The Meyer inequality for the Ornstein-Uhlenbeck operator in L^{1} and probabilistic proof of Stein’s L^{p} multiplier theorem

Ichiro Shigekawa (Kyoto)

ABSTRACT We discuss the Meyer inequality for the Ornstein-Uhlenbeck operator in L^{1}.

The Doob type inequality plays an important role. We also give a probabilistic proof of Stein’s multiplier theorem.

1. Introduction

The Sobolev space on the Wiener space is a fundamental framework of the Malliavin calculus. This space is defined in terms of the Ornstein-Uhlenbeck operator. We can introduce several kinds of norms in the Sobolev space. The equivalence of norms is important and Meyer [8] proved the equivalence of two kinds of norms; one is defined by means of gradient and the other by means of the square root of the Ornstein-Uhlenbeck operator. The equivalence is discussed in the L^{p} setting for 1 < p < ∞. In the paper, we shall discuss it in L^{1} setting. We remark that this kind of problem was already discussed by Bakry [1]. He dealt with this problem by using H^{1} martingale theory. Our approach is to connect his method to the Doob inequality. So our method is still probabilistic.

We only consider the Ornstein-Uhlenbeck operator, but a part of our argument is valid for any symmetric diffusion semigroups if we can use the stochastic calculus.

Let us start with giving a precise formulation. Let L be the Ornstein-Uhlenbeck operator on R^{d}:

(1.1)  L = ∑_{i=1}^{d} \frac{∂^{2}}{(∂x^{i})^{2}} − ∑_{i=1}^{d} x^{i} \frac{∂}{∂x^{i}} .

We assumed that our space is of finite dimension. Since every estimate is independent of the dimension, our result is still valid for an infinite dimensional space, i.e., an abstract Wiener space. Define the Gaussian measure μ by

(1.2)  μ(dx) = \frac{1}{(2π)^{d/2}} e^{−|x|^{2}/2}dx.

The Ornstein-Uhlenbeck semigroup {T_{t}} generated by L is given by

(1.3)  T_{t}f(x) = ∫_{R^{n}} f(e^{−t}x + \sqrt{1 − e^{−2t}}y)μ(dy).

It is easy to see that the semigroup is strongly continuous in L^{p}(R^{d}, μ) for p ∈ [1, ∞). When p = 2, the semigroup {T_{t}} is symmetric and thereby the generator is selfadjoint.

The Meyer equivalence, which consists of two inequalities, states that for any fixed p ∈ (1, ∞), it holds that

(1.4)  ∥\sqrt{1 − L}f∥_{p} < ∼ ∥∇f∥_{p} + ∥f∥_{p} < ∼ ∥\sqrt{1 − L}f∥_{p}, ∀f ∈ C_{0}^{∞}.

Here ∥ · ∥_{p} denotes the L^{p} norm and A < ∼ B means that there exists a constant c depending only on p such that A ≤ cB. In the sequel, we use this convention without mentioning.

We are interested in whether this inequality holds when p = 1. Our main result is as follows:

Theorem 1.1. For α ≥ 0, it holds that

(1.5)  ∥\sqrt{α − L}f∥_{1} < ∼ ∥∇f∥_{L log_{+} L(μ)} + \sqrt{α}∥f∥_{L^{1}}

and

(1.6)  ∥∇f∥_{1} < ∼ ∥\sqrt{α − L}f∥_{L log_{+} L(μ)}.

Here, L log_{+} L(μ) is the set of all functions such that

(1.7)  ∫_{R^{d}} |f| log_{+} |f|dμ < ∞.

We will give the definition of the norm ∥ · ∥_{L log_{+} L(μ)} later.

In the section 2, we prepare basic tools. Our method relies on the martingale theory; in particular, two inequalities, the Burkholder inequality and the Doob inequality, are used crucially. We also need H^{1} martingale theory. Using these, we give a proof of Theorem 1.1. In the section 3, we discuss a multiplier theorem in L^{p} setting (including the case p = 1). This theorem was proved by Stein [10], but we give a probabilistic proof.

2. The Meyer inequality

We first recall several inequalities for martingales which we need to give a proof. Let (Z_{t}) be a non-negative submartingale with the Doob-Meyer decomposition

Z_{t} = Z_{0} + M_{t} + A_{t}

where (M_{t}) is a martingale with M_{0} = 0 and (A_{t}) is a non-decreasing process with A_{0} = 0. We always assume that every process is cadlag (i.e., right continuous with left hand limit). The following inequality is a special case of the Langlart-Lépingle-Pratelli inequality [5]: for any 0 < p < ∞, there exists a constant c_{p} such that

(2.1)  E[A_{∞}^{p}] ≤ c_{p}E[sup_{t} Z_{t}^{p}].

Here we have to assume that (Z_{t}) is previsible when p < 1.

We also need the following inequality, which is a generalization of the Doob inequality: there exists a constant c > 0 such that

(2.2)  E[sup_{t} Z_{t}] ≤ c{E[Z_{∞} log_{+} Z_{∞}] + 1}.

where log_{+} x = (log x)∨0. We define the space L log_{+} L to be the set of all functions F satisfying

E[|F | log_{+} |F |] < ∞.

E[|F | log_{+} |F |] is not a norm but we can define a norm ∥ · ∥_{L log_{+} L} equivalent to this:

∥F ∥_{L log_{+} L} := inf{λ > 0; E[Φ(|F |/λ)] ≤ 1}

where Φ is defined by

Φ(x) = ∫_{0}^{x} log(1 + t)dt.

Hence (2.2) means that if Z_{∞} ∈ L log_{+} L, then sup_{t} Z_{t} ∈ L^{1}. We use this inequality in the following form.

(2.3)  E[sup_{t} Z_{t}] ≤ c∥Z_{∞}∥_{L log_{+} L}.

We introduce the Ornstein-Uhlenbeck precess (X_{t}) as a solution of the following stochastic differential equation (SDE) in R^{d}:

dX_{t} = \sqrt{2}dw_{t} − X_{t}dt

where {w_{t}} is a d-dimensional Brownian motion. Since we need a stationary process, we assume that the law of X_{0} is μ.

We introduce another Brownian motion (B_{t}) that is independent of (w_{t}). But our Brownian motion is different from usual one up to constant, i.e., the generator of (B_{t}) is \frac{d^{2}}{da^{2}} (not \frac{1}{2} \frac{d^{2}}{da^{2}} ). The starting point of (B_{t}) is denoted by N. E_{N} denotes the expectation with respect to this law. Let τ be the hitting time of (B_{t}) to the point 0. Then the following identity is fundamental:

(2.4)  E_{N}[∫_{0}^{τ} j(B_{s}, X_{s})ds] = ∫_{0}^{∞}(N ∧ a)da ∫_{R^{d}} j(a, x)μ(dx)

where j is a function on R × R^{d}.

We use stochastic analysis, in particular, the martingale theory. We introduce several martingales, which play an crucial role in our later argument. Take any α ≥ 0 and let \{Q_{t}^{(α)}\} be a semigroup generated by \sqrt{α − L}. \{Q_{t}^{(α)}\} is realized by means of a subordination of {T_{t}}:

Q_{t}^{(α)} = ∫_{0}^{∞} e^{−αs}T_{s}λ_{t}(ds)

where λ_{t} is a probability measure on [0, ∞) with the Laplace transform

∫_{0}^{∞} e^{−γs}λ_{t}(ds) = e^{−\sqrt{γ}t}, γ ≥ 0.

For a vector valued C^{∞} function f with compact support, we set u(x, a) = Q_{a}^{(α)}f(x). Then, u satisfies

\{ \begin{matrix}u(x, 0) = f(x), \\ L_{x}u(x, a) + \frac{∂^{2}u}{∂a^{2}}(x, a) − αu(x, a) = 0. \end{matrix}

Hence, by the Itô formula, we can see

(2.5) Q_{B_{t∧τ}}^{(α)} f(X_{t∧τ}) − Q_{B_{0}}^{(α)}f(X_{0}) = Mf^{→}(t) + Mf^{↑}(t) + α ∫_{0}^{t∧τ} Q_{B_{s}}^{(α)}f(X_{s})ds

where Mf^{→} and Mf^{↑} are martingales defined by

(2.6)  Mf^{→}(t) = ∫_{0}^{t∧τ} \frac{∂}{∂a}Q_{B_{s}}^{(α)}f(X_{s})dB_{s}

and

(2.7)  Mf^{↑}(t) = \sqrt{2} ∫_{0}^{t∧τ} ∇Q_{B_{s}}^{(α)}f(X_{s}) · dw_{s}

where · stands for the inner product in R^{d}. Note that |Q_{B_{t∧τ}}^{(α)} f(X_{t∧τ})|^{2} is a submartingale:

|Q_{B_{t∧τ}}^{(α)} f(X_{t∧τ})|^{2} (2.8)

= |Q_{B_{0}}^{(α)}f(X_{0})|^{2} + ∫_{0}^{t∧τ} Q_{B_{s}}^{(α)}f(X_{s}) · dMf^{→}(s) + ∫_{0}^{t∧τ} Q_{B_{s}}^{(α)}f(X_{s}) · dMf^{↑}(s)

+ α ∫_{0}^{t∧τ} |Q_{B_{s}}^{(α)}f(X_{s})|^{2}ds + ⟨Mf^{→}⟩_{t} + ⟨Mf^{↑}⟩_{t}.

Here ⟨ ⟩ denotes the quadratic variation of the martingale. In addition, we can show that |Q_{B_{t∧τ}}^{(α)} f(X_{t∧τ})| is a submartingale as well.

Proposition 2.1. |Q_{B_{t∧τ}}^{(α)} f(X_{t∧τ})| is a submartingale.

Proof: Set

u(a, x) = Q_{a}^{(α)}f(x).

Then, (2.5) can be written as

u(X_{t∧τ} , B_{t∧τ}) = u(B_{0}, X_{0}) + M_{f}(t) + α ∫_{0}^{t∧τ} u(X_{s}, B_{s})ds

where we set M_{f} = Mf^{→} + Mf^{↑}. Then

d|u|^{2} = 2(u, dM_{f}) + 2α|u|^{2}dt + d⟨M_{f}⟩

Here we omit the variables of u. Taking any ε > 0.

d(|u|^{2} + ε)^{1/2} = \frac{1}{2}(|u|^{2} + ε)^{−1/2}2u · dM_{f} + \frac{1}{2}(|u|^{2} + ε)^{−1/2}(2α|u|^{2}dt + d⟨M_{f}⟩_{t})

+ \frac{1}{2} \frac{1}{2}(−\frac{1}{2})(|u|^{2} + ε)^{−3/2}4⟨u · dM_{f}, u · dM_{f}⟩

= (|u|^{2} + ε)^{−1/2}u · dM_{f} + α(|u|^{2} + ε)^{−1/2}|u|^{2}dt

+ \frac{1}{2}(|u|^{2} + ε)^{−3/2}\{(|u|^{2} + ε)d⟨M_{f}⟩ − ⟨u · dM_{f}, u · dM_{f}⟩\}.

The third term in the right hand side is non-negative. To see this, we express u and M_{f} by components (recall that f is a vector valued function), say, u = (u_{1}, …, u_{n}) and M_{f} = ((M_{f}^{1}, …, M_{f}^{n}). Then

⟨u · dM_{f}, u · dM_{f}⟩

= ∑_{i,j}⟨u_{i}dM_{f}^{i}, u_{j}dM_{f}^{j}⟩

= ∑_{i,j} u_{i}u_{j}⟨dM_{f}^{i}, dM_{f}^{j}⟩

≤ ∑_{i,j} \sqrt{u_{i}^{2}⟨dM_{f}^{j}, dM_{f}^{j}⟩}\sqrt{u_{j}^{2}⟨dM_{f}^{i}, dM_{f}^{i}⟩} (∵ Kunita-Watanabe’s inequality)

≤ \sqrt{∑_{i,j} u_{i}^{2}⟨dM_{f}^{j}, dM_{f}^{j}⟩}\sqrt{∑_{i,j} u_{j}^{2}⟨dM_{f}^{i}, dM_{f}^{i}⟩} (∵ Schwarz’ inequality)

= |u|^{2}d⟨M_{f}⟩_{t}.

Here, the inequality above means that the difference is an increasing process. Now we can see that the third term is non-negative. This show that (Z_{t}) is a submartingale. □

Similarly, taking ∇f in place of f (so our function takes values in the Hilbert space H), it holds that

Q_{B_{t∧τ}}^{(α+1)}∇f(X_{t∧τ}) − Q_{B_{0}}^{(α+1)}∇f(X_{0})

= Nf^{→}(t) + Nf^{↑}(t) + (α + 1) ∫_{0}^{t∧τ} Q_{B_{s}}^{(α+1)}∇f(X_{s})ds

where

NDf^{→}(t) = ∫_{0}^{t∧τ} \frac{∂}{∂a}Q_{B_{s}}^{(α+1)}∇f(X_{s})dB_{s}

and

NDf^{↑}(t) = ∫_{0}^{t∧τ} ∇Q_{B_{s}}^{(α+1)}∇f(X_{s}) · dw_{s}.

On the other hand, let us recall the following commutation relation between ∇ and L:

∇L = L∇ − ∇.

This implies

∇Q_{a}^{(α)} = Q_{a}^{(α+1)}∇.

By noting this,

\frac{∂}{∂a}Q_{a}^{(α+1)}∇f(x) = Q_{a}^{(α+1)}\sqrt{α + 1 − L}∇f(x) = ∇Q_{a}^{(α)}\sqrt{α − L}f(x)

we have

(2.9)  ⟨NDf^{→}⟩_{t} = ∫_{0}^{t∧τ} |∇Q_{B_{s}}^{(α)}\sqrt{α − L}f(X_{s})|^{2}ds = ⟨M\sqrt{α−L}f^{↑}⟩_{t}.

Lemma 2.2. We have

(2.10)  ∥Mf^{→}∥_{BMO} ≤ ∥f∥_{∞},
(2.11)  ∥Mf^{↑}∥_{BMO} ≤ ∥f∥_{∞}.

Proof: We recall that

|Q_{B_{t∧τ}}^{(α)} f(X_{t∧τ})|^{2} = |Q_{B_{0}}^{(α)}f(X_{0})|^{2} + a martingale + A_{t}

where (A_{t}) is an increasing process with A_{0} = 0. By (2.8), we easily see that (A_{t} − ⟨Mf^{→}⟩_{t}) is increasing. Hence, for any stopping time T, it holds that

E[|Mf^{→}(∞) − Mf^{→}(T −)|^{2}|F_{T}] = E[|Mf^{→}(∞) − Mf^{→}(T)|^{2}|F_{T}]

= E[⟨Mf^{→}⟩_{∞} − ⟨Mf^{→}⟩_{T} |F_{T}]

≤ E[A_{∞} − A_{T} |F_{T}]

= E[|Q_{B_{τ}}^{(α)}f(X_{τ})|^{2} − |Q_{B_{T ∧τ}}^{(α)} f(X_{T ∧τ})|^{2}|F_{T}]

≤ E[|Q_{B_{τ}}^{(α)}f(X_{τ})|^{2}|F_{T}]

≤ E[|f(X_{τ})|^{2}|F_{T}]

≤ ∥f∥_{∞}^{2}.

Now, by the definition of BMO martingale (see, e.g., [4]), we have (2.10). (2.11) can be proved similarly. □

Lemma 2.3. We have

(2.12) lim_{N→∞} E_{N}[Mf^{→}(∞)^{2}] = \{ \begin{matrix}\frac{1}{2}∥f∥_{2}^{2}, & α > 0, \\ \frac{1}{2}∥f − J_{0}f∥_{2}^{2}, & α = 0, \end{matrix}
(2.13)  lim_{N→∞} E_{N}[Mf^{↑}(∞)^{2}] = \{ \begin{matrix}\frac{1}{2}∥f∥_{2}^{2} − \frac{1}{2}α⟨(α − L)^{−1}f, f⟩, & α > 0, \\ \frac{1}{2}∥f − J_{0}f∥_{2}^{2}, & α = 0. \end{matrix}

Proof: By (2.4), (2.6) and (2.7) we have

(2.14) E_{N}[Mf^{→}(∞)^{2}] = 2E[∫_{0}^{τ} \frac{∂u}{∂a}(B_{s}, X_{s})^{2}ds]

= 2 ∫_{0}^{∞}(N ∧ a)da ∫_{E} \frac{∂u}{∂a}(a, x)^{2}μ(dx)

= 2 ∫_{0}^{∞}(N ∧ a)da ∫_{E} \sqrt{α − L}u(a, x)^{2}μ(dx)

= 2 ∫_{0}^{∞}(N ∧ a)da ∫_{E}(α − L)u(a, x)u(a, x)μ(dx)

= 2 ∫_{0}^{∞}(N ∧ a)da ∫_{E}{αu(a, x)^{2} + |∇u(a, x)|^{2}}μ(dx)

= 2E_{N}[∫_{0}^{τ} {αu(B_{s}, X_{s})^{2} + |∇u(B_{s}, X_{s})|^{2}}ds]

= 2E_{N}[∫_{0}^{τ} αu(B_{s}, X_{s})^{2}ds] + E_{N}[Mf^{↑}(∞)^{2}].

Further, by letting N → ∞,

lim_{N→∞} E_{N}[Mf^{→}(∞)^{2}] = 2 ∫_{0}^{∞} ada ∫_{E} \sqrt{α − L}u(a, x)^{2}μ(dx)

= 2 ∫_{0}^{∞} ada ∫_{0}^{∞}(α + λ)e^{−2a\sqrt{α+λ}}d(E_{λ}f, f)

= 2 ∫_{0}^{∞}\{∫_{0}^{∞} a(α + λ)e^{−2a\sqrt{α+λ}}da\}d(E_{λ}f, f)

= \{ \begin{matrix}\frac{1}{2}∥f∥_{2}^{2}, & α > 0, \\ \frac{1}{2}∥f − J_{0}f∥_{2}^{2}, & α = 0. \end{matrix}

This proves (2.12).

As for Mf^{↑}(∞), we have
lim_{N→∞} E_{N}[∫_{0}^{τ} αQ_{B_{s}}^{(α)}f(X_{s})^{2}ds] = ∫_{0}^{∞} ada ∫_{0}^{∞} αe^{−2a\sqrt{α+λ}}d(E_{λ}f, f)

= ∫_{0}^{∞} \frac{α}{4(α + λ)}d(E_{λ}f, f)

= \frac{α}{4}((α − L)^{−1}f, f).

Combining this with (2.14), we easily have (2.13). □ Proposition 2.4. It holds that

(2.15)  ∥Df∥ < ∼ ∥\sqrt{α − L}f∥_{L log_{+} L(μ)}.

Proof: We note that the identity (2.12) holds for martingales taking values in a Hilbert space. Hence,

|⟨Df, G⟩| = lim_{N→∞} |E[(NDf^{→}(∞), NG^{→}(∞))_{H^{∗}}]|

< ∼ lim_{N→∞} ∥NDf^{→}∥_{H^{1}}∥NG^{→}∥_{BMO}

< ∼ lim_{N→∞} ∥NDf^{→}∥_{H^{1}}∥G∥_{∞} (∵ (2.10))

which implies

∥Df∥_{1} < ∼ lim_{N→∞} ∥NDf^{→}∥_{H^{1}}.

What remains is to estimate ∥NDf^{→}∥_{H^{1}}. Using (2.9), we have

∥NDf^{→}∥_{H^{1}} = E_{N}[sup_{t} |NDf^{→}(t)|]

< ∼ E_{N}[⟨NDf^{→}⟩_{∞}^{1/2}]

= E_{N}[⟨M\sqrt{α−L}f^{↑}⟩_{∞}^{1/2}]

< ∼ E_{N}[sup_{t}\{|Q_{B_{t∧τ}}^{(α)} \sqrt{α − L}f(X_{t∧τ})|^{2} − |Q_{B_{0}}^{(α)}\sqrt{α − L}f(X_{0})|^{2}\}^{1/2}]

≤ E_{N}[sup_{t} |Q_{B_{t∧τ}}^{(α)} \sqrt{α − L}f(X_{t∧τ})|]

≤ ∥Q_{B_{τ}}^{(α)}\sqrt{α − L}f(X_{τ})∥_{L log_{+} L(P_{N})} = ∥\sqrt{α − L}f(X_{τ})∥_{L log_{+} L(P_{N})} = ∥\sqrt{α − L}f∥_{L log_{+} L(μ)}.

Thus we have

∥Df∥_{1} < ∼ ∥\sqrt{α − L}f∥_{L log_{+} L(μ)}

which is equivalent to (2.15). □

Next we show the reversed inequality.

Proposition 2.5. It holds that

∥\sqrt{α − L}f∥_{1} < ∼ ∥∇f∥_{L log_{+} L(μ)} + \sqrt{α}∥f∥_{L^{1}}.

Proof: By the Itô formula,

Q_{B_{t∧τ}}^{(α+1)}∇f(X_{t∧τ}) − Q_{B_{0}}^{(α+1)}∇f(X_{0})

= ∫_{0}^{t∧τ} \frac{∂}{∂a}Q_{B_{s}}^{(α+1)}∇f(X_{s})dB_{s} + \sqrt{2} ∫_{0}^{t∧τ} ∇Q_{B_{s}}^{(α+1)}∇f(X_{s}) · dw_{s}

+ (α + 1) ∫_{0}^{t∧τ} Q_{B_{s}}^{(α+1)}∇f(X_{s})ds.

Set

Z_{t} = |Q_{B_{t∧τ}}^{(α+1)}∇f(X_{t∧τ})|.

Then, again by the Itô formula,

Z_{t}^{2} = Z_{0}^{2} + a martingale + A_{t}

where an increasing process (A_{t}) is given by

A_{t} = 2 ∫_{0}^{t∧τ} |\frac{∂}{∂a}Q_{B_{s}}^{(α+1)}∇f(X_{s})|^{2}ds + 2 ∫_{0}^{t∧τ} |∇Q_{B_{s}}^{(α+1)}∇f(X_{s})|^{2}ds
+ 2(α + 1) ∫_{0}^{t∧τ} |Q_{B_{s}}^{(α+1)}∇f(X_{s})|^{2}ds

By using the commutation relation to the first term in the right hand side, we have

∫_{0}^{t∧τ} |\frac{∂}{∂a}Q_{B_{s}}^{(α+1)}∇f(X_{s})|^{2}ds = ∫_{0}^{t∧τ} |\sqrt{α + 1 − L}Q_{B_{s}}^{(α+1)}∇f(X_{s})|^{2}ds

= ∫_{0}^{t∧τ} |∇Q_{B_{s}}^{(α)}\sqrt{α − L}f(X_{s})|^{2}ds

= ⟨M\sqrt{α−L}f^{↑}⟩_{t}.

Now it is clear that

(2.16)  A_{t} ≥ ⟨M\sqrt{α−L}f^{↑}⟩_{t}.

By the Langlart-Lépingle-Pratelli inequality (2.1), we have

E[A_{∞}^{1/2}] ≤ E[sup_{t}{Z_{t}^{2}}^{1/2}] ≤ E[sup_{t} Z_{t}].

This and (2.16) implies that

(2.17)  ⟨M\sqrt{α−L}f^{↑}⟩_{∞}^{1/2} ≤ E[sup_{t} Z_{t}].
Now we have, for g = \sqrt{α − L}f,
|(g, h)| = | lim_{N→∞} E[Mg^{↑}, Mh^{↑}] + α((α − L)^{−1}g, h)|

≤ lim_{N→∞} |E[Mg^{↑}, Mh^{↑}]| + α|(\sqrt{α − L}^{−1}f, h)|

< ∼ lim_{N→∞} ∥Mg^{↑}∥_{H^{1}}∥Mh^{↑}∥_{BMO} + α∥\sqrt{α − L}^{−1}f∥_{L^{1}}∥h∥_{L^{∞}}

< ∼ ( lim_{N→∞} ∥Mg^{↑}∥_{H^{1}} + α∥\sqrt{α − L}^{−1}f∥_{L^{1}})∥h∥_{L^{∞}} (∵ (2.11))

which implies

∥g∥_{L^{1}} < ∼ ( lim_{N→∞} ∥Mg^{↑}∥_{H^{1}} + α∥\sqrt{α − L}^{−1}f∥_{L^{1}}).

By noting the following expression of \sqrt{α − L}^{−1} (α > 0),

\sqrt{α − L}^{−1} = \frac{1}{Γ(1/2)} ∫_{0}^{∞} t^{−1/2}e^{−αt}T_{t}dt

and the fact that {T_{t}} is a contraction semigroup in L^{1}, we easily obtain

∥α\sqrt{α − L}^{−1}f∥_{L^{1}} ≤ \sqrt{α}∥f∥_{L^{1}}.

It remains to estimate Mg^{↑}. We recall that (Z_{t}) is a submartingale. Combining this with Doob’s inequality (2.3),

∥Mg^{↑}∥_{H^{1}} = E_{N}[sup_{t} |Mg^{↑}(t)|]

< ∼ E_{N}[⟨Mg^{↑}⟩_{∞}^{1/2}] (∵ Burkholder’s inequality)

< ∼ E_{N}[sup_{t} |Z_{t}|]

< ∼ ∥Z_{∞}∥_{L log_{+} L(P_{N})}.

By letting N → ∞, we obtain

lim_{N→∞} ∥Mg^{↑}∥_{H^{1}} < ∼ lim_{N→∞} ∥Z_{∞}∥_{L log_{+} L(P_{N})}

= lim_{N→∞} ∥Q_{0}^{(α)}∇f(X_{τ})∥_{L log_{+} L(P_{N})}

= ∥∇f∥_{L log_{+} L(μ)}.

Thus we have

∥\sqrt{α − L}f∥_{1} < ∼ ∥∇f∥_{L log_{+} L(μ)} + \sqrt{α}∥f∥_{L^{1}}.

This completes the proof. □

3. L^{p} multiplier theorem

In this section, we discuss the L^{p} multiplier theorem for the Ornstein-Uhlenbeck operator. Our main interest is in the L^{1} theory, but we start with L^{p} multiplier (1 < p < ∞). E. Stein [10] proved the multiplier theorem for symmetric diffusion operators. In this note, we will give a probabilistic proof. We only consider the Ornstein-Uhlenbeck operator, but our argument is valid for any symmetric diffusion semigroups if we can use the stochastic calculus.

Since −L is non-negative definite, we can have a spectral decomposition of −L as follows:

(3.1)  −L = ∫_{0}^{∞} λdE_{λ}.

If ϕ is a bonded function on [0, ∞), then

(3.2)  ϕ(−L) = ∫_{0}^{∞} ϕ(λ)dE_{λ}

defines a bonded linear operator in L^{2}(R^{d}, μ). We are interested in when this operator is bounded in L^{p}(R^{d}, μ) for p ≥ 1. As to this problem, Stein obtained a sufficient condition:

Theorem 3.1 (Stein). Suppose we are given a ϕ in the following form:

(3.3)  ϕ(λ) = λ ∫_{0}^{∞} e^{−2tλ}m(t)dt

Here, m is a bounded measurable function on [0, ∞). Then, the multiplier ϕ(−L) is a bounded operator on L^{p} for p > 1 and its operator norm depends only on p and ∥m∥_{∞}.

An important example is (−L)^{−iα} (α ∈ R) which plays an fundamental role in the interpolation theory.

We give a probabilistic proof of this theorem. Here we remark that Varopoulos [11] gave a probabilistic proof of the multiplier theorem. But his case does not cover all ϕ with (3.3).

We fix a positive constant K > 0 and take any f ∈ C_{0}^{∞}(R^{n}), C_{0}^{∞}(R^{n}) being the set of all C^{∞} function with compact support. Let {T_{t}} be the semigroup generated by L. Now we set u(x, t) = T_{K−t}f(x) for t ∈ [0, K]. In the previous section, we used the Cauchy semigroup (i.e., subordination), but here, we need the original semigroup. Then u satisfies the following differential equation

(3.4)  \{ \begin{matrix}u(x, 0) = T_{K}f(x) \\ \frac{∂}{∂t}u(x, t) + L_{x}u(x, 0) = 0. \end{matrix}

Hence, by Ito’s formula, we have

(3.5)  du(X_{t}, t) = \sqrt{2}(DT_{K−t}f(X_{t}), dw_{t})

where D is the gradient operator: D = (\frac{∂}{∂x^{1}} , …, \frac{∂}{∂x^{d}}). We define a martingale M_{f} as follows:

(3.6)  M_{f}(t) = \sqrt{2} ∫_{0}^{t}(DT_{K−s}f(X_{s}), dw_{s})
= T_{K−t}f(X_{t}) − T_{K}f(X_{0}).

This martingale plays a fundamental role in the proof.

Proof of the Theorem 3.1: First of all, let us obtain the expression ϕ(−L).

(3.7)  ∫_{0}^{∞} m(t)dt ∫_{E}(DT_{t}f(x), DT_{t}g(x))μ(dx)
= ∫_{0}^{∞} m(t)dt ∫_{E}(−L)T_{t}f(x)T_{t}g(x)μ(dx)
= ∫_{0}^{∞} m(t)dt ∫_{E}(−L)T_{2t}f(x)g(x)μ(dx)
= ∫_{0}^{∞} m(t)dt ∫_{0}^{∞} λe^{−2tλ}d(E_{λ}f, g)
= ∫_{0}^{∞} λd(E_{λ}f, g) ∫_{0}^{∞} e^{−2tλ}m(t)dt
= (ϕ(−L)f, g).

We introduce a martingale (M_{t})_{0≤t≤K} defined by

M_{t} = ∫_{0}^{t} m(K − s)(DT_{K−s}f(X_{s}), dw_{s}).

Then, its quadratic variation is given by

⟨M⟩_{t} = ∫_{0}^{t} m(K − s)^{2}|DT_{K−s}f(X_{s})|^{2}ds.

On the other hand, we have

E_{μ}[{ ∫_{0}^{K} m(K − t)(DT_{K−t}f(X_{t}), dw_{t})}g(X_{K})]
= E_{μ}[{∫_{0}^{K} m(K − t)(DT_{K−t}f(X_{t}), dw_{t})}
× \{T_{K}g(X_{0}) + \sqrt{2} ∫_{0}^{K}(DT_{K−t}g(X_{t}), dw_{t})\}]
= \sqrt{2}E_{μ}[∫_{0}^{K} m(t)(DT_{K−t}f(X_{t}), DT_{K−t}g(X_{t}))dt]
= \sqrt{2} ∫_{0}^{K} m(K − t)dt ∫_{E}(DT_{K−t}f(x), DT_{K−t}g(x))μ(dx)
= \sqrt{2} ∫_{0}^{K} m(t)dt ∫_{E}(DT_{t}f(x), DT_{t}g(x))μ(dx).

Combining this with (3.7), we have

\sqrt{2}(ϕ(−L)f, g) = lim_{K→∞} E_{μ}[\{∫_{0}^{K} m(K − t)(DT_{K−t}f(X_{t}), dw_{t})\}g(X_{K})].

Moreover the right hand side is estimated as follows:

E_{μ}[{ ∫_{0}^{K} m(K − t)(DT_{K−t}f(X_{t}), dw_{t})}g(X_{K})]

= E_{μ}[M_{K}g(X_{K})]

≤ E_{μ}[|M_{K}|^{p}]^{1/p}E_{μ}[|g(X_{K})|^{q}]^{1/q}

< ∼ E_{μ}[⟨M⟩_{K}^{p/2}]^{1/p}∥g∥_{q} (∵ Burkholder’s inequality)

= E_{μ}[{∫_{0}^{K} m(K − t)^{2}|DT_{K−t}f(X_{t})|^{2}dt}^{p/2}]^{1/p}∥g∥_{q}

≤ ∥m∥_{∞}E_{μ}[{∫_{0}^{K} |DT_{K−t}f(X_{t})|^{2}dt}^{p/2}]^{1/p}∥g∥_{q}

< ∼ ∥m∥_{∞}E_{μ}[{ sup_{0≤t≤K}|∫_{0}^{t}(DT_{K−t}f(X_{t}), dw_{t})|^{p}]^{1/p}∥g∥_{q}

(∵ Burkholder’s inequality)

< ∼ ∥m∥_{∞}E_{μ}[{|∫_{0}^{K}(DT_{K−t}f(X_{t}), dw_{t})|^{p}]^{1/p}∥g∥_{q} (∵ Doob’s inequality)

= ∥m∥_{∞}E_{μ}[|u(X_{K}, K) − u(X_{0}, 0)|^{p}]^{1/p}∥g∥_{q}

= ∥m∥_{∞}E_{μ}[|f(X_{K}) − T_{K}f(X_{0})|^{p}]^{1/p}∥g∥_{q} < ∼ ∥m∥_{∞}∥f∥_{p}∥g∥_{q}.

The above estimate is independent of K, we have eventually ∥ϕ(−L)f∥_{p} ≤ ∥f∥_{p} which is the desired result. □

The above proof works even in the case p = 1 except for Doob’s inequality. When p = 1, Doob’s inequality is of the form (2.3). Therefore, we have

E_{μ}[ sup_{0≤t≤K} |M_{f}(t)|] < ∼ ∥M_{f}(K)∥_{L log_{+} L(P)}
= ∥f(X_{K}) − T_{K}f(X_{0})∥_{L log_{+} L(P)}
≤ ∥f(X_{K})∥_{L log_{+} L(P)} + ∥T_{K}f(X_{0})∥_{L log_{+} L(P)}
≤ ∥f∥_{L log_{+} L(μ)} + ∥T_{K}f∥_{L log_{+} L(μ)}
≤ 2∥f∥_{L log_{+} L(μ)}. (∵ T_{K} is contraction in L log_{+} L)

Therefore we have the following theorem in the case p = 1.

Theorem 3.2. Under the same conditions in Theorem 3.1, ϕ(−L) is a bounded operator from L log^{+}L to L^{1} and its operator norm depends only on ∥m∥_{∞}.

References

[1] D. Bakry, Etude probabiliste des transformées de Riesz et de l’espace H^{1} sur les sphères, Séminaire de Prob. XVIII, pp. 197–218, Lecture Notes in Math., vol. 1059, Springer-Verlag, Berlin-Heidelberg-New York, 1983.

[2] D. Bakry, Transformations de Riesz pour les semigroupes symétriques, Premiere partie: etude de la dimension 1, Séminaire de Prob. XIX, pp. 130–144, Lecture Notes in Math., vol. 1123, Springer-Verlag, Berlin-Heidelberg-New York, 1985.

[3] D. Bakry, Transformations de Riesz pour les semigroupes symétriques, Seconde partie: etude sous la condition Γ_{2} ≥ 0, Séminaire de Prob. XIX, pp. 145– 174, Lecture Notes in Math., vol. 1123, Springer-Verlag, Berlin-Heidelberg-New York, 1985.

[4] C. Dellacherie and P. A. Meyer, “Probabilities and Potential B,” North-Holland, Amsterdam, 1982.

[5] E. Lenglart, D. Lépingle and M. Pratelli, Présentation unifiée de certaines inégalités de théorie des martingales, Séminaire de Prob. XIV, Lecture Notes in Math., vol. 784, 26–48. Springer-Verlag, Berlin-Heidelberg-New York, 1980.

[6] P. A. Meyer, Démonstration probabiliste de certaines inégalités de Littlewood-Paley, Séminaire de Prob. X, Lecture Notes in Math., vol. 511, 125–183, Springer-Verlag, Berlin-Heidelberg-New York, 1976.

[7] P. A. Meyer, Retour sur la théorie de Littlewood-Paley, Séminaire de Prob. XV, Lecture Notes in Math., vol. 850, 151–166, Springer-Verlag, Berlin-Heidelberg-New York, 1981.

[8] P. A. Meyer, Notes sur les processus d’Ornstein-Uhlenbeck, Séminaire de Prob. XVI, ed. par J. Azema et M. Yor, Lecture Notes in Math., vol. 920 (1982), 95–133, Springer-Verlag, Berlin-Heidelberg-New York.

[9] P. A. Meyer, Quelques results analytiques sur le semigroupe d’Ornstein-Uhlenbeck en dimension infinie, Theory and application of random fields, Proceedings of IFIP-WG 7/1 Working conf. at Bangalore, ed. by G. Kallianpur, Lecture Notes in Cont. and Inform. Sci., vol. 49 (1983), 201–214, Springer-Verlag, Berlin-Heidelberg-New York.

[10] E. M. Stein, “Topics in harmonic analysis related to Littlewood-Paley theory,” Annals of Math. Study no. 63, Princeton University Press, Princeton, 1970.

[11] N. T. Varopoulos, Aspects of probabilistic Littlewood-Paley theory, J. Funct. Anal., 38 (1980), 25–60.

Ichiro SHIGEKAWA Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-01, Japan. E-mail:ichiro@kusm.kyoto-u.ac.jp