∫_{0}^{t} ∥N(s)∥^{p}∥B∥^{p} \frac{ϵ^{2p}}{m^{p}} ds]t^{p−1}\} \\ ≦ C(p) (\frac{ϵ}{\sqrt{m}})^{p} ∫_{0}^{t} E[∥N(s)∥^{p}]ds. \end{matrix}

On the other hand, let H be the 2n-dimensional Cameron-Martin space ([13]). It is easily seen that

DΞ_{m}^{ϵ}(s)[h] = ∑_{j=1}^{2n} \frac{ϵ}{\sqrt{m}}A_{j}h^{j}(s) for h ∈ H,
D^{r}Ξ_{m}^{ϵ}(s) = 0 for r ≧ 2.

Then we see that

(4.6)  \begin{matrix}E[∥ ∫_{0}^{t} N(s)d(DΞ_{m}^{ϵ})(s)∥^{p}] \\ ≦ C(p) (\frac{ϵ}{\sqrt{m}})^{p} ∫_{0}^{t} E[∥N(s)∥^{p}]ds. \end{matrix}

Since, by (3.2),

D^{r}M_{m}^{ϵ}(t) =δ_{r0}I + ∫_{0}^{t} D^{r}M_{m}^{ϵ}(s)dΞ_{m}^{ϵ}(s)
+ r ∫_{0}^{t} D^{r−1}M_{m}^{ϵ}(s)d(DΞ_{m}^{ϵ})(s),

we have

(4.7)  \begin{matrix}∑_{r=0}^{s} E[∥D^{r}M_{m}^{ϵ}(t)∥^{p}] \\ ≦ C(p, s)(1 + ∑_{r=0}^{s} (\frac{ϵ}{\sqrt{m}})^{p} ∫_{0}^{t} E[∥D^{r}M_{m}^{ϵ}(s)∥^{p}]ds). \end{matrix}

By using the equivalences of the Sobolev norms ([5] Theorem V-8.4), (4.7) is rewritten as follows:

(4.8)  ∥M_{m}^{ϵ}(t)∥_{p,s}^{p} ≦ C(p, s)(1 + (\frac{ϵ}{\sqrt{m}})^{p} ∫_{0}^{t} ∥M_{m}^{ϵ}(s)∥_{p,s}^{p}ds).

By using the lemma of Gronwall, we obtain (4.3) easily. Similarly we obtain (4.4). □

Lemma 4.3. For any k ∈ ℕ, α ∈ ℤ_{+}^{2n} and q > 1, there exists s ∈ ℕ such that

(4.9)  sup_{x∈ℝ^{2n}}(1 + |x|)^{k}∥∂^{α}δ_{x}(x(1))∥_{q,−s} ≦ ∞.

Proof. For any κ > 0, there exists s ∈ ℕ such that

∥(1 + | · | − Δ)^{−s}ϕ(·)∥_{∞} ≦ C(s, κ)∥(1 + | · | − Δ)^{−κ}ϕ(·)∥_{2} for ϕ ∈ S(ℝ^{2n})

where ∥ · ∥_{∞} and ∥ · ∥_{2} are the L_{∞}-norm and L_{2}-norm on ℝ^{2n}, respectively. By combining this with Theorem 2.1 of [13], we have

∥∂^{α}δ_{x}(x(1))∥_{q,−s} ≦ C(q, s, k)∥(1 + | · | − Δ)^{−κ}∂^{α}δ_{x}(·)∥_{2}.

Furthermore the L_{2}-norm is estimated as follows:

(1 + |x|^{2})^{k}∥(1 + | · | − Δ)^{−κ}∂^{α}δ_{x}(·)∥_{2}

= (1 + |x|^{2})^{k} sup_{\begin{matrix}g∈S(ℝ^{2n}) \\ ∥g∥_{2}≦1 \end{matrix}} ∫_{ℝ^{2n}}(1 + |ξ|^{2} − Δ_{ξ})^{−κ}∂_{ξ}^{α}δ_{x}(ξ)g(ξ)dξ

= sup_{g} |(1 + |x|^{2})^{k}∂_{x}^{α}(1 + |x|^{2} − Δ_{x})^{−κ}g(x)|

= sup_{g} |(1 + |x|^{2})^{k} ∫_{ℝ^{2n}} \frac{e^{iξx}}{(2π)^{n}} ξ^{α}(1 + |ξ|^{2} − Δ_{ξ})^{−κ}ĝ(ξ)dξ|

≦ C sup_{g} ( ∫_{ℝ^{2n}} |(1 + |ξ|^{2})^{n}(1 − Δ_{ξ}^{k})ξ^{α}(1 + |ξ|^{2} − Δ_{ξ})^{−κ}ĝ(ξ)|^{2}dξ)^{\frac{1}{2}}

≦ C sup_{g} ( ∫_{ℝ^{2n}} |(1 + |ξ|^{2} − Δ_{ξ})^{κ(k,α,n)−κ}ĝ(ξ)|^{2}dξ)^{\frac{1}{2}}

for some κ(k, α, n) ≧ 0 determined by k, α and n. Thus, if κ is large enough, we have

(4.10)  (1 + |x|^{2})^{k}∥∂^{α}δ_{x}(x(1))∥_{q,−s} ≦ C(q, s, k, α).

Hence we conclude (4.9).□

Now we prove Theorem 2.2.

Proof of Theorem 2.2. We prove the theorem as λ → −∞ only. The case as λ → +∞ can be proved similarly.

We consider (i). We take m ∈ ℕ arbitrary. We exchange the order of differetiations and integrations in (4.1) formally:

(4.11)  \begin{matrix}x^{β}∂^{α}E[S_{0}(1)^{k}e^{imλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ] \\ = ∫_{(ℝ^{2n})^{m−1}} dx_{1} ⋯ dx_{m−1}Φ_{k,α,β}^{ϵ}(m, λ, x_{1}, ⋯, x_{m−1}, x) \end{matrix}

where (4.12)

Φ_{k,α,β}^{ϵ}(m, λ, x_{1}, ⋯, x_{m−1}, x)

= ∑_{k_{1}+···+k_{m}=k} \frac{k!}{k_{1}! ⋯ k_{m}!}E[ (\frac{S_{0}(1)}{m})^{k_{1}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x_{1}} (\frac{x(1)}{\sqrt{2m}}) ]

× E[ (\frac{S_{x_{1}}(1)}{m})^{k_{2}} e^{iλS_{x_{1}} (1)}M_{m}^{ϵ}(1)δ_{x_{2}} (x_{1} + \frac{x(1)}{\sqrt{2m}}) ]

× ⋯ × E[ (\frac{S_{x_{m−2}}(1)}{m})^{k_{m−1}} e^{iλS_{x_{m−2}} (1)}M_{m}^{ϵ}(1)δ_{x_{m−1}} (x_{m−2} + \frac{x(1)}{\sqrt{2m}}) ]

× x^{β}∂^{α}E[ (\frac{S_{x_{m−1}}(1)}{m})^{k_{m}} e^{iλS_{x_{m−1}} (1)}M_{m}^{ϵ}(1)δ_{x} (x_{m−1} + \frac{x(1)}{\sqrt{2m}}) ].

We will estimate the right hand side of (4.11). Let ∥·∥ be the Hilbert-Schmidt norm defined in Section 2. By using a property of δ-functions and making the change of variables \sqrt{2m}x_{j} → x_{j},

(4.13) \begin{matrix}∫_{(ℝ^{2n})^{m−1}} dx_{1} ⋯ dx_{m−1}∥Φ_{k,α,β}^{ϵ}(m, λ, x_{1}, ⋯, x_{m−1}, x)∥ \\ ≦ ∑_{k_{1}+···+k_{m}=k} \frac{k!}{k_{1}! ⋯ k_{m}!} \\ × ∫_{(ℝ^{2n})^{m−1}} dx_{1} ⋯ dx_{m−1}∥E[ (\frac{S_{0}(1)}{m})^{k_{1}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1) \\ × δ_{\sqrt{2m}x_{1}}(x(1))]∥(2m)^{n} \\ × ∥E[ (\frac{S_{x_{1}}(1)}{m})^{k_{1}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{\sqrt{2m}x_{2}}(\sqrt{2m}x_{1} + x(1))]∥(2m)^{n} \\ × ⋯ × ∥E[ (\frac{S_{x_{m−2}}(1)}{m})^{k_{m−1}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1) \\ × δ_{\sqrt{2m}x_{m−1}}(\sqrt{2m}x_{m−2} + x(1))]∥(2m)^{n} \\ × ∥x^{β}∂^{α}E[ (\frac{S_{x_{m−1}}(1)}{m})^{k_{m}} e^{iλS_{x_{m−1}} (1)}M_{m}^{ϵ}(1)δ_{x} (x_{m−1} + \frac{x(1)}{\sqrt{2m}}) ]∥ \\ = ∑_{k_{1}+···+k_{m}=k} \frac{k!m^{k}}{k_{1}! ⋯ k_{m}!} \end{matrix}

× ∫_{(ℝ^{2n})^{m−1}} dx_{1} ⋯ dx_{m−1}∥E[ (\frac{S_{0}(1)}{m^{2}})^{k_{1}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x_{1}}(x(1))]∥

× ∥E[ (\frac{S_{x_{1}/\sqrt{2m}}(1)}{m^{2}})^{k_{2}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x_{2}}(x_{1} + x(1))]∥

× ⋯ × ∥E[ (\frac{S_{x_{m−2}/\sqrt{2m}}(1)}{m^{2}})^{k_{m−1}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x_{m−1}}(x_{m−2} + x(1))]∥

× ∥x^{β}∂^{α}E[ (\frac{S_{x_{m−1}/\sqrt{2m}}(1)}{m^{2}})^{k_{m}} e^{iλS_{x_{m−1}/\sqrt{2m}}(1)}M_{m}^{ϵ}(1)δ_{x} (\frac{x_{m−1} + x(1)}{\sqrt{2m}}) ]∥.

Since

(1 + |x_{1}|) · (1 + |x_{2} − x_{1}|) · · · · · (1 + |x_{j} − x_{j−1}|) ≧ 1 + |x_{j}|

for any j ≧ 1, if we set R_{n} = ∫_{ℝ^{2n}}(1 + |x|)^{−2n−1}dx, (4.14)

the right hand side of (4.13)

≦ ∑_{k_{1}+···+k_{m}=k} \frac{k!m^{k}}{k_{1}! ⋯ k_{m}!} ∫_{(ℝ^{2n})^{m−1}} dx_{1} ⋯ dx_{m−1}

× ∥(1 + |x_{1}|)^{2n+2k+|α|+|β|+1}E[ (\frac{S_{0}(1)}{m^{2}})^{k_{1}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x_{1}}(x(1))]∥

× ∥(1 + |x_{2} − x_{1}|)^{2n+2k+|α|+|β|+1}E[ (\frac{S_{x_{1}/\sqrt{2m}}(1)}{m^{2}(1 + |x_{1}|)^{2}} )^{k_{1}} e^{iλS_{0}(1)}

× M_{m}^{ϵ}(1)δ_{x_{2}}(x_{1} + x(1))]∥

× ⋯ × ∥(1 + |x_{m−1} − x_{m−2}|)^{2n+2k+|α|+|β|+1}E[ (\frac{S_{x_{m−2}/\sqrt{2m}}(1)}{m^{2}(1 + |x_{m−2}|)^{2}} )^{k_{m−1}}

× e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x_{m−1}}(x_{m−2} + x(1))]∥

× ∥\frac{x^{β}}{(1 + |x_{m−1}|)^{|α|+|β|}} ∂^{α}E[ (\frac{S_{x_{m−1}/\sqrt{2m}}(1)}{m^{2}(1 + |x_{m−1}|)^{2}} )^{k_{m}} e^{iλS_{x_{m−1}/\sqrt{2m}}(1)}

× M_{m}^{ϵ}(1)δ_{x} (\frac{x_{m−1} + x(1)}{\sqrt{2m}}) ]∥

× \frac{1}{(1 + |x_{1}|)^{2n+1}} · \frac{1}{(1 + |x_{2} − x_{1}|)^{2n+1}} · · · · · \frac{1}{(1 + |x_{m−1} − x_{m−2}|)^{2n+1}}

≦ ∑_{k_{1}+···+k_{m}=k} \frac{k!m^{k}}{k_{1}! ⋯ k_{m}!}

× \{ ∏_{j=1}^{m−1} R_{n} sup_{x,x′} ∥(1 + |x′ − x|)^{2n+2k+|α|+|β|+1}E[ (\frac{S_{x/\sqrt{2m}}(1)}{m^{2}(1 + |x|)^{2}} )^{k_{j}}

× e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x′}(x + x(1))]∥}

× sup_{x,x′} ∥\frac{x^{′β}}{(1 + |x|)^{|α|+|β|}} ∂_{x′}^{α}E[ (\frac{S_{x/\sqrt{2m}}(1)}{m^{2}(1 + |x|)^{2}} )^{k_{m}} e^{iλS_{x/\sqrt{2m}}(1)}

× M_{m}^{ϵ}(1)δ_{x′} (\frac{x + x(1)}{\sqrt{2m}}) ]∥.

The middle factor under the summation in the right hand side of (4.14) is estimated as follows:

(4.15) \begin{matrix}∥(1 + |x′ − x|)^{2n+2k+|α|+|β|+1}E[ (\frac{S_{x/\sqrt{2m}}(1)}{m^{2}(1 + |x|)^{2}} )^{l} e^{iλS_{0}(1)} \\ × M_{m}^{ϵ}(1)δ_{x′}(x + x(1))]∥ \\ ≦ ∑_{h=0}^{l} \frac{l! · (1 + |x′ − x|)^{2n+2k+|α|+|β|+1}}{h!(l − h)!} (\frac{∑_{j=1}^{n} |(x^{n+j}x^{′j} − x^{j}x^{′n+j})|}{\sqrt{m}m^{2}(1 + |x|)^{2}})^{h} \\ × ∥E[ (\frac{S_{0}(1)}{m^{2}(1 + |x|)^{2}} )^{l−h} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x′}(x + x(1))]∥ \\ ≦ (\frac{4}{m})^{l} (1 + |x′ − x|)^{2n+3k+|α|+|β|+1} \\ × ∑_{h=0}^{l} ∥E[ (\frac{S_{0}(1)}{m})^{h} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x′}(x + x(1))]∥. \end{matrix}

On the other hand, since

∂_{x′}^{α}E[Φδ_{ax′}(x + x(1))] = (−a)^{|α|}E[Φ∂^{α}δ_{ax′}(x + x(1))]

for any a > 0 and α ∈ ℤ_{+}^{2n},

∂_{x′}^{α}E[ (\frac{S_{x/\sqrt{2m}}(1)}{m^{2}(1 + |x|)^{2}} )^{l} e^{iλS_{x/\sqrt{2m}}(1)}M_{m}^{ϵ}(1)δ_{x′} (\frac{x + x(1)}{\sqrt{2m}}) ]

= ∑_{α_{1}+α_{2}+α_{3}=α} ∑_{h=0}^{l} \frac{l!}{h!(l − h)!}∂_{x′}^{α_{1}} (\frac{\sqrt{2} ∑_{j=1}^{n}(x^{n+j}x^{′j} − x^{j}x^{′n+j})}{m^{2}(1 + |x|)^{2}})^{h}

× ∂_{x′}^{α_{2}} exp (iλ\sqrt{2} ∑_{j=1}^{n}(x^{n+j}x^{′j} − x^{j}x^{′n+j})) (−1)^{|α_{3}|}(2m)^{n+ \frac{|α_{3}|}{2}}

× E[ (\frac{S_{0}(1)}{m^{2}(1 + |x|)^{2}} )^{l−h} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)

× ∂^{α_{3}}δ_{\sqrt{2m}x′}(x + x(1))].

Thus the last factor under the summation of the right hand side of (4.4) is estimated as follows: (4.16)

∥\frac{x^{′β}}{(1 + |x|)^{|α|+|β|}} ∂_{x′}^{α}E[ (\frac{S_{x/\sqrt{2m}}(1)}{m^{2}(1 + |x|)^{2}} )^{l} e^{iλS_{x/\sqrt{2m}}(1)}M_{m}^{ϵ}(1)δ_{x′} (\frac{x + x(1)}{\sqrt{2m}}) ]∥

≦ ∑_{α_{1}+α_{2}+α_{3}=α} ∑_{h=0}^{l} \frac{l!}{h!(l − h)!} (\frac{|x′|}{1 + |x|})^{|β|} (\frac{2\sqrt{2}n|x||x′|}{m^{2}(1 + |x|)^{2}} )^{h} (\frac{\sqrt{2}λ|x|}{1 + |x|})^{|α_{2}|}

× (2m)^{n+|α_{3}|/2}∥E[ (\frac{S_{0}(1)}{m^{2}(1 + |x|)^{2}} )^{l−h} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)

× ∂^{α_{3}}δ_{\sqrt{2m}x′}(x + x(1))]∥ ≦ 2^{|α|+2k+n}m^{n+|α|/2}n^{k}λ^{|α|}(1 + |\sqrt{2m}x′ − x|)^{|β|+k}

× ∑_{\begin{matrix}h≦l \\ α′≦α \end{matrix}} ∥E[ (\frac{S_{0}(1)}{m})^{h} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)∂^{α′}δ_{\sqrt{2m}x′}(x + x(1))]∥.

By combining (4.14), (4.15) and (4.16), we obtain (4.17)

sup_{x} ∫_{(ℝ^{2n})^{m−1}} dx_{1} ⋯ dx_{m−1}∥Φ_{k,α,β}^{ϵ}(m, λ, x_{1}, ⋯, x_{m−1}, x)∥

≦ ∑_{k_{1}+···+k_{m}=k} \frac{k!m^{k}}{k_{1}! ⋯ k_{m}!} (\frac{4}{m})^{k} 2^{|α|+2k+n}m^{n+|α|/2}n^{k}λ^{|α|}

× \{ ∏_{j=1}^{m−1} R_{n} ∑_{h=0}^{k_{j}} sup_{x} ∥(1 + |x|)^{2n+3k+|α|+|β|+1}E[ (\frac{S_{0}(1)}{m})^{h}

× e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x}(x(1))]∥}

× \{ ∑_{\begin{matrix}0≦h≦k_{m} \\ α′≦α \end{matrix}} sup_{x} ∥(1 + |x|)^{|β|+k}E[ (\frac{S_{0}(1)}{m})^{h} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)

× ∂^{α′}δ_{x}(x(1))]∥} ≦ 2^{|α|+4k+n}m^{n+k|α|/2}λ^{|α|}n^{k}

× \{R_{n} ∑_{h=0}^{k} sup_{x} ∥(1 + |x|)^{2n+3k+|α|+|β|+1}E[ (\frac{S_{0}(1)}{m})^{h} e^{iλS_{0}(1)}

× M_{m}^{ϵ}(1)δ_{x}(x(1))]∥}^{m−1}

× \{ ∑_{\begin{matrix}0≦h≦k \\ α′≦α \end{matrix}} sup_{x} ∥(1 + |x|)^{|β|+k}E[ (\frac{S_{0}(1)}{m})^{h} e^{iλS_{0}(1)}

× M_{m}^{ϵ}(1)∂^{α′}δ_{x}(x(1))]∥} ≦ 2^{|α|+4k+n}m^{n+k+|α|/2}λ^{|α|}n^{k}

× {C sup_{x} |(1 + |x|)^{2n+3k+|α|+|β|+1}E[e^{iλS_{0}(1)}δ_{x}(x(1))]|

+ C(p, s)∥e^{iλS_{0}(1)}(M_{m}^{ϵ}(1) − I)∥_{p,s}

× sup_{x}(1 + |x|)^{2n+3k+|α|+|β|+1}∥δ_{x}(x(1))∥_{q,−s}

+ C(p, s) ∑_{h=1}^{k} ∥ (\frac{S_{0}(1)}{m})^{h} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)∥_{p,s}

× sup_{x}(1 + |x|)^{2n+3k+|α|+|β|+1}∥δ_{x}(x(1))∥_{q,−s}}^{m−1}

\begin{matrix}× \{C(p, s) ∑_{h=0}^{k} ∥ (\frac{S_{0}(1)}{m})^{h} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)∥_{p,s} \\ × ∑_{α′≦α} sup_{x}(1 + |x|)^{|β|+k}∂^{α′}∥δ_{x}(x(1))∥_{q,−s}\} \end{matrix}

where p, q > 1, 1/p + 1/q = 1, s ∈ ℕ. Furthermore, ∥e^{iλS_{0}(1)}(M_{m}^{ϵ}(1) − I)∥_{p,s}

≦ ∥e^{iλS_{0}(1)}∥_{2p,s}∥M_{m}^{ϵ}(1) − I∥_{2p,s}
≦ C(p)λ^{N(s)}∥M_{m}^{ϵ}(1) − I∥_{2p,s}

where N(s) is an integer which is determined by the number s. Similarly,

∥S_{0}(1)^{h}e^{iλS_{0}(1)}M_{m}^{ϵ}(1)∥_{p,s} ≦ C(p, h)λ^{N(s)}∥M_{m}^{ϵ}(1)∥_{p,s}.

Thus by Lemma 4.2, we have

(4.18)  \begin{matrix}∥e^{iλS_{0}(1)}(M_{m}^{ϵ}(1) − I)∥_{p,s} \\ ≦ \frac{ϵ}{\sqrt{m}}λ^{N(s)}C(p, s) exp \{(\frac{ϵ}{\sqrt{m}})^{p}C(p, s)\} \end{matrix}

and

(4.19)  \begin{matrix}∥S_{0}(1)^{h}e^{iλS_{0}(1)}M_{m}^{ϵ}(1)∥_{p,s} \\ ≦ λ^{N(s)}C(p, s, h) exp \{(\frac{ϵ}{\sqrt{m}})^{p}C(p, s)\} . \end{matrix}

By applying Lemma 4.3, (4.18) and (4.19) to (4.17), we have (4.20)

sup_{x∈ℝ^{2n}} ∫_{(ℝ^{2n})^{m−1}} dx_{1} ⋯ dx_{m−1}∥Φ_{k,α,β}^{ϵ}(m, λ, x_{1}, ⋯, x_{m−1}, x)∥

≦ 2^{|α|+4k+n}m^{n+k+|α|/2}λ^{|α|}n^{k}

× {C sup_{x∈ℝ^{2n}} |(1 + |x|)^{2n+3k+|α|+|β|+1}E[e^{iλS_{0}(1)}δ_{x}(x(1))]|

+ \frac{ϵ}{\sqrt{m}}λ^{N(s)}C(p, s, α, β, k) exp ((\frac{ϵ}{\sqrt{m}})^{p}C(p, s))

+ \frac{1}{m}λ^{N(s)}C(p, s, α, β, k) exp ((\frac{ϵ}{\sqrt{m}})^{p}C(p, s)) \}^{m−1}

× \{λ^{N(s)}C(p, s, α, β, k) exp ((\frac{ϵ}{\sqrt{m}})^{p}C(p, s)) \}.

Since the right hand side of (4.20) is finite, the exchanging of the order of differentiations and integrations in (4.11) is justified and we obtain

(4.21)  \begin{matrix}sup_{x∈ℝ^{2n},0<ϵ≦ϵ_{0}} ∥x^{β}∂^{α}E[S_{0}(1)^{k}e^{imλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥ \\ ≦ 2^{|α|+4k+n}m^{n+k+|α|/2}λ^{|α|}n^{k} \\ × \{C sup_{x∈ℝ^{2n}} |(1 + |x|)^{2n+3k+|α|+|β|+1}E[e^{iλS_{0}(1)}δ_{x}(x(1))]| \\ + \frac{1}{\sqrt{m}}λ^{N(s)}C(p, s, α, β, k, ϵ_{0})\}^{m−1} \\ × \{λ^{N(s)}C(p, s, α, β, k, ϵ_{0})\}. \end{matrix}

If we replace (λ, m) with (λζ/[λ], [λ]) where [λ] is the integral part of λ,

(4.22)  \begin{matrix}sup_{x∈ℝ^{2n},0<ϵ≦ϵ_{0}} ∥x^{β}∂^{α}E[S_{0}(1)^{k}e^{iλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥ \\ ≦ 2^{|α|+4k+n}[λ]^{n+k+|α|/2} (\frac{λ}{[λ]}ζ)^{|α|} n^{k} \\ × \{C sup_{x∈ℝ^{2n}} |(1 + |x|)^{2n+3k+|α|+|β|+1}E[e^{iλS_{0}(1)}δ_{x}(x(1))]| \\ + \frac{1}{\sqrt{[λ]}} (\frac{λ}{[λ]}ζ)^{N(s)} C(p, s, α, β, k, ϵ_{0})\}^{[λ]−1} \\ × \{ (\frac{λ}{[λ]}ζ)^{N(s)} C(p, s, α, β, k, ϵ_{0})\}. \end{matrix}

Hence we have

(4.23)

\overline{lim}_{λ→∞} \frac{1}{λ} log sup_{x∈ℝ^{2n}} ∥x^{β}∂^{α}E[S_{0}(1)^{k}e^{iλζS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥

= \overline{lim}_{λ→∞} log \{C sup_{x∈ℝ^{2n}} |(1 + |x|)^{2n+3k+|α|+|β|+1}

× E[e^{iλζS_{0}(1)/[λ]}δ_{x}(x(1))]|

+ \frac{1}{\sqrt{[λ]}} (\frac{λ}{[λ]}ζ)^{N(s)} C(p, s, α, β, k, ϵ_{0})\}

= log \overline{lim}_{λ→∞} \{C sup_{x∈ℝ^{2n}} |(1 + |x|)^{2n+3k+|α|+|β|+1}

× E[e^{iλζS_{0}(1)/[λ]}δ_{x}(x(1))]|}.

Since λ coth λ ≧ 1 for ∀λ ∈ ℝ,

(4.24)

sup_{x∈ℝ^{2n}} |(1 + |x|)^{h}E[e^{iλS_{0}(1)}δ_{x}(x(1))]|

= sup_{x∈ℝ^{2n}} |(1 + |x|)^{h} (\frac{1}{2π} )^{n} (\frac{λ/2}{sinh(λ/2)})^{n} exp (−\frac{|x|^{2}λ}{4} coth \frac{λ}{2}) |

≦ C(h) (\frac{λ}{sinh(λ)})^{n} ,

and so

\overline{lim}_{λ→∞} \frac{1}{λ} log sup_{x∈ℝ^{2n},0<ϵ≦ϵ_{0}} ∥x^{β}∂^{α}E[S_{0}(1)^{k}e^{iλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥

≦ log C(k, α, β) \overline{lim}_{λ→∞} (\frac{λζ/[λ]}{sinh λζ/[λ]})^{n}

= log C(k, α, β) (\frac{ζ}{sinh ζ} )^{n} .

Consequently we obtain (4.25)

\overline{lim}_{λ→∞} \frac{1}{λ} log sup_{x∈ℝ^{2n},0<ϵ≦ϵ_{0}} ∥x^{β}∂^{α}E[S_{0}(1)^{k}e^{iλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥

≦ \overline{lim}_{ζ→∞} \overline{lim}_{λ→∞} \frac{1}{ζλ} log sup_{x∈ℝ^{2n},0<ϵ≦ϵ_{0}} ∥x^{β}∂^{α}E[S_{0}(1)^{k}e^{iζλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥

≦ \overline{lim}_{ζ→∞} \frac{1}{ζ} log C(k, α, β) (\frac{ζ}{sinh ζ} )^{n}

= \overline{lim}_{ζ→∞} \frac{n}{ζ} log C(k, α, β) (\frac{ζ}{sinh ζ} )

= −n

which completes the proof of (i). For the proof of (ii), we use the following (4.26), instead of (4.1): for any m ∈ ℕ and ϵ > 0,

(4.26) \begin{matrix}E[S_{0}(1)^{k}e^{imλS_{0}(1)}(M^{ϵ}(1) − I)δ_{x} (\frac{x(1)}{\sqrt{2}}) ] \\ = ∑_{k_{1}+···+k_{m}=k} ∑_{l=1}^{m} \frac{k!}{k_{1}! ⋯ k_{m}!} \\ × ∫_{ℝ^{2n}} dx_{1}E[ (\frac{S_{0}(1)}{m})^{k_{1}} e^{iλS_{0}(1)}M_{m}^{ϵ}(1)δ_{x_{1}} (\frac{x(1)}{\sqrt{2m}}) ] \\ × ⋯ × ∫_{ℝ^{2n}} dx_{l−1}E[ (\frac{S_{x_{l−2}}(1)}{m})^{k_{l−1}} e^{iλS_{x_{l−1}} (1)}M_{m}^{ϵ}(1) \\ × δ_{x_{l−1}} (x_{l−2} + \frac{x(1)}{\sqrt{2m}}) ] \end{matrix}

× ∫_{ℝ^{2n}} dx_{l}E[ (\frac{S_{x_{l−1}}(1)}{m})^{k_{l}} e^{iλS_{x_{l−1}} (1)}(M_{m}^{ϵ}(1) − I)

× δ_{x_{l}} (x_{l−1} + \frac{x(1)}{\sqrt{2m}}) ]

× ∫_{ℝ^{2n}} dx_{l+1}E[ (\frac{S_{x_{l}}(1)}{m})^{k_{l+1}} e^{iλS_{x_{l}} (1)}δ_{x_{l+1}} (x_{l} + \frac{x(1)}{\sqrt{2m}}) ]

× ⋯ × ∫_{ℝ^{2n}} dx_{m−1}E[ (\frac{S_{x_{m−2}}(1)}{m})^{k_{m−1}} e^{iλS_{x_{m−2}} (1)}

× δ_{x_{m−1}} (x_{m−2} + \frac{x(1)}{\sqrt{2m}}) ]

× E[ (\frac{S_{x_{m−1}}(1)}{m})^{k_{m}} e^{iλS_{x_{m−1}} (1)}δ_{x} (x_{m−1} + \frac{x(1)}{\sqrt{2m}}) ].

This identity is easily proven from (4.1). □

5. Proof of Theorem 2.1 and properties of the heat kernels

In this section, we prove Theorem 2.1. Furthermore we prove some properties of the kernel. We name the right hand side of (2.12) as follows:

(5.1)  \begin{matrix}h_{t}((x, u, r), (x′, u′, r′)) \\ = p_{t}^{n−2q+2}((x, u)^{−1}(x′, u′))e_{t}^{−}(r, r′)P \\ + p_{t}^{n−2q}((x, u)^{−1}(x′, u′))e_{t}^{+}(r, r′)Q \\ + q_{t}^{n−2q}((x, u)^{−1}(x′.u′), r + r′)Q \\ =: h_{t}^{(1)}((x, u, r), (x′, u′, r′)) \\ + h_{t}^{(2)}((x, u, r), (x′, u′, r′)) \\ + h_{t}^{(3)}((x, u, r), (x′, u′, r′)). \end{matrix}

First we prove that h_{t}^{(i)} has the following good regularity properties.

Proposition 5.1. If i ≠ 3 or q ≠ 0 (resp. i = 3 and q = 0), h_{t}^{(i)}(X, X′) is the restriction to D̄ of a rapidly decreasing function of X for each fixed (t, X′) ∈ (0, ∞) × D̄ (resp. (0, ∞) × D) and the restriction to D̄ of a rapidly decreasing function of X′ for each fixed (t, X) ∈ (0, ∞) × D̄ (resp. (0, ∞) × D) uniformly with respect to t ∈ [t_{0}, t_{1}] for any 0 < t_{0} ≦ t_{1}. Moreover h_{t}^{(i)}(X, X′) is a smooth function of (t, X, X′) ∈ (0, ∞) × D × D.

Proof. For any β, γ ∈ ℤ_{+}^{2n} and k_{1}, k_{2}, k_{3}, k_{4} ∈ ℤ_{+}, we exchange the order of differentiations and integrate by parts in q_{t} formally:

(5.2)  x^{γ}u^{k_{1}}r^{k_{2}}∂_{x}^{β} \frac{∂^{k_{3}}}{∂u^{k_{3}}} \frac{∂^{k_{4}}}{∂r^{k_{4}}} q_{t}^{α}(x, u, r)

= a linear combination of { ∫_{−∞}^{∞} dλ · λ^{l_{1}} exp(iλu − tλα)

× x^{γ}∂_{x}^{β}E[S_{0}(t)^{l_{2}}e^{−iλS_{0}(t)}M(t)δ_{x} (\frac{x(t)}{\sqrt{2}}) ]

× r^{l_{3}}e^{rλ} ∫_{\sqrt{t/2}(r/t+λ)}^{∞} dμe^{−μ^{2}}, ∫_{−∞}^{∞} dλ · λ^{l_{1}} exp(iλu − tλα)

× x^{γ}∂_{x}^{β}E[S_{0}(t)^{l_{2}}e^{−iλS_{0}(t)}M(t)δ_{x} (\frac{x(t)}{\sqrt{2}}) ]

× r^{l_{3}}e^{rλ} exp (−\frac{t}{2}(\frac{r}{t} + λ)^{2}) \}_{0≦l_{1},l_{2},l_{3}≦k}

=: q_{−}_{t}(x, u, r),

where k = 1 + k_{1} + k_{2} + k_{3} + k_{4}. By using (2.21), the right hand side of (4.2) is estimated as follows:

(5.3)

sup_{\begin{matrix}(x,u)∈ℝ^{2n+1} \\ r≧r_{0} \\ t∈[t_{0},t_{1}] \end{matrix}} ∥q_{−}_{t}(x, u, r)∥

≦ C(k, t_{0}, t_{1}) ∑_{0≦l_{1},l_{2},l_{3}≦k} ∫_{−∞}^{∞} dλ · |λ^{l_{1}}| sup_{t∈[t_{0},t_{1}]} { exp(−tλα)

× sup_{x∈ℝ^{2n}} ∥x^{γ}∂_{x}^{β}E[S_{0}(t)^{l_{2}}e^{−iλS_{0}(t)}M(t)δ_{x} (\frac{x(t)}{\sqrt{2}}) ]∥

× sup_{r≧r_{0}} |r|^{l_{3}}e^{rλ}( ∫_{\sqrt{t/2}(r/t+λ}^{∞} dμe^{−μ^{2}} + exp (−\frac{t}{2}(\frac{r}{t} + λ)^{2})\}

≦ C(k, γ, β, t_{0}, t_{1}, ϵ, r_{0}) ∑_{0≦l_{1},l_{2},l_{3}≦2k} [ ∫_{0}^{∞} dλ · λ^{l_{1}} sup_{t∈[t_{0},t_{1}]} {e^{−tλα}

× e^{λ(−n+ϵ)t} exp(−\frac{t}{4}λ^{2})\}

+ ∫_{−∞}^{0} dλ · (−λ)^{l_{1}} sup_{t∈[t_{0},t_{1}]}{e^{−tλα}e^{r_{0}λ}e^{−λ(−n+ϵ)t}}]

for any r_{0} ≧ 0 and ϵ > 0. The right hand side of (5.3) is finite if and only if α < n or α ≦ n and r_{0} > 0 . In this case the formal computations in (5.2) are justified and q_{t}^{α}(x, u, r) is the restriction to ℝ^{2n+1} × [r_{0}, ∞) of a rapidly decreasing function uniformly with respect to t ∈ [t_{0}, t_{1}] for any 0 < t_{0} ≦ t_{1} and a smooth function of (t, x, u, r) ∈ (0, ∞) × ℝ^{2n+1} × [r_{0}, ∞). By similar methods, we can prove that p_{t}^{α}(x, u) is a rapidly decreasing function of (x, u) ∈ ℝ^{2n+1} and a smooth function of (t, x, u) ∈ (0, ∞) × ℝ^{2n+1}. □

Now, for any f ∈ L_{2}^{p,q}(D), we define

(5.4)  \begin{matrix}(H_{t}^{(i)}f)(x, u, r) := ∫D̄ h_{t}^{(i)}((x, u, r), (x′, u′, r′))f(x′, u′, r′)dx′du′dr′ \\ (x, u, r) ∈ D̄, i = 1, 2, 3. \end{matrix}

For this H_{t}^{(i)}, we have the following (cf. [10] [11]):

Theorem 5.1. H_{t}^{(i)}, i = 1, 2, 3, are bounded operators on L_{2}^{p,q}(D). Furthermore, for fixed f ∈ Λ_{0}^{p,q}(D), the following holds:

(i) for T > 0, there is a constant C depending only on f and T such that for t ∈ (0, T], ∥H_{t}^{(i)}f∥ ≦ C, i = 1, 2, 3;

(ii) H_{t}^{(1)}f ⟶ P f, H_{t}^{(2)}f ⟶ Qf, H_{t}^{(3)}f ⟶ 0 in L_{2}^{p,q}(D) and uniformly as

t ⟶ 0; (iii) H_{t}^{(i)}f, i = 1, 2, 3, are the restrictions of rapidly decreasing forms; (iv) H_{t}^{(i)}f, i = 1, 2, 3, are differentiable in t;

(v) H_{t}f := H_{t}^{(1)}f + H_{t}^{(2)}f + H_{t}^{(3)}f ∈ Dom □; (vi) (∂/∂t + □) H_{t}f = 0; (vii) □H_{t}f = H_{t}□f.

Proof. For any r_{0} ≧ 0, we set D_{r_{0}} := {(x, u, r) ∈ D; r ≧ r_{0}}. Then, for any r_{0} ≧ 0 and f ∈ L_{2}^{p,q}(D), we have

(5.5)

∫_{D_{r_{0}}} dX∥(H_{t}^{(i)}f)(X)∥^{2} ≦ ∫_{D_{r_{0}}} dX ∫_{D} dY ∥h_{t}^{(i)}(X, Y)∥ ∫_{D} dY ∥h_{t}^{(i)}(X, Y)∥∥f(Y)∥^{2}

≦ ( sup_{X∈D_{r_{0}}} ∫_{D} dY ∥h_{t}^{(i)}(X, Y)∥) ∫_{D} dY ∥f(Y)∥^{2}

× ∫_{D_{r_{0}}} dX∥h_{t}^{(i)}(X, Y)∥

≦ ( sup_{X∈D_{r_{0}}} ∫_{D} dY ∥h_{t}^{(i)}(X, Y)∥) ( sup_{Y ∈D} ∫_{D_{r_{0}}} dX∥h_{t}^{(i)}(X, Y)∥)

× ∫_{D} ∥f(Y)∥^{2}

≦ (∫_{D_{r_{0}}} dY ∥h_{t}^{(i)}(0, Y)∥)^{2} ∫_{D} dY ∥f(Y)∥^{2}.

Hence H_{t}^{(i)} is a bounded linear operator from L_{2}^{p,q}(D) to L_{2}^{p,q}(D_{r_{0}}) if r_{0} > 0 or (i, q) ≠ (3, 0). Especially, if (i, q) ≠ (3, 0), H_{t}^{(i)} is a bounded linear operator on L_{2}^{p,q}(D). In the followings, let f ∈ Λ_{0}^{p,q}(D). We will prove (i)-(vii) in order.

(i): From (5.5), it is enough to show that

(5.6)  sup_{ϵ∈(0,T]} ∫_{D} ∥h_{ϵ^{2}}^{(i)}(0, Y)∥dY < ∞.

To do this for i = 1 and 2, it is enough to show that

(5.7)  sup_{ϵ∈(0,T]} ∫_{ℝ^{2n+1}} ∥p_{ϵ^{2}}^{α}(x, u)∥dxdu < ∞

for any α ∈ ℤ. We use scaling property of the Brownian motion and make the change of the variables (x → ϵx, u → ϵu, ϵλ → λ):

(5.8)

∫_{ℝ^{2n+1}} ∥p_{ϵ^{2}}^{α}(x, u)∥dxdu

= ∫_{ℝ^{2n+1}} dxdu∥ ∫_{−∞}^{∞} dλ\frac{1}{2π} exp(iλu − ϵλα − \frac{λ^{2}}{2} )

× E[e^{−iϵλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥

≦ C sup_{(x,u)∈ℝ^{2n+1}}(1 + |x|^{2})^{2n}(1 + u^{2})∥ ∫_{−∞}^{∞} dλ exp(iλu − ϵλα − \frac{λ^{2}}{2} )

× E[e^{−iϵλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥

= C sup_{(x,u)∈ℝ^{2n+1}}(1 + |x|^{2})^{2n}∥ ∫_{−∞}^{∞} dλe^{iλu} (1 − \frac{∂^{2}}{∂λ^{2}} )

× \{ exp (−ϵλα − \frac{λ^{2}}{2} ) E[e^{−iϵλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]\}∥

≦ C(α)(1 + ϵ^{2}) ∫_{−∞}^{∞} dλ(1 + |λ|^{2}) exp (−ϵλα − \frac{λ^{2}}{2} )

× ∑_{k=0}^{2} sup_{x∈ℝ^{2n}}(1 + |x|^{2})^{2n}∥E[S_{0}(1)^{k}e^{−iϵλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥

≦ C(α, T).

The last inequality follows from Theorem 2.2 (i). From (5.8), we obtain (5.7). For i = 3, we use the following inequality instead of (5.5): if R is the distance from supp f to bD,

(5.9)  ∥H_{t}^{(3)}f∥_{2} ≦ C (∫_{ℝ^{2n+1}} dxdu ∫_{R}^{∞} dr∥q_{t}^{n−2q}(x, u, r)∥) ∥Qf∥_{2}.

As in (5.8), we use scaling property and make the change of variables (x → ϵx, u → ϵu, r → ϵr, ϵλ → λ):

(5.10)  \begin{matrix}∫_{ℝ^{2n+1}} dxdu ∫_{R}^{∞} dr∥q_{ϵ^{2}}^{α}(x, u, r)∥ \\ = \frac{1}{π^{n+3/2}} ∫_{ℝ^{2n+1}} dxdu ∫_{R/ϵ}^{∞} dr∥ ∫_{−∞}^{∞} dλ · λ exp(iλu − ϵα) \end{matrix}

× E[e^{−iϵλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]e^{rλ} ∫_{(r+λ)/\sqrt{2}}^{∞} dμe^{−μ^{2}} ∥

≦ C ∫_{R/ϵ}^{∞} dr sup_{(x,u)∈ℝ^{2n+1}}(1 + |x|^{2})^{2n}(1 + u^{2})∥ ∫_{−∞}^{∞} dλ · λ exp(iλu − ϵλα)

× E[e^{−iϵλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]e^{rλ} ∫_{(r+λ)/\sqrt{2}}^{∞} dμe^{−μ^{2}} ∥

= C ∫_{R/ϵ}^{∞} dr sup_{(x,u)∈ℝ^{2n+1}}(1 + |x|^{2})^{2n}∥ ∫_{−∞}^{∞} dλe^{iλu} (1 − \frac{∂^{2}}{∂λ^{2}} ) \{λe^{−ϵλα}

× E[e^{−iϵλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]e^{rλ} ∫_{(r+λ)/\sqrt{2}}^{∞} dμe^{−μ^{2}}\}∥

≦ C(α)(1 + ϵ^{2}) ∫_{R/ϵ}^{∞} dr(1 + r^{2}) ∫_{−∞}^{∞} dλ(1 + λ^{2})e^{−ϵα}

× ∑_{k=0}^{2} sup_{x∈ℝ^{2n}}(1 + |x|^{2})^{2n}∥E[S_{0}(1)^{k}e^{−iϵλS_{0}(1)}M^{ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥

× e^{rλ} (∫_{(r+λ)/\sqrt{2}}^{∞} dμe^{−μ^{2}} + exp (−\frac{1}{2}(r + λ)^{2}))

≦ C(α, T, η) ∫_{R/ϵ}^{∞} dr(1 + r^{2}) ∫_{−∞}^{∞} dλ(1 + λ^{2}) exp(−ϵλα − ϵ|λ|(n − η)

+ rλ) (∫_{(r+λ)/\sqrt{2}}^{∞} dμe^{−μ^{2}} + exp (−\frac{1}{2}(r + λ)^{2}))

for any η > 0. The last inequality follows from Theorem 2.2 (i). Futhermore,

∫_{R/ϵ}^{∞} dr(1 + r^{2}) ∫_{−∞}^{∞} dλ(1 + λ^{2}) exp(−ϵλα − ϵ|λ|(n − η) + rλ)

× exp(−\frac{1}{2}(r + λ)^{2})

≦ C(α, T, η) exp (−\frac{R^{2}}{4ϵ^{2}} ) ,

and

∫_{R/ϵ}^{∞} dr(1 + r^{2}) ∫_{0}^{∞} dλ(1 + λ^{2}) exp(−ϵλα − ϵ|λ|(n − η) + rλ)

× ∫_{(r+λ)/\sqrt{2}}^{∞} dμe^{−μ^{2}}

≦ C(α, T, η) exp (−\frac{R^{2}}{4ϵ^{2}} ) .

For 0 < η < R/T^{2} and α ≦ n, we have

∫_{R/ϵ}^{∞} dr(1 + r^{2}) ∫_{−∞}^{0} dλ(1 + λ^{2}) exp(−ϵλα − ϵ|λ|(n − η) + rλ)

× ∫_{(r+λ)/\sqrt{2}}^{∞} dμe^{−μ^{2}}

= ∫_{R/ϵ}^{∞} dr(1 + r^{2}) exp (−\frac{r^{2}}{2} ) ∫_{0}^{∞} dμ exp(−μ^{2} − \sqrt{2}rμ)

× ∫_{0}^{∞} dλ(1 + λ^{2}) exp (−\frac{λ^{2}}{2} + λ(\sqrt{2}μ + ϵ(α − n + η)))

≦ C(α, T, η) ∫_{R/ϵ}^{∞} dr(1 + r^{2}) exp (−\frac{r^{2}}{2} ) ∫_{0}^{∞} dμ exp (\sqrt{2}μ(ϵ(α − n + η) − r))

+ \frac{ϵ^{2}}{2} (α − n + η)^{2})

≦ C(α, T, η) ∫_{R/ϵ}^{∞} dr(1 + r^{2}) exp (−\frac{r^{2}}{2} ) ∫_{0}^{∞} dμ exp (−\sqrt{2}ϵμ(\frac{R}{T^{2}} − η))

≦ C(α, T, η, R)\frac{1}{ϵ} ∫_{R/ϵ}^{∞} dr(1 + r^{2}) exp (−\frac{r^{2}}{2} )

≦ C(α, T, η, R) exp (−\frac{R^{2}}{4ϵ^{2}} ) .

Hence if α ≦ n, we have

(5.11)  \begin{matrix}∫_{ℝ^{2n+1}} dxdu ∫_{R}^{∞} dr∥q_{ϵ^{2}}^{α}(x, u, r)∥ \\ ≦ C(α, T, R) exp (−\frac{R^{2}}{4ϵ^{2}} ) \end{matrix}

for any ϵ ∈ [0, T]. This lead to (i) for i = 3.

(ii): To prove (ii) for i = 1 and 2, it is enough to show that

(5.12)  \begin{matrix}∫_{D} p_{t}^{α}((x, u)^{−1}(x′, u′))\frac{1}{\sqrt{2πt}} exp (−\frac{(r − r′)^{2}}{2t} ) \\ × f(x′, u′, r′)dx′du′dr′, \\ ⟶ f(x, u, r), \end{matrix}

and

(5.13)  \begin{matrix}∫_{D} p_{t}^{α}((x, u)^{−1}(x′, u′))\frac{1}{\sqrt{2πt}} exp (−\frac{(r + r′)^{2}}{2t} ) \\ × f(x′, u′, r′)dx′du′dr′ \\ ⟶ 0 \end{matrix}

in L_{2}^{p,q}(D) and uniformly as t → 0. We set

p_{t}^{0,α}(x, u) := ∫_{−∞}^{∞} dλf_{t}^{0,α}(x, u, λ) exp (−\frac{t}{2}λ^{2}) ,
p_{t}^{−,α}(x, u) := p_{t}^{α}(x, u) − p_{t}^{0,α}(x, u)I.

As in (5.5), we have

(5.14)  \begin{matrix}∫_{D} ∥ ∫_{D} p_{t}^{−,α}((x, u)^{−1}(x′u′))\frac{1}{\sqrt{2πt}} exp (−\frac{(r ± r′)^{2}}{2t} ) \\ × f(x′u′r′)dx′du′dr′∥^{2}dxdudr \\ ≦ C (∫_{ℝ^{2n+1}} ∥p_{t}^{−,α}(x, u)∥dxdu)^{2} ∥f∥_{2}^{2}. \end{matrix}

On the other hand, it is easy to see that (5.15)

sup_{(x,u,r)∈D} ∥ ∫_{D} p_{t}^{−,α}((x, u)^{−1}(x′u′))\frac{1}{\sqrt{2πt}} exp (−\frac{(r ± r′)^{2}}{2t} )

× f(x′, u′, r′)dx′du′dr′∥ ≦ ∫_{ℝ^{2n+1}} ∥p_{t}^{−,α}(x, u)∥dxdu∥f∥_{∞}.

If we set t = ϵ^{2}, as in (5.8), we have

(5.16)  \begin{matrix}∫_{ℝ^{2n+1}} ∥p_{ϵ^{2}}^{−,α}(x, u)∥dxdu \\ ≦ C(T, α) ∑_{k=0}^{2} sup_{λ∈ℝ,x∈ℝ^{2n}}(1 + |x|^{2})^{2n}∥E[S_{0}(1)^{k}e^{−iλS_{0}(1)} \\ × (M^{ϵ}(1) − I)δ_{x}(\frac{x(1)}{\sqrt{2}})]∥ \\ ≦ C(T, α)ϵ \end{matrix}

for 0 < ϵ ≦ T. Here we used Theorem 2.2 (ii). Hence

(5.17)  \begin{matrix}∫_{D} p_{t}^{−,α}((x, u)^{−1}(x′, u′))\frac{1}{\sqrt{2πt}} exp (−\frac{(r ± r′)^{2}}{2t} ) \\ × f(x′, u′, r′)dx′du′dr′ \\ ⟶ 0 \end{matrix}

in L_{2}^{p,q}(D) and uniformly as t → 0. The following facts are proved in Stanton [10]:

(5.18)  \begin{matrix}∫_{D} p_{t}^{0,α}((x, u)^{−1}(x′, u′))\frac{1}{\sqrt{2πt}} exp (−\frac{(r − r′)^{2}}{2t} ) \\ × f(x′, u′, r′)dx′du′dr′ \\ ⟶ f(x, u, r), \end{matrix}

and

(5.19)  \begin{matrix}∫_{D} p_{t}^{0,α}((x, u)^{−1}(x′, u′))\frac{1}{\sqrt{2πt}} exp (−\frac{(r + r′)^{2}}{2t} ) \\ × f(x′, u′, r′)dx′du′dr′ \\ ⟶ 0 \end{matrix}

in L_{2}^{p,q}(D) and uniformly as t → 0. From (5.16-18), we conclude (5.12) and (5.13). Next we will consider H_{t}^{(3)}f. If R is the distance from supp f to bD,

(5.20)  \begin{matrix}sup_{(x,u,r)∈D} ∥(H_{t}^{(3)}f)(x, u, r)∥ \\ ≦ (∫_{ℝ^{2n+1}} dxdu ∫_{R}^{∞} dr∥q_{t}^{(n−2q)}(x, u, r)∥) sup_{(x,u,r)∈D} ∥Qf(x, u, r)∥. \end{matrix}

Thus, by (5.9) and (5.20), to prove (ii) for i = 3, it is enough to show that

(5.21)  ∫_{ℝ^{2n+1}} dxdu ∫_{R}^{∞} dr∥q_{t}^{α}(x, u, r)∥ ⟶ 0 as t → 0

for α ≦ n. We set t = ϵ^{2} and calculate like (5.10). Then for α ≦ n and 0 < ϵ ≦ T, we have

(5.22)  ∫_{ℝ^{2n+1}} dxdu ∫_{R}^{∞} dr∥q_{ϵ^{2}}^{α}(x, u, r)∥ ≦ C(α, T, R) exp (−\frac{R^{2}}{4ϵ^{2}} ) ,

which proves (5.21).

(iii)(iv): Since f has compact support, we can easily prove (iii) and (iv) by using Proposition 5.1.

(v): From (5.1), we easily see that

(5.23)  h_{t}^{(i)}((x, u, 0), (x′, u′, r′)) = 0.

On the other hand, (5.24)

(\frac{∂}{∂r} + i\frac{∂}{∂u}) [\frac{1}{\sqrt{2πt}}\{ exp (−\frac{(r − r′)^{2}}{2t} ) + exp (−\frac{(r + r′)^{2}}{2t} ) \}

× ∫_{−∞}^{∞} dλ · f_{t}^{α}(x, u, r) exp (−\frac{t}{2}λ^{2})

− \frac{2}{\sqrt{π}} ∫_{−∞}^{∞} dλ · λf_{t}^{α}(x, u, λ)e^{(r+r′)λ}

× ∫_{\sqrt{t/2}((r+r′)/t+λ)}^{∞} dμe^{−μ^{2}}] ↾_{r=0} = 0

Hence we have

(5.25)  (\frac{∂}{∂r} − i\frac{∂}{∂u}) (h_{t}^{(2)} + h_{t}^{(3)})((x, u, 0), (x′, u′, r′)) = 0.

(vi): By Feynman-Kac formula, we see that

(5.26)  \frac{∂}{∂t}f_{t}^{α}((x, u)^{−1}(x′, u′), λ) = −□_{λ}^{α}f_{λ}^{α}((x, u)^{−1}(x′, u′), λ)

where

□_{λ}^{α} = − \frac{1}{4} ∑_{j=1}^{n} \{(\frac{∂}{∂x^{j}} − 2iλy^{j})^{2} + (\frac{∂}{∂y^{j}} + 2iλx^{j})^{2}\}

− ∑_{j=1}^{n} \frac{A_{j}}{\sqrt{2}} (\frac{∂}{∂x^{j}} − 2iλy^{j}) − ∑_{j=1}^{n} \frac{A_{n+j}}{\sqrt{2}} (\frac{∂}{∂y^{j}} + 2iλx^{j})

+ B + λα.

By using the fact that f_{t}^{α}((x, u), λ) has good regularity, we have

(5.27)  \begin{matrix}\frac{∂}{∂t}h_{t}^{(i)}((x, u, r), (x′, u′, r′)) \\ = −□h_{t}^{(i)}((x, u, r), (x′, u′, r′)), t > 0, (x, u, r) ∈ D̄, \end{matrix}

for each (x′, u′, r′) ∈ D̄ and i = 1, 2, 3 ((x′, u′, r′) ∈ D for q = 0 and i = 3). (vii): By using Markovian property of the Brownian motion, we can prove that

(5.28)  \begin{matrix}□_{λ}^{α} ∫_{H_{n}} f_{t}^{α}((x, u)^{−1}(x′, u′), λ)g(x′, u′)dx′du′ \\ = ∫_{H_{n}} f_{t}^{α}((x, u)^{−1}(x′, u′), λ)□_{λ}^{α}g(x′, u′)dx′du′ \end{matrix}

for any rapidly decreasing section on H_{n} to Λ^{p,q}(T^{∗}D) ↾_{H_{n}} where □_{λ}^{α} is that of (5.26). Then we can easily prove (vii).

Finally we prove that H_{t}^{(3)} is a bounded operator on L_{2}^{p,0}(D). By the above (i-vii) and the proof of Proposition 2.1, we see that

(5.29)  H_{t}f = e^{−t□}f for f ∈ Λ_{0}^{p,q}(D̄).

For any r_{0} > 0, since both H_{t} and e^{−t□} are bounded operators from L_{2}^{p,0}(D) to L_{2}^{p,0}(D_{r_{0}}), we have

(H_{t}f)(X) = (e^{−t□}f)(X) a.e. X ∈ D_{r_{0}}

for any f ∈ L_{2}^{p,q}(D). Hence we have

(5.30)  H_{t}f = e^{−t□}f for any f ∈ L_{2}^{p,q}(D).

Since e^{−t□} is a bounded operator on L_{2}^{p,0}(D), H_{t} is also a bounded operator on L_{2}^{p,0}(D).□

By Proposition 5.1 and Theorem 5.1, we see that the integral operator involved by the kernel h_{t}(X, Y) satisfies the conditions of a fundamental solution in Section 2. Hence we conclude Theorem 2.1.

6. Short time asymptotic behavior

In this section, we examine the behaviour of h_{t} on the diagonal of D̄ × D̄ as t → 0 (cf. Stanton [11] §6, Beals-Stanton [2]). By (5.1), we have

(6.1) h_{t}((x, u, r), (x, u, r))

= p_{t}^{n−2q+2}(0)\frac{1}{\sqrt{2πt}}P − p_{t}^{n−2q+2}(0)\frac{1}{\sqrt{2πt}} exp (−\frac{2r^{2}}{t} ) P

+ p_{t}^{n−2q}(0)\frac{1}{\sqrt{2πt}}Q + p_{t}^{n−2q}(0)\frac{1}{\sqrt{2πt}} exp (−\frac{2r^{2}}{t} ) Q

+ q_{t}^{n−2q}(0, 2r)Q =: k_{t}^{n−2q+2}(0)P − k_{t}^{n−2q+2}(r)P

+ k_{t}^{n−2q}(0)Q + k_{t}^{n−2q}(r)Q + q_{t}^{n−2q}(0, 2r)Q

where

k_{t}^{α}(r) = p_{t}^{α}(0)\frac{1}{\sqrt{2πt}} exp (−\frac{2r^{2}}{t} ) .

We examine the behaviour of k_{t}^{α}(r) for α ∈ ℤ and q_{t}^{α}(0, 2r) for α ≦ n as t → 0. Proposition 6.1. For any α ∈ ℤ,

(6.2)  lim_{t→0} t^{n+1}k_{t}^{α}(0) = \frac{1}{2π^{n+1}} I.

Proof. k_{t}^{α}(0) is decomposed as follows:

(6.3)  \begin{aligned}k_{t}^{α}(0) & = p_{t}^{α}(0)\frac{1}{\sqrt{2πt}} \\ & = p_{t}^{0,α}(0)\frac{1}{\sqrt{2πt}}I + p_{t}^{−,α}(0)\frac{1}{\sqrt{2πt}}. \end{aligned}

As in Stanton [11] Proposition 6.1,

(6.4)  lim_{t→0} t^{n+1}p_{t}^{0,α}(0)\frac{1}{2πt} = \frac{1}{2π^{n+1}} .

For the second term of the right hand side of (6.3), we take t = ϵ^{2} and use scaling property of the Brownian motion and Theorem 2.2 (ii): for any ϵ ∈ (0, T],

(6.5)  \begin{matrix}∥ϵ^{2(n+1)}p_{ϵ^{2}}^{−,α}(0)\frac{1}{\sqrt{2πϵ^{2}}} ∥ \\ = ∥ϵ^{2} ∫_{−∞}^{∞} dλ\frac{1}{2π} exp (−ϵ^{2}λα − \frac{ϵ^{2}}{2} λ^{2}) \\ × E[e^{−iϵ^{2}λS_{0}(1)}(M^{ϵ}(1) − I)δ_{0} (\frac{x(1)}{\sqrt{2}}) ]∥ \\ ≦ C(T)ϵ^{2}. \end{matrix}

Hence we have

(6.6) lim_{t→0} t^{n+1}p_{t}^{−,α}(0)\frac{1}{\sqrt{2πt}} = 0.

By (6.4) and (6.6), we conclude (6.2).□

Next we consider the boundary correction term: Proposition 6.2. For any α ∈ ℤ, r > 0 and k ∈ ℤ,

(6.7) lim_{t→0} t^{k}k_{t}^{α}(r) = 0.

For any α ≦ n, r > 0 and k ∈ ℤ,

(6.8)  lim_{t→0} t^{k}q_{t}^{α}(0, 2r) = 0.

Proof. We easily see that

(6.9)  k_{t}^{α}(r) = k_{t}^{α}(0) exp (−\frac{2r^{2}}{t} )

and lim_{t→0} t^{k} exp(−2r^{2}/t) = 0. Thus we conclude (6.7) from (6.2). Since

e^{2rλ} ∫_{\sqrt{t/2}(2r/t+λ)}^{∞} dμe^{−μ^{2}} ≦ \{ \begin{matrix}exp(−\frac{2r^{2}}{t}) ∫_{\sqrt{t/2}λ}^{∞} dμe^{−μ^{2}}, \\ exp(−\frac{r^{2}}{t} + \sqrt{2}rλ) ∫_{\sqrt{t/2}(r/\sqrt{2}t+λ)}^{∞} dμe^{−μ^{2}}, \end{matrix}

(6.10)

∥q_{t}^{α}(0, 2r)∥ ≦ C\{ exp (−2r^{2}/t) ∫_{0}^{∞} dλ · λ∥f_{t}^{α}(0, λ)∥ ∫_{\sqrt{t/2}λ}^{∞} dμe^{−μ^{2}}

+ exp (−\frac{r^{2}}{t} ) ∫_{−∞}^{0} dλ · (−λ)∥f_{t}^{α}(0, λ)∥ exp(\sqrt{2}rλ)

× ∫_{\sqrt{t/2}(r/\sqrt{2}t+λ)}^{∞} dμe^{−μ^{2}}\}

≦ C′ exp(−\frac{r^{2}}{t})\{ ∫_{0}^{∞} dλ · λ∥f_{t}^{α}(0, λ)∥ exp(−\frac{tλ^{2}}{4} )

+ ∫_{−∞}^{0} dλ · (−λ)∥f_{t}^{α}(0, λ)∥e^{\sqrt{2}rλ}\}.

As in (5.15), we take t = ϵ^{2}(≦ T^{2}) and use scaling property of the Brownian motion and Theorem 2.2 (i): for any ϵ ∈ (0, T],

(6.11) \begin{matrix}∫_{0}^{∞} dλ · λ∥f_{ϵ^{2}}^{α}(0, λ)∥ exp(−\frac{ϵ^{2}λ^{2}}{4} ) \\ = Cϵ^{−2n} ∫_{0}^{∞} dλ · λe^{−ϵ^{2}λα}∥E[e^{−iϵ^{2}λS_{0}(1)}M^{ϵ}(1)δ_{0} (\frac{x(1)}{\sqrt{2}}) ]∥ exp(−\frac{ϵ^{2}λ^{2}}{4} ) \\ ≦ C(T, α)ϵ^{−2(n+1)}. \end{matrix}

For 0 < η < \sqrt{2}r/T^{2} and α ≦ n,

(6.12) \begin{matrix}∫_{−∞}^{0} dλ · (−λ)∥f_{ϵ^{2}}^{α}(0, λ)∥e^{\sqrt{2}rλ} \\ = Cϵ^{−2n} ∫_{−∞}^{0} dλ · (−λ)e^{−ϵ^{2}λα}∥E[e^{−iϵ^{2}λS_{0}(1)}M^{ϵ}(1)δ_{0} (\frac{x(1)}{\sqrt{2}}) ]∥e^{\sqrt{2}rλ} \\ ≦ C(α, T, η)ϵ^{−2(n+2)}. \end{matrix}

Hence, if α ≦ n, for any k ∈ ℤ,

(6.13)  \begin{matrix}∥t^{k}q_{t}^{α}(0, 2r)∥ \\ ≦ C(T, α) exp (−\frac{r^{2}}{t} ) t^{−(n+2)+k} ⟶ 0 as t → 0. \end{matrix}

□

Next result suggest the short time asymptotic behaviour of the trace of the heat semigroup by Beals and Stanton [2]:

Proposition 6.3. (i) For any α ∈ ℤ,

(6.14) lim_{t→0} t^{n+ \frac{1}{2}} ∫_{0}^{∞} drk_{t}^{α}(r) = \frac{1}{4\sqrt{2}π^{n+ \frac{1}{2}}} I.

(ii) For any α < n, (6.15)

lim_{t→0} t^{n+1} ∫_{0}^{∞} drq_{t}^{α}(0, 2r)

= \frac{1}{2π^{n+1}} ∫_{0}^{∞} dτ (\frac{τ}{sinh τ} )^{n} e^{ατ} I.

Proof. (i) follows from (6.2) and (6.9). We consider (ii). q_{t}^{α}(0, 2r) is decomposed as follows:

(6.16)  q_{t}^{α}(0, 2r) = q_{t}^{0,α}(0, 2r)I + q_{t}^{−,α}(0, 2r).

As is shown in Stanton [11] Theorem 2.2 (i),

(6.17) lim_{t→0} t^{n+1} ∫_{0}^{∞} drq_{t}^{0,α}(0, 2r) = \frac{1}{2π^{n+1}} ∫_{0}^{∞} dτ (\frac{τ}{sinh τ} )^{n} e^{ατ} .

On the other hand,

(6.18)

∥ ∫_{0}^{∞} drq_{t}^{−,α}(0, 2r)∥
≦ C ∫_{−∞}^{∞} dλ|λ|∥f_{t}^{−,α}(0, λ)∥

× ∫_{\sqrt{t/2}λ}^{∞} dμe^{−μ^{2}} ∫_{0}^{\sqrt{t/2}μ−tλ/2} dre^{2rλ}

= C ∫_{−∞}^{∞} dλ∥f_{t}^{−,α}(0, λ)∥

× |\frac{\sqrt{π}}{2} exp(−\frac{t}{2}λ^{2}) − ∫_{\sqrt{t/2}λ}^{∞} dμe^{−μ^{2}} |

≦ C′\{ ∫_{0}^{∞} dλ · exp (−\frac{t}{4}λ^{2}) ∥f_{t}^{−,α}(0, λ)∥

+ ∫_{−∞}^{0} dλ ∥f_{t}^{−,α}(0, λ)∥}.

Now we calculate as in (6.11) and (6.12) by using Theorem 2.2 (ii): for ϵ ∈ [0, T],

(6.19)  ∫_{0}^{∞} dλ · exp (−\frac{ϵ^{2}}{4} λ^{2}) ∥f_{ϵ^{2}}^{−α}(0, λ)∥ ≦ C(T)ϵ^{−2n}

and

(6.20)  ∫_{−∞}^{0} dλ ∥f_{ϵ^{2}}^{−,α}(0, λ)∥ ≦ C(T)ϵ^{−2n−1}.

Hence we have

(6.21) ∥t^{n+1} ∫_{0}^{∞} drq_{t}^{−,α}(0, 2r)∥ ≦ C(T)\sqrt{t} ⟶ 0 as t → 0.

From (6.17) and (6.21), we conclude (6.15).□

7. The analogue of the Siegel domain with a nondegenerate indefinite Levi form

For 0 ≦ κ ≦ n, let D_{κ} = {(z, w) ∈ ℂ^{n} × ℂ : Im w > ∑_{j=1}^{n} ϵ_{j}|z^{j}|^{2}} where ϵ_{j} = 1 for 1 ≦ j ≦ κ and ϵ_{j} = −1 for κ < j ≦ n. This is an analogue of the Siegel domain D with a nondegenerate indefinite Levi form having κ positive and n − κ negative eigenvalues. In [10], Stanton states that her results also apply to this domain in the (0, q)-form case. In the general (p, q)-form case, we need a few remarks to apply our methods (cf. Remark 7.1 (i) below). Our main theorem is Theorem 7.2 bellow.

We consider the Hermitian metric for which {Z_{1}, Z_{2}, ⋯, Z_{n+1}} is an orthonormal basis of T^{(1,0)}(D_{κ}), where

(7.1) Z_{j} = \frac{∂}{∂z^{j}} + 2iϵ_{j}z̄^{j} \frac{∂}{∂w} , j = 1, 2, ⋯, n, Z_{n+1} = i\sqrt{2}\frac{∂}{∂w} .

The dual basis is given by

(7.2) ω^{j} = dz^{j}, ω^{n+1} = \frac{1}{i\sqrt{2}}(dw − 2i ∑ ϵ_{j}z̄^{j}dz^{j}).

Then ¯∂ and ¯∂^{∗} can be represented as follows:

(7.3)

¯∂ = ∑_{j=1}^{n+1} ext(ω̄^{j})Z̄_{−}_{j} + \sqrt{2} ∑_{j=1}^{n} ϵ_{j}ext(ω^{j})ext(ω̄^{j})int(ω̄^{n+1}),
¯∂^{∗} = − ∑_{j=1}^{n+1} int(ω^{j})Z_{−}_{j} + \sqrt{2} ∑_{j=1}^{n} ϵ_{j}ext(ω^{n+1})int(ω^{j})int(ω̄^{j}),

on S^{p,q}(D̄_{κ}).

We use as coordinates on D_{κ} (z, u, r) where u = Re w and r = Im w − ∑_{j=1}^{n} ϵ_{j}|z^{j}|^{2}. Then we can regard D_{κ} as product of the boundary bD_{κ} and ℝ^{+}. We identify bD_{κ} with the Heisenberg group H_{n}. The group law on H_{n} is

(7.4)  \begin{aligned}(z′, u′)(z, u) & = (z′ + z, u′ + u + 2 ∑_{j=1}^{n} ϵ_{j}(x^{′n+j}x^{j} − x^{′j}x^{n+j})) \\ & for (z′, u′), (z, u) ∈ H_{n}, \end{aligned}

where z^{j} = x^{j} + ix^{n+j}. The metric we gave is the product metric of an invariant metric on H_{n} and the standard metric on ℝ^{+}. In terms of these coordinates,

(7.5) Z_{j} = \frac{∂}{∂z^{j}} + iϵ_{j}z̄^{j} \frac{∂}{∂u}, j = 1, 2, ⋯, n,  Z_{n+1} = \frac{1}{\sqrt{2}} (\frac{∂}{∂r} + i\frac{∂}{∂u}) .

Then we have

Z_{j} = 1/2 (X_{j} − iX_{n+j})

where

(7.6) X_{j} = \frac{∂}{∂x^{j}} + 2ϵ_{j}x^{n+j} \frac{∂}{∂u},  X_{n+j} = \frac{∂}{∂x^{n+j}} − 2ϵ_{j}x^{j} \frac{∂}{∂u}, j = 1, 2, ⋯, n.

If we define the projections P and Q by

P = ext(ω̄^{n+1})int(ω^{n+1}) and Q = int(ω^{n+1})ext(ω̄^{n+1}),

the ¯∂-Neumann boundary condition can be rewritten as

(7.7)  P f ↾_{bD_{κ}}= Q (\frac{∂}{∂r} − i\frac{∂}{∂u}) f ↾_{bD_{κ}}= 0 for f ∈ S^{p,q}(D̄_{κ}).

The ¯∂-Laplacian □ is expressed on S^{p,q}(D̄_{κ}) as follows:

(7.8)  □ = −\frac{1}{4} ∑_{j=1}^{2n} X_{−}_{j}^{2} − \frac{1}{2} (\frac{∂^{2}}{∂u^{2}} + \frac{∂^{2}}{∂r^{2}} ) + iU \frac{∂}{∂u} − ∑_{j=1}^{2n} \frac{A_{j}}{\sqrt{2}}X_{−}_{j} − B

where

U = ∑_{j=1}^{n} ϵ_{j}[int(ω^{j}), ext(ω̄^{j})],
A_{j} = ϵ_{j}(ext(ω^{n+1})int(ω̄^{j}) − ext(ω^{j})int(ω̄^{n+1}), j = 1, 2, ⋯, n,
A_{n+j} = iϵ_{j}(ext(ω^{n+1})int(ω̄^{j}) + ext(ω^{j})int(ω̄^{n+1})), j = 1, 2, ⋯, n,
B = −2 ∑_{j,k=1}^{n} ϵ_{j}ϵ_{k}ext(ω^{j})ext(ω̄^{j})int(ω^{k})int(ω̄^{k})
− 2 ∑_{j=1}^{n}(int(ω^{j})ext(ω̄^{j}) − ext(ω^{j})int(ω̄^{j}))ext(ω^{n+1})int(ω̄^{n+1}).

Remark 7.1. (i) The coefficient U of ∂/∂u in (7.8) does not commute with A_{j} and B. This is the only difficult point of the problem in the domain D_{κ} (cf. Lemma 7.2 and Theorem 7.1).

(ii) As in Remark 2.2, □ acts diagonally on (0, q)-forms and generally preserves the orthogonal decomposition Λ^{p,q}(T^{∗}D) = Ran P ⊕ Ran Q.

Now we consider the heat equation (1.1) for D_{κ}. Let (W_{0}^{2(n+1)}, P) be a 2(n+1)- dimensional Wiener space as in Section 3. We consider the following diffusion process (X(t), U(t), R(t)):

(7.9)  \{ \begin{matrix}X^{j}(t) & = x^{j} + \frac{x^{j}(t)}{\sqrt{2}}, j = 1, 2, ⋯, 2n, \\ U(t) & = u + u(t) + S_{x}^{κ}(t) − iϕ(t), \\ R(t) & = r + B(t) + ϕ(t) \end{matrix}

where

S_{x}^{κ}(t) = ∑_{j=1}^{n} ϵ_{j} ∫_{0}^{t} \sqrt{2}(X^{n+j}(s) ∘ dx^{j}(s) − X^{j}(s) ∘ dx^{n+j}(s)),
B(t) = ∫_{0}^{t} sgn(r + r(s))dr(s)

and

ϕ(t) = lim_{η↘0} \frac{1}{2η} ∫_{0}^{t} I_{(−η,η)}(r + r(s))ds.

We define the End(Λ^{p,q}(T^{∗}D_{κ})) -valued process M(t, x(·), u(·)) by the solution of the SDE

(7.10)  \{ \begin{matrix}dM(t) & = M(t)dΞ(t), \\ M(0) & = I \end{matrix}

where

Ξ(t) = −iUu(t) + ∑_{j=1}^{2n} A_{j}x^{j}(t) + Bt.

Let K(t) be the End(Λ^{p,q}(T^{∗}D_{κ})) -valued process as in (3.5).

Then the heat kernel h_{t} is expected to be represented as follows

(7.12)  h_{t}(X, X′) = ∫_{−∞}^{∞} dλH_{t,λ}(X, X′)

where

(7.13)  \begin{matrix}H_{t,λ}((x, u, r), (x′, u′, r′)) \\ = \frac{1}{2π} e^{iλ(u′−u)}E^{x}[e^{−iλS_{x}^{κ}(t)}E^{u}[e^{−iλu(t)}M(t, x(·), u(·))]δ_{x′}(X(t))] \\ × E[e^{−λϕ(t)}K(t)|R(t) = r′]r_{t}(r, r′) \end{matrix}

where E^{u} is the expectation with respect to the 1-dimensional Brownian motion u(·) and E^{x} is the generalized expectation of the generalized Wiener functional with respect to the 2n-dimensional Brownian motion x(·).

For any ϵ > 0, m ∈ ℕ and λ ∈ ℝ, we define the End(Λ^{p,q}(T^{∗}D_{κ})) -valued processes m_{m}^{λ}(t) and M_{m}^{λ,ϵ}(t) by the solutions of

\{ \begin{matrix}dm_{m}^{λ}(t) & = m_{m}^{λ}(t)( − \frac{λ}{m}U)dt, \\ m_{m}^{λ}(0) & = I \end{matrix} (7.14)

and

\{ \begin{matrix}dM_{m}^{λ,ϵ}(t) & = M_{m}^{λ,ϵ}(t)dΞ_{m}^{λ,ϵ}(t), \\ M_{m}^{λ,ϵ}(0) & = I \end{matrix} (7.15)

respectively, where

Ξ_{m}^{λ,ϵ}(t) = ∑_{j=1}^{2n} \frac{ϵ}{\sqrt{m}}A_{j}x^{j}(t) + \frac{ϵ^{2}}{m}Bt − λ\frac{ϵ^{2}}{m}Ut.

When m = 1, we omit the subscript m: Ξ_{1}^{λ,ϵ}(t) =: Ξ^{λ,ϵ}(t), M_{1}^{λ,ϵ}(t) =: M^{λ,ϵ}(t), m_{1}^{λ}(t) =: m^{λ}(t). When m = ϵ = 1, we also omit the superscript ϵ: Ξ_{1}^{λ,1}(t) =: Ξ^{λ}(t), M_{1}^{λ,1}(t) =: M^{λ}(t).

Then H_{t,λ}(X, X′) is rewritten as follows:

Lemma 7.2. H_{t,λ}(X, X′) is expressed as follows:

(7.16)

H_{t,λ}((x, u, r), (x′, u′, r′))

= f_{t}((x, u)^{−1}(x′, u′), λ) exp (−\frac{t}{2}λ^{2}) e_{t}^{−}(r, r′)P

+ f_{t}((x, u)^{−1}(x′, u′), λ) exp (−\frac{t}{2}λ^{2}) e_{t}^{+}(r, r′)Q

− \frac{2}{\sqrt{π}} λf_{t}((x, u)^{−1}(x′, u′), λ)e^{rλ} ∫_{\sqrt{t/2}(r/t+λ)}^{∞} dμ e^{−μ^{2}}Q

where

f_{t}(x, u, λ) = \frac{1}{2π} e^{iλu}E[e^{−iλS_{0}^{κ}(t)}M^{λ}(t)δ_{x} (\frac{x(t)}{\sqrt{2}}) ].

Proof. Generally for a square integrable continuous σ(x(s), u(s) : s ≦ t)-adapted process Φ(·), it holds that

E^{u} [∫_{0}^{t} Φ(s)dx^{j}(s)] = ∫_{0}^{t} E^{u}[Φ(s)]dx^{j}(s), 1 ≦ j ≦ 2n.

Then, by the uniqueness of the solution of (7.15), we obtain

(7.17)  M^{λ}(t) = E^{u} [e^{−iλu(t)}M(t, x(·), u(·))] exp (\frac{t}{2}λ^{2}) .

Hence we have

(7.18)  \begin{matrix}E^{x}[e^{−λS_{x}^{κ}(t)}E^{u}[e^{−iλu(t)}M(t, x(·), u(·))]δ_{x′}(x(t))] \\ = exp (−iλ ∑_{j=1}^{n} 2ϵ_{j}(x^{j}x^{′j} − x^{j}x^{′n+j}) − \frac{t}{2}λ^{2}) \\ × E[e^{−iλS_{0}^{κ}(t)}M^{λ}(t)δ_{x′−x} (\frac{x(t)}{\sqrt{2}}) ]. \end{matrix}

By combining (7.13),(7.18) and (3.14), we conclude (7.16). □

We first note the following theorem (cf. Theorem 2.2).

Theorem 7.1. We take α, β ∈ ℤ_{+}^{2n}, k ∈ ℤ_{+} and ϵ_{0} > 0 arbitrary.

(i) If q ≠ n − κ, (7.19)

\overline{lim}_{λ→−∞} \frac{1}{|λ|} log sup_{\begin{matrix}x∈ℝ^{2n} \\ 0<ϵ≦ϵ_{0} \end{matrix}} ∥x^{β}∂_{x}^{α} \frac{∂^{k}}{∂λ^{k}} E[e^{−iλS_{0}^{κ}(1)}

× M^{λ/ϵ^{2},ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]Q∥

≦ −2.

Generally (7.20)

\overline{lim}_{λ→±∞} \frac{1}{|λ|} log sup_{\begin{matrix}x∈ℝ^{2n} \\ 0<ϵ≦ϵ_{0} \end{matrix}} ∥x^{β}∂_{x}^{α} \frac{∂^{k}}{∂λ^{k}} E[e^{−iλS_{0}^{κ}(1)}

× M^{λ/ϵ^{2},ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ]∥

≦ 0.

(ii) If q ≠ n − κ, (7.21)

\overline{lim}_{λ→−∞} \frac{1}{|λ|} log sup_{\begin{matrix}x∈ℝ^{2n} \\ 0<ϵ≦ϵ_{0} \end{matrix}} ∥x^{β}∂_{x}^{α} \frac{∂^{k}}{∂λ^{k}} E[e^{−iλS_{0}^{κ}(1)}

× (M^{λ/ϵ^{2},ϵ}(1) − m^{λ}(1))δ_{x} (\frac{x(1)}{\sqrt{2}}) ]Q/ϵ∥

≦ −2.

Generally, (7.22)

\overline{lim}_{λ→±∞} \frac{1}{|λ|} log sup_{\begin{matrix}x∈ℝ^{2n} \\ 0<ϵ≦ϵ_{0} \end{matrix}} ∥x^{β}∂_{x}^{α} \frac{∂^{k}}{∂λ^{k}} E[e^{−iλS_{0}^{κ}(1)}

× (M^{λ/ϵ^{2},ϵ}(1) − m^{λ}(1))δ_{x} (\frac{x(1)}{\sqrt{2}}) ]/ϵ∥

≦ 0.

Corollary. We take α, β ∈ ℤ_{+}^{2n} and ϵ_{0} arbitrary. If q ≠ n − k, (7.23)

\overline{lim}_{λ→−∞} sup_{t_{0}≦t≦t_{1}} \frac{1}{t|λ|} log sup_{x∈ℝ^{2n}} ∥x^{β}∂_{x}^{α} \frac{∂^{k}}{∂λ^{k}} E[e^{−iλS_{0}^{κ}(t)}

× M^{λ}(t)δ_{x} (\frac{x(t)}{\sqrt{2}}) ]Q∥

≦ −2.

Generally, (7.24)

\overline{lim}_{λ→±∞} sup_{t_{0}≦t≦t_{1}} \frac{1}{t|λ|} log sup_{x∈ℝ^{2n}} ∥x^{β}∂_{x}^{α} \frac{∂^{k}}{∂λ^{k}} E[e^{−iλS_{0}^{κ}(t)}

× M^{λ}(t)δ_{x} (\frac{x(t)}{\sqrt{2}}) ]∥

≦ 0.

The proof of Theorem 7.1 proceeds just like the proof of Theorem 2.2, using Lemmas 7.3 and 7.4 below: we omit the details. Lemma 7.3. For any m ∈ ℕ and ϵ > 0,

(7.25)  \begin{matrix}E[e^{iλS_{0}^{κ}(1)}M^{λ/ϵ^{2},ϵ}(1)δ_{x} (\frac{x(1)}{\sqrt{2}}) ] \\ = ∫_{ℝ^{2n}} dx_{1}E[e^{iλS_{0}^{κ}(1)/m}M_{m}^{λ/ϵ^{2},ϵ}(1)δ_{x_{1}} (\frac{x(1)}{\sqrt{2m}}) ] \\ × ∫_{ℝ^{2n}} dx_{2}E[e^{iλS_{x_{1}}^{κ} (1)/m}M_{m}^{λ/ϵ^{2},ϵ}(1)δ_{x_{2}} (x_{1} + \frac{x(1)}{\sqrt{2m}}) ] \\ × ⋯ × ∫_{ℝ^{2n}} dx_{m−1}E[e^{iλS_{x_{m−2}} (1)/m}M_{m}^{λ/ϵ^{2},ϵ}(1) \\ × δ_{x_{m−1}} (x_{m−2} + \frac{x(1)}{\sqrt{2m}}) ] \\ × E[e^{iλS_{x_{m−1}} (1)/m}M_{m}^{λ/ϵ^{2},ϵ}(1)δ_{x} (x_{m−1} + \frac{x(1)}{\sqrt{2m}}) ]. \end{matrix}

Lemma 7.4. (i) For any p > 1, s ∈ ℤ_{+}, h ∈ ℤ_{+} and λ > 0,

(7.26)  \begin{matrix}∥\frac{∂^{h}}{∂λ^{h}} M_{m}^{λ/ϵ^{2},ϵ}(1)∥_{p,s} \\ ≦ C(p, s, h)m^{−h} exp \{C(p, s)( (\frac{ϵ^{2}}{m})^{p/2} + (\frac{|λ|}{m}) )\} \end{matrix}

and

(7.27)  \begin{matrix}∥\frac{∂^{h}}{∂λ^{h}} (M_{m}^{λ/ϵ^{2},ϵ}(1) − m_{m}^{λ}(1))∥_{p,s} \\ ≦ C(p, s, h)m^{−h− \frac{1}{2}} exp \{C(p, s, h)( (\frac{ϵ^{2}}{m})^{p/2} + (\frac{|λ|}{m})^{p} + \frac{|λ|}{m})\}. \end{matrix}

(ii) If q ≠ n − κ,

(7.28)  ∥\frac{∂^{h}}{∂λ^{h}} m_{m}^{λ}(1)Q∥ ≦ Cm^{−h} exp \{\frac{|λ|(n − 2)}{m}\}

for any λ ≦ 0. Generally,

(7.29)  ∥\frac{∂^{h}}{∂λ^{h}} m_{m}^{λ}(1)∥ ≦ Cm^{−h} exp \{\frac{|λ|n}{m}\} .

By using Theorem 7.1, we can prove the following theorem: Theorem 7.2. The unique fundamental solution H_{t} of the heat equation for the ¯∂-Neumann problem on (p, q)-forms on D has the following smooth kernel:

(7.30)  \begin{matrix}h_{t}((x, u, r).(x′, u′, r′)) \\ = p_{t}((x, u)^{−1}(x′, u′))e_{t}^{−}(r, r′)P \\ + p_{t}((x, u)^{−1}(x′, u′))e_{t}^{+}(r, r′)Q \\ + q_{t}((x.u)^{−1}(x′.u′), r + r′)Q \end{matrix}

where

p_{t}(x, u) = ∫_{−∞}^{∞} dλ f_{t}(x, u, λ) exp (−\frac{t}{2}λ^{2}) ,
q_{t}((x, u), r) = −\frac{2}{\sqrt{π}} ∫_{−∞}^{∞} dλ · λf_{t}(x, u, λ)e^{rλ} ∫_{\sqrt{t/2}(r/t+λ)}^{∞} dμ e^{−μ^{2}},

and f_{t}(x, u, λ) is defined in (7.16).

As in Section 5, we can state properties of the heat kernel more precisely. We set

(7.31)  \begin{aligned}h_{t}^{(1)}((x, u, r), (x′, u′, r′)) & = p_{t}((x, u)^{−1}(x′, u′))e_{t}^{−}(r, r′)P, \\ h_{t}^{(2)}((x, u, r), (x′, u′, r′)) & = p_{t}((x, u)^{−1}(x′, u′))e_{t}^{+}(r, r′)Q, \\ h_{t}^{(3)}((x, u, r), (x′, u′, r′)) & = q_{t}((x.u)^{−1}(x′.u′), r + r′)Q. \end{aligned}

Then we have the following: Proposition 7.1. If i ≠ 3 or q ≠ n − κ (resp. i = 3 and q = n − κ), h_{t}^{(i)}(X, X′) is the restriction to D̄_{κ} of a rapidly decreasing function of X for each fixed (t, X′) ∈ (0, ∞) × D̄_{κ} (resp. (0, ∞) × D_{κ}) and the restriction to D̄_{κ} of a rapidly decreasing function of X′ for each fixed (t, X) ∈ (0, ∞) × D̄_{κ} (resp. (0, ∞) × D_{κ}) uniformly with respect to t ∈ [t_{0}, t_{1}] for any 0 < t_{0} < t_{1}. Moreover h_{t}^{(i)}(X, X′) is a smooth function of (t, X, X′) ∈ (0, ∞) × D_{κ} × D_{κ}.

For any f ∈ L_{2}^{p,q}(D_{κ}), we define

(7.32)  (H_{t}^{(i)}f)(X) := ∫_{D_{κ}} h_{t}^{(i)}(X, Y)f(Y)dY,  X ∈ D_{κ}, i = 1, 2, 3.

For this H_{t}^{(i)}, we have the following theorem: Theorem 7.3. H_{t}^{(i)}, i = 1, 2, 3, are bounded operators on L_{2}^{p,q}(D_{κ}). Furthermore, for fixed f ∈ Λ_{0}^{p,q}(D_{κ}), the followings hold:

(i) for T > 0, there is a constant C depending only on f and T such that for t ∈ (0, T], ∥H_{t}^{(i)}f∥ ≦ C, i = 1, 2, 3;

(ii) H_{t}^{(1)}f ⟶ P f, H_{t}^{(2)}f ⟶ Qf, H_{t}^{(3)}f ⟶ 0 in L_{2}^{p,q}(D_{κ}) and uniformly

as t ⟶ 0; (iii) H_{t}^{(i)}f, i = 1, 2, 3, are the restrictions of rapidly decreasing forms; (iv) H_{t}^{(i)}f, i = 1, 2, 3, are differentiable in t;

(v) H_{t}f := H_{t}^{(1)}f + H_{t}^{(2)}f + H_{t}^{(3)}f ∈ Dom □;

(vi) (∂/∂t + □) H_{t}f = 0;

(vii) □H_{t}f = H_{t}□f.

Finally we investigate the behavior of h_{t}(X, X′) on the diagonal of D̄_{κ} × D̄_{κ} as t → 0. As in (6.1), we have

(7.32)  \begin{matrix}h_{t}((x, u, r), (x, u, r)) \\ = k_{t}(0) − k_{t}(r)P + k_{t}(r)Q + q_{t}(0, 2r)Q \end{matrix}

where

k_{t}(r) = p_{t}(0)\frac{1}{\sqrt{2πt}} exp (−\frac{2r^{2}}{t} ) .

Let L be the Levi form on bD_{κ} and L^{q} be its extension to Λ^{q}(T^{(1,0)}D_{κ} ∩ ℂT_{z}(bD_{κ})) (cf. Beals-Stanton [2] p.407):

L_{X}^{q}(Z_{j_{1}} ∧ ⋯ ∧ Z_{j_{q}}, Z_{k_{1}} ∧ ⋯ ∧ Z_{k_{q}}) = (∑_{l=1}^{q} 2ϵ_{j_{l}}) δ_{j_{1}k_{1}} ⋯ δ_{j_{q}k_{q}}

for 1 ≦ j_{1} < ⋯ < j_{q} ≦ n and 1 ≦ k_{1} < ⋯ < k_{q} ≦ n. Then we have the followings: Proposition 7.2. (i)

(7.33)  lim_{t→0} t^{n+1}k_{t}(0) = \frac{1}{2π^{n+1}} I.

(ii) For any r > 0 and k ∈ ℤ,

(7.34)  lim_{t→0} t^{k}k_{t}(r) = 0

and

(7.35)  lim_{t→0} t^{k}q_{t}(0, 2r)Q = 0.

(iii)

(7.36)  lim_{t→0} t^{n+1/2} ∫_{0}^{∞} drk_{t}(r) = \frac{1}{4\sqrt{2}π^{n+1/2}} I.

(iv) When q ≠ n − κ, we have

(7.37)

lim_{t→0} t^{n+1} ∫_{0}^{∞} dr · tr[q_{t}(0, 2r)Q]

= ( \begin{matrix}n + 1 \\ p \end{matrix} ) \frac{1}{2π^{n+1}} ∫_{0}^{∞} dτtr[exp(−τL^{q})]

× exp(−τtr^{−}L) det {τ|L|(1 − e^{−τ|L|})^{−1}} . where tr^{−}L is the sum of the negative parts of the eigenvalues of L and |L| := (L^{∗}L)^{1/2} = (L^{2})^{1/2}. Proof. The proofs of (i),(ii) and (iii) are similar to the corresponding results in Section 6. As in Proposition 6.3 (ii), we can prove the following: when q ≠ n − κ,

(7.38)

lim_{t→0} t^{n+1} ∫_{0}^{∞} drq_{t}(0, 2r)Q

= ∑_{0∨(q+κ−n)≦l≦κ∧q} \frac{1}{2π^{n+1}} ∫_{0}^{∞} dτ (\frac{τ}{sinh τ} )^{n} e^{−(n−2κ−2q+4l)τ} Q_{l}

where Q_{l} is the projection onto the space

\begin{matrix}span\{ω^{j_{1}} ∧ ⋯ ∧ ω^{j_{l}} ∧ ω̄^{k_{l+1}} ∧ ⋯ ∧ ω̄^{k_{q}} \\ |j_{1}, ⋯, j_{l} ≦ κ < k_{l+1}, ⋯, k_{q} ≦ n\}. \end{matrix}

By taking the trace of the both hand sides in the equality (7.38), we have (7.37). □

Acknowledgement

The author would like to express his gratitude to Professor N. Ikeda for many

helpful advices.

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Department of Mathematics Faculty of Science Osaka University Toyonaka, Osaka 560 Japan

Present address:

Department of Mathematics Faculty of Science Himeji Institute of Technology Shosha 2167, Himeji 671-22 Japan