Advanced Studies in Pure Mathematics **, 200* ** pp. 1–18

Orlicz norm equivalence for the Ornstein-Uhlenbeck operator

Ichiro Shigekawa

Dedicated to Professor Kiyosi Ito on the occasion of his 88th birthday

Abstract.

The Meyer equivalence on an abstract Wiener space states that the L^{p}-norm of square root of the Ornstein-Uhlenbeck operator is equivalent to L^{p}-norm of the Malliavin derivative. We prove the equivalence in the framework of Orlicz space. We also discuss the logarithmic Sobolev inequality in L^{p} setting and higher order logarithmic Sobolev inequality.

§1. Introduction

Let (B, H, μ) be an abstract Wiener space: B is a separable real Banach space, H is a separable real Hilbert space which is embedded densely and continuously in B and μ is a Gaussian measure with

∫_{B} exp\{\sqrt{−1}_{B^{∗}}⟨l, x⟩_{B}\}μ(dx) = exp\{−\frac{1}{2}|l|_{H^{∗}}^{2}\}, l ∈ B^{∗} ↪ H^{∗}.

On an abstract Wiener space, the Ornstein-Uhlenbeck semigroup is defined as

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

The generator of the semigroup {T_{t}} is called the Ornstein-Uhlenbeck operator and we denote it by L. Then the following Meyer equivalence is well-known: for any 1 < p < ∞, there exists positive constants C_{1} and C_{2} such that

C_{1}\{∥Df∥_{p} + ∥f∥_{p}\} ≤ ∥\sqrt{1 − L}f∥_{p} ≤ C_{2}\{∥Df∥_{p} + ∥f∥_{p}\}. (2)

Partially supported by the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research (A) 14204008, 2002.

Here D is the Malliavin H-derivation and ∥ ∥_{p} is the L^{p}-norm. The constants C_{1} and C_{2} depend only on p.

In this paper we show that similar inequalities hold in the framework of Orlicz space, i.e., the above inequalities hold for the Orlicz norm in place of L^{p}-norm. Typical example we are in mind is L^{p} log^{β} L. As an application, we discuss the logarithmic Sobolev inequality in L^{p} setting and higher order logarithmic Sobolev inequality.

§2. Orlicz space

In this section, we review the Orlicz space (see, e.g., [1] or [8] for details). First we need the notion of Young function. Young function is a function Φ defined as

Φ(x) = ∫_{0}^{x} ϕ(t)dt, x ≥ 0 (3)

where ϕ is a non-negative, right continuous, non-decreasing function. If, in addition, ϕ satisfies ϕ(0) = 0, ϕ(t) > 0 for t > 0, ϕ(∞) = ∞, then Φ is called a nice Young function or N-function. Define ψ by

ψ(u) = inf{t; ϕ(t) > u}.

ψ is right continuous and non-decreasing. The function Ψ defined by

Ψ(y) = ∫_{0}^{y} ψ(u)du, y ≥ 0

is called a complementary function. The following properties are fundamental.

(4)  xy ≤ Φ(x) + Ψ(y),
(5)  xϕ(x) = Φ(x) + Ψ(ϕ(x)).

(4) is called the Young inequality.

The Orlicz space associated with Φ is defined as follows. Let (M, m) be a measure space and Φ be a nice Young function. Define a norm ∥ ∥_{Φ} by

∥f∥_{Φ} := inf{λ > 0 ; ∫_{M} Φ(|f|/λ)dm ≤ 1}. (6)

L^{Φ}(m) is the set of all measurable functions f which satisfy ∥f∥_{Φ} < ∞. We call L^{Φ}(m) an Orlicz space. It is a Banach space with the norm ∥ ∥_{Φ}. If Φ satisfies the Δ_{2} condition, i.e., there exists a constant C such that Φ(2x) ≤ CΦ(x), then the dual space is identified with L^{Ψ}(m), Ψ being the complementary function of Φ.

We introduce some classes of functions.

Definition 2.1. For non-negative constant α, we define a set of functions L(α), U(α) as follows:

(i) ϕ ∈ L(α) ⟺^{def} αϕ(t) ≤ tϕ′(t), ∀t > 0.

(ii) ϕ ∈ U(α) ⟺^{def} tϕ′(t) ≤ αϕ(t), ∀t > 0.

The following inequality for semimartingales is important in our later argument.

Let (Z_{t}) (t ∈ [0, ∞]) be a non-negative submartingale. We assume that (Z_{t}) is right continuous and has left hand limits. By the Doob-Meyer decomposition theorem, (Z_{t}) can be decomposed as

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

where (M_{t}) is a martingale and (A_{t}) is an increasing process. We assume that (A_{t}) is continuous and A_{0} = 0. If Φ ∈ U(α), then the following inequality holds (see [4, Theorem VI.99]):

E[Φ(A_{∞})] ≤ E[Φ(αZ_{∞})]. (7)

Further, a generalization of the Doob’s inequality also holds. It is stated as follows (see [4, Chapter VI, Section 3]). We assume that Φ ∈ L(α) for an α > 1. Then, setting Z_{t}^{∗} := sup_{s≤t} Z_{s}, it holds that

E[Φ(Z_{∞}^{∗})] ≤ E[Φ(αZ_{∞})]. (8)

From this inequality, we can have the following maximal ergodic inequality.

∫_{B} Φ(sup_{t≥0} |T_{t}f(x)|)μ(dx) ≤ ∫_{B} Φ(α|f(x)|)μ(dx). (9)

Here {T_{t}} is the Ornstein-Uhlenbeck semigroup on an abstract Wiener space (B, H, μ).

§3. Littlewood-Paley inequality

Let (B, H, μ) be an abstract Wiener space and K be a separable Hilbert space. {T_{t}} is the Ornstein-Uhlenbeck semigroup on L^{p}(E, μ; K) defined by (1). For α > 0, set

T_{t}^{(α)} = e^{−αt}T_{t}.

Then the generator of {T_{t}^{(α)}} is L − α. We further define a semigroup {Q_{t}^{(α)}} by subordination as follows:

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

Here λ_{t} is a probability measure on [0, ∞) whose Laplace transform is given by

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

When α = 0, Q_{t}^{(0)} is simply denoted by Q_{t} and called the Cauchy semigroup. For F ∈ L^{Φ}(B, μ; K), it holds that

∥Q_{t}^{(α)}F ∥_{Φ} ≤ e^{−\sqrt{α}t}∥F ∥_{Φ}

and {Q_{t}^{(α)}} is a strongly continuous semigroup on L^{Φ}. The generator will be denoted by −\sqrt{α − L}.

We denote by P(K) a set of all functions f: B → K which can be expressed as

f(x) = ∑_{i} p_{i}(⟨l_{1}, x⟩), …, ⟨l_{n}, x⟩)k_{i}

where p_{i} is a polynomial on ℝ^{n} and k_{1}, …, k_{n} ∈ K, l_{1}, …, l_{n} ∈ B^{∗}. For f ∈ P(K), define

g^{→}f(x, t) = |∂_{t}Q_{t}^{(α)}(x, f)|_{K},
g^{↑}f(x, t) = |DQ_{t}^{(α)}(x, f)|_{HS},
gf(x, t) = \sqrt{g^{→}f(x, t)^{2} + g^{↑}f(x, t)^{2}}.

Here Q_{a}^{(α)}(x, f) = Q_{a}^{(α)}f(x) and the norm | |_{HS} denotes the Hilbert-Schmidt norm. g^{→}f, g^{↑}f, gf all depend on α but we fix it throughout the argument and suppress it for simplicity. We further define

G^{→}f(x) = {∫_{0}^{∞} tg^{→}f(x, t)^{2}dt}^{1/2},
G^{↑}f(x) = {∫_{0}^{∞} tg^{↑}f(x, t)^{2}dt}^{1/2},
Gf(x) = {∫_{0}^{∞} tgf(x, t)^{2}dt}^{1/2}.

We call them Littlewood-Paley G-functions.

Our aim in this section is to prove the following theorem.

Theorem 3.1. Assume that Φ ∈ L(α) ∩ U(β) for constants 1 < α < β. Further assume that ϕ is either convex or concave. Then we have

(10)  ∥Φ(Gf)∥_{1} ≲ ∥Φ(|f|)∥_{1},
(11)  ∥Φ(|f|)∥_{1} ≲ ∥Φ(G^{→}f)∥_{1}.

In the above theorem, A ≲ B stands for A ≤ CB for a positive constant C that is independent of f. We use this convention in the sequel without mentioning.

We give a probabilistic proof. To do this, we take the Ornstein-Uhlenbeck process (X_{t}) on B, i.e., the diffusion process generated by L. We also take a process (B_{t}) on ℝ generated by \frac{d^{2}}{da^{2}} . We assume that the initial distribution of (X_{t}) is the stationary measure μ so that the process becomes stationary. We denote the starting point of the Brownian motion (B_{t}) by N. E_{N} stands for the integration with respect to this measure. Later we let N → ∞.

Now, for f ∈ P(K), set u(x, a) = Q_{a}^{(α)}(x, f). Then u(x, a) satisfies

\{ \begin{matrix}u(x, 0) = f(x) \\ L_{x}u(x, a) + ∂_{a}^{2}u(x, a) − αu(x, a) = 0. \end{matrix} (12)

Define a stopping time τ by

τ = inf{t > 0 | B_{t} = 0}.

Then we can think of u(X_{t}, B_{t}) for t ≤ τ. Set

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

Then (M_{t}) is a martingale with M_{0} = Q_{B_{0}}f(X_{0}). The quadratic variation is given as

⟨M⟩_{t} = 2 ∫_{0}^{t∧τ} gf^{2}(X_{s}, B_{s})ds. (13)

Therefore we have

(14)  d|u|^{2} = 2(u, dM) + 2α|u|^{2}dt + ⟨dM, dM⟩
= 2(u, dM) + (2α|u|^{2} + 2gf^{2})dt

Now set

Z_{t} = |u(X_{t∧τ} , B_{t∧τ})|^{2}.

(Z_{t}) is a non-negative submartingale. To compute Φ(\sqrt{Z_{t}}), we approximate it as follows. Take any ε > 0 and set F (x) = Φ(\sqrt{x + ε}). Recall that ϕ is either convex or concave. We divide into tow cases.

(i) ϕ is concave.

We need the following proposition.

Proposition 3.2. Assume Φ ∈ L(α) (α > 1). Then it holds that, for u, v ≥ 0,

Φ(v) ≤ \frac{1}{α − 1}(\frac{1}{2}ϕ′(u)v^{2} + Φ(u)). (15)

Proof. From the assumption, αΦ(x) ≤ xϕ(x) holds. Since ϕ is concave, Φ(x) ≥ \frac{1}{2}xϕ(x) which leads to α ≤ 2. Hence (15) clearly holds when u ≥ v.

If v ≥ u, we have

|\{(x, y) ; 0 ≤ x ≤ u, ϕ(x) ≤ y ≤ ϕ(u)\}| ≤ \frac{1}{2}uϕ(u)
|{(x, y) ; 0 ≤ x ≤ u, ϕ(u) ≤ y ≤ ϕ(v)}| ≤ u(ϕ(v) − ϕ(u)) ≤ uϕ′(u)(v − u)
|\{(x, y) ; u ≤ x ≤ v, ϕ(x) ≤ y ≤ ϕ(v)\}| ≤ \frac{1}{2}(v − u)^{2}ϕ′(u).

These are easily obtained by observing the graph.

Summing up three terms of the left-hand side and Φ(v), we have vϕ(v). Therefore

\frac{1}{2}uϕ(u) + uϕ′(u)(v − u) + \frac{1}{2}(v − u)^{2}ϕ′(u) + Φ(v) ≥ vϕ(v) ≥ αΦ(v).

Hence we have

(α − 1)Φ(v) ≤ \frac{1}{2}uϕ(u) + ϕ′(u)(v − u)(u + \frac{1}{2}v − \frac{1}{2}u)

= \frac{1}{2}uϕ(u) + \frac{1}{2}ϕ′(u)(v^{2} − u^{2})

≤ Φ(u) + \frac{1}{2}ϕ′(u)v^{2}

which is the desired result. Q.E.D.

Orlicz norm equivalence 7

The derivatives of F (x) = Φ(\sqrt{x + ε}) are

F′(x) = Φ′(\sqrt{x + ε})\frac{1}{2\sqrt{x + ε}},
F′′(x) = Φ′′(\sqrt{x + ε})\frac{1}{4(x + ε)} + Φ′(\sqrt{x + ε})\frac{1}{2}(−\frac{1}{2})\frac{1}{\sqrt{x + ε}^{3}} .

By the Itô formula,

dΦ(\sqrt{Z_{t} + ε}) = \frac{Φ′(\sqrt{Z_{t} + ε})}{2\sqrt{Z_{t} + ε}}dZ_{t}

+ \frac{1}{2}\{\frac{Φ′′(\sqrt{Z_{t} + ε})}{4(Z_{t} + ε)} − \frac{1}{4} \frac{Φ′(\sqrt{Z_{t} + ε})}{\sqrt{Z_{t} + ε}^{3}}\}⟨dZ, dZ⟩

= \frac{ϕ(\sqrt{|u|^{2} + ε})}{2\sqrt{|u|^{2} + ε}}\{2(u, dM) + 2(α|u|^{2} + gf^{2})dt\}

+ \frac{1}{2}\{\frac{ϕ′(\sqrt{|u|^{2} + ε})}{4(|u|^{2} + ε)} − \frac{1}{4} \frac{ϕ(\sqrt{|u|^{2} + ε})}{\sqrt{|u|^{2} + ε}^{3}}\}⟨dZ, dZ⟩

= \frac{ϕ(\sqrt{|u|^{2} + ε})}{\sqrt{|u|^{2} + ε}}(u, dM) + \frac{ϕ(\sqrt{|u|^{2} + ε})}{\sqrt{|u|^{2} + ε}}(α|u|^{2} + gf^{2})dt

+ \frac{1}{8} \frac{1}{|u|^{2} + ε}\{ϕ′(\sqrt{|u|^{2} + ε}) − \frac{ϕ(\sqrt{|u|^{2} + ε})}{\sqrt{|u|^{2} + ε}}\}⟨dZ, dZ⟩.

Now we note ⟨dZ, dZ⟩ ≤ 4|u|^{2}⟨dM, dM⟩ = 8|u|^{2}gf^{2}dt. Further ϕ′(t) ≤ ϕ(t)/t since ϕ is concave. We therefore have

\frac{ϕ(\sqrt{|u|^{2} + ε})}{\sqrt{|u|^{2} + ε}}gf^{2}dt + \frac{1}{8} \frac{1}{|u|^{2} + ε}\{ϕ′(\sqrt{|u|^{2} + ε}) − \frac{ϕ(\sqrt{|u|^{2} + ε})}{\sqrt{|u|^{2} + ε}}\}⟨dZ, dZ⟩

≥ \frac{ϕ(\sqrt{|u|^{2} + ε})}{\sqrt{|u|^{2} + ε}}gf^{2}dt + \{ϕ′(\sqrt{|u|^{2} + ε}) − \frac{ϕ(\sqrt{|u|^{2} + ε})}{\sqrt{|u|^{2} + ε}}\}gf^{2}dt

= ϕ′(\sqrt{|u|^{2} + ε})gf^{2}dt. Integrating from 0 to τ and taking expectation, we have ∥Φ(\sqrt{|f|^{2} + ε})∥_{1} ≥ E_{N}[∫_{0}^{τ} ϕ′(\sqrt{|u|^{2} + ε})gf^{2}dt]. (16)

We will give an estimate from below of the right-hand side. We note that f^{∗}(x) := sup_{t≥0} |T_{t}f(x)| ≥ sup_{a≥0} |u(x, a)|.

E_{N}[∫_{0}^{τ} ϕ′(\sqrt{|u|^{2} + ε})gf^{2}dt] = ∥∫_{0}^{∞} ϕ′(\sqrt{|u|^{2} + ε})gf^{2}( · , a)(a ∧ N)da∥_{1}

≥ ∥∫_{0}^{∞} ϕ′(\sqrt{f^{∗2} + ε})gf^{2}( · , a)(a ∧ N)da∥_{1}.

(∵ ϕ′ is non-increasing)

Combining this with (16) and letting N → ∞

∥Φ(\sqrt{|f|^{2} + ε})∥_{1} ≥ ∥∫_{0}^{∞} ϕ′(\sqrt{f^{∗2} + ε})gf^{2}( · , a)ada∥_{1}
= ∥ϕ′(\sqrt{f^{∗2} + ε})Gf^{2}∥_{1}

Now we use the inequality Φ(v) ≤ \frac{1}{α−1}(\frac{1}{2}ϕ′(u)v^{2} + Φ(u)) in Proposition 3.2 and get

∥Φ(Gf)∥_{1} ≲ ∥ϕ′(\sqrt{f^{∗2} + ε})Gf^{2}∥_{1} + ∥Φ(\sqrt{f^{∗2} + ε})∥_{1}
≤ ∥Φ(\sqrt{|f|^{2} + ε})∥_{1} + ∥Φ(\sqrt{f^{∗2} + ε})∥_{1}.

Letting ε → 0 and using the maximal ergodic inequality (9), we have

∥Φ(Gf)∥_{1} ≲ ∥Φ(|f|)∥_{1} + ∥Φ(f^{∗})∥_{1} ≲ ∥Φ(|f|)∥_{1}.

This completes the proof in the case that ϕ is concave.

(ii) ϕ is convex.

Set Φ̃(x) = Φ(\sqrt{x}). Then Φ̃ is convex. In fact, by differentiating, we have

\frac{d}{dx}Φ(\sqrt{x}) = Φ′(\sqrt{x})\frac{1}{2\sqrt{x}} = \frac{ϕ(\sqrt{x})}{2\sqrt{x}}.

The function is increasing since ϕ is convex and so the convexity of Φ̃ follows. Further Φ̃ ∈ U(α/2) since

\frac{xΦ̃′(x)}{Φ̃(x)} = \frac{xΦ′(\sqrt{x})}{2\sqrt{x}Φ(\sqrt{x})} = \frac{\sqrt{x}Φ′(\sqrt{x})}{2Φ(\sqrt{x})}.

The submartingale Z_{t} = |u(X_{t∧τ} , B_{t∧τ})|^{2} is decomposed as a sum of a martingale and an increasing process as in (14). By using (7), we get

(17) E_{N}[Φ̃(∫_{0}^{τ} gf(X_{s}, B_{s})^{2}ds)] ≲ E_{N}[Φ̃(Z_{∞})] = E_{N}[Φ̃(|f(X_{τ})|^{2})]
= E_{N}[Φ(|f(X_{τ})|)] = ∥Φ(|f|)∥_{1}

Now we introduce H-functions as follows.

H^{→}f(x) = {∫_{0}^{∞} tQ_{t}g^{→}f(x, t)^{2}dt}^{1/2},
H^{↑}f(x) = {∫_{0}^{∞} tQ_{t}g^{↑}f(x, t)^{2}dt}^{1/2},
Hf(x) = {∫_{0}^{∞} tQ_{t}gf(x, t)^{2}dt}^{1/2}.

Then we have

∥Φ(Hf)∥_{1} = ∥Φ̃(Hf^{2})∥_{1}

= lim_{N→∞}∥Φ̃(∫_{0}^{∞} Q_{a}gf( · , a)^{2}(a ∧ N)da)∥_{1}

= lim_{N→∞} ∫_{B} Φ̃(E_{N}[∫_{0}^{τ} gf^{2}(X_{t}, B_{t})dt|X_{τ} = x])μ(dx)

≤ lim_{N→∞} ∫_{B} E_{N}[Φ̃(∫_{0}^{τ} gf^{2}(X_{t}, B_{t})dt)|X_{τ} = x])μ(dx)

≤ lim_{N→∞} E_{N}[Φ̃(∫_{0}^{τ} gf^{2}(X_{t}, B_{t})dt)]

≲ ∥Φ(|f|)∥_{1}. (∵ (17))

It is well-known that Gf is dominated by Hf (see [7]) and so (10) follows. (11) can be shown by a standard duality argument. This completes the proof of Theorem 3.1.

Using this theorem, the Meyer equivalence in Orlicz space, which is of our main interest, follows easily. In fact, the same proof as in L^{p} setting works (see e.g., [9]).

Theorem 3.3. Assume that Φ ∈ L(α) ∩ U(β) for 1 < α < β and that ϕ is either convex or concave. Then there exist positive constants C_{1} and C_{2} such that

C_{1}\{∥Df∥_{Φ} + ∥f∥_{Φ}\} ≤ ∥\sqrt{1 − L}f∥_{Φ} ≤ C_{2}\{∥Df∥_{Φ} + ∥f∥_{Φ}\}. (18)

§4. Examples

We give some example of nice Young functions that satisfy the condition of Theorem 3.3. For indicies p > 1, β ∈ ℝ, k ≥ 1, we set

ϕ_{p,β,k}(x) = x^{p−1} log^{pβ}(k + x), x ≥ 0 (19) and define

Φ_{p,β,k}(x) = ∫_{0}^{x} ϕ_{p,β,k}(y)dy. (20)

We regards this as a Young function. The function does not satisfy the condition of Young function in general since β might be negative. We see when it is a Young function. To avoid complexity, we simply denote ϕ and Φ in place of ϕ_{p,β,k} and Φ_{p,β,k}, respectively. Differentiating ϕ, we have

ϕ′(x) = (p − 1)x^{p−2} log^{pβ}(k + x) + pβx^{p−1}\{log^{pβ−1}(k + x)\}\frac{1}{k + x}

= x^{p−2} log^{pβ}(k + x)\{p − 1 + pβ \frac{x}{(k + x) log(k + x)}\}.

We look for the condition so that ϕ′ is positive. To do this, set

f(x) = \frac{x}{(k + x) log(k + x)}. (21)

If k = 1, f takes its maximum 1 at x = 0. If k > 1, f takes its maximum at x = α where α is the solution of k log(k + x) − x = 0. We can see that f(α) ≤ \frac{1}{1+log k}. Therefore, in all cases of k, it holds that

0 ≤ \frac{x}{(k + x) log(k + x)} ≤ \frac{1}{1 + log k} . (22)

Now it is easy to see that Φ is a nice Young function if p(1+ \frac{β}{1 + log k} ) ≥

1. Further we easily have the following proposition.

Proposition 4.1. ϕ satisfies following inequalities:

(p − 1)ϕ(x) ≤ xϕ′(x) ≤ (p − 1 + \frac{pβ}{1 + log k} )ϕ(x), for β ≥ 0, (23)

(p − 1 + \frac{pβ}{1 + log k} )ϕ(x) ≤ xϕ′(x) ≤ (p − 1)ϕ(x), for β < 0. (24)

Similar inequalities hold for Φ. To see this, we need the following proposition.

Proposition 4.2. For positive constant α, it holds that

(i) if ϕ ∈ L(α), then Φ ∈ L(α + 1), (ii) if ϕ ∈ U(α), then Φ ∈ U(α + 1).

Orlicz norm equivalence 11

Proof. Suppose ϕ ∈ L(α), i.e., αϕ(t) ≤ tϕ′(t). By integrating both hands, we have αΦ(x) ≤ ∫_{0}^{x} tϕ′(t)dt. (25)

On the other hand, since (tϕ(t))′ = ϕ(t) + tϕ′(t), we have

xϕ(x) = ∫_{0}^{x}{ϕ(t) + tϕ′(t)}dt

and hence

xϕ(x) − Φ(x) = ∫_{0}^{x} tϕ′(t)dt.

This combined with (25) leads

xϕ(x) ≥ (α + 1)Φ(x).

(ii) can be shown similarly. Q.E.D. Now the following proposition easily follows. Proposition 4.3. The following inequalities hold:

pΦ(x) ≤ xΦ′(x) ≤ p(1 + \frac{β}{1 + log k} )Φ(x), (β ≥ 0), (26)

p(1 + \frac{β}{1 + log k} )Φ(x) ≤ xΦ′(x) ≤ pΦ(x), (β < 0). (27)

Lastly, we will see the asymptotic behavior of the complementary function Ψ. We use the notation f ∼ g when lim_{x→∞} \frac{f(x)}{g(x)} = 1 holds.

Proposition 4.4. Assume p > 1 and let q be the conjugate exponent of p: \frac{1}{p} + \frac{1}{q} = 1. Then it holds that

(28)  pΦ(x) ∼ x^{p} log^{pβ} x,
(29)  (q − 1)^{qβ}qΨ(x) ∼ x^{q} log^{−qβ} x.

Proof. By the l’Hôpital theorem, we have

lim_{x→∞} \frac{pΦ(x)}{x^{p} log^{pβ} x} = lim_{x→∞} \frac{pϕ(x)}{px^{p−1} log^{pβ} x + x^{p}pβ(log^{pβ} x)/x}

= lim_{x→∞} \frac{px^{p−1} log^{pβ}(k + x)}{px^{p−1} log^{pβ} x + pβx^{p−1} log^{pβ−1} x}

12 I. Shigekawa

= lim_{x→∞} \frac{log^{pβ}(k + x)}{log^{pβ} x + β log^{pβ−1} x}

= 1

which shows (28).

As for Ψ, we have, by the l’Hôpital theorem,

lim_{x→∞} \frac{qΨ(x)}{x^{q} log^{−qβ} x}

= lim_{x→∞} \frac{qΨ(ϕ(x))}{ϕ(x)^{q} log^{−qβ} ϕ(x)}

= lim_{x→∞} \frac{qΨ′(ϕ(x))ϕ′(x)}{qϕ(x)^{q−1}ϕ′(x) log^{−qβ} ϕ(x) + ϕ(x)^{q}(−qβ)\{log^{−qβ−1} ϕ(x)\}ϕ′(x)/ϕ(x)}

= lim_{x→∞} \frac{x}{ϕ(x)^{q−1} log^{−qβ} ϕ(x) − βϕ(x)^{q−1} log^{−qβ−1} ϕ(x)}

= lim_{x→∞} \frac{x}{ϕ(x)^{q−1} log^{−qβ} ϕ(x)(1 − β/ log ϕ(x))}

= lim_{x→∞} \frac{x}{ϕ(x)^{q−1} log^{−qβ} ϕ(x)}

= lim_{x→∞} \frac{x}{\{x^{p−1} log^{pβ}(k + x)\}^{q−1} log^{−qβ}(x^{p−1} log^{pβ}(k + x))}

= lim_{x→∞} \frac{x}{x^{(p−1)(q−1)} log^{pβ(q−1)}(k + x)\{(p − 1) log x + pβ log log(k + x)\}^{−qβ}}

= lim_{x→∞} \frac{\{(p − 1) log x + pβ log log(k + x)\}^{qβ}}{log^{pβ(q−1)}(k + x)} (∵ (p − 1)(q − 1) = 1)

= lim_{x→∞}\{\frac{\{(p − 1) log x + pβ log log(k + x)}{log(k + x)}\}^{qβ} (∵ q = p(q − 1))

= (p − 1)^{qβ} which shows (29). Q.E.D.

We denotes the Orlicz space L^{Φ} associated with Φ by L^{p} log^{pβ} L. We do not specify k since it does not affect the asymptotic behavior at infinity. Since the Wiener measure is finite, L^{p} log^{pβ} L is independent of k. The above theorem means that the dual space of L^{p} log^{pβ} L is L^{q} log^{−qβ} L.

§5. Logarithmic Sobolev inequality

The logarithmic Sobolev inequality in L^{p} setting was discussed by D. Bakry-P. A. Meyer [3] and higher order Logarithmic Sobolev inequality was discussed by G. F. Feissner [5] and R. A. Adams [2]. They all used the interpolation theorem. Here we take a different approach.

The following logarithmic Sobolev inequality holds for the Ornstein-Uhlenbeck process.

E [f^{2} log(f^{2}/∥f∥_{2}^{2})] ≤ 2E [|Df|^{2}] .

Here E [ ] stands for the integration with respect to μ. Hereafter we use this notation. Recall that ∫_{B}(Df, Dg)_{H^{∗}}dμ is the Dirichlet form associated with the Ornstein-Uhlenbeck process. We remark that the following argument works for the diffusion Dirichlet form satisfying the logarithmic Sobolev inequality if we assume the Dirichlet form is of the gradient type.

We introduce a new Young function. Set

(30) θ(x) = {x^{2} log(e + x^{2})}^{(p−2)/4} log^{pβ/4}(k + x^{2} log(e + x))

and define

Θ(x) = ∫_{0}^{x} θ(y)dy. (31)

Then we have the following proposition.

Proposition 5.1. For sufficient large k if necessary, there exists a positive constant K such that

(32)  x^{p} log^{p/2}(e + x^{2}) log^{pβ/2}(k + x^{2} log(e + x^{2}))
≤ KΘ(x)^{2} log(e + Θ(x)^{2})

Proof. We divide the proof into two cases.

(a) β ≥ 0, k = 1.

Let us see the asymptotic behavior as x → 0.

LHS ∼ x^{p} · x^{(pβ/2)2} = x^{p(1+β)}.

On the other hand,

θ(x) ∼ x^{(p−2)/2+(pβ)/2} = x^{p(β+1)/2−1}.

and hence

Θ(x) ∼ \frac{2}{p(β + 1)}x^{p(β+1)/2}

Θ(x)^{2} ∼ \frac{4}{p^{2}(β + 1)^{2}} x^{p(β+1)}.

Thus both hands have the same asymptotic behavior.

As x → ∞,

LHS ∼ x^{p}2^{p/2}(log^{p/2} x)2^{pβ/2} log^{pβ/2} x = 2^{p(β+1)/2}x^{p} log^{p(β+1)/2} x.

On the other hand,

θ(x) ∼ x^{(p−2)/2}2^{(p−2)/4}(log^{(p−2)/4} x)2^{pβ/4} log^{pβ/4} x
Θ(x) ∼ (2/p)2^{(p+pβ−2)/4}x^{p/2} log^{(p+pβ−2)/4} x
Θ(x)^{2} log(e + Θ(x)^{2}) ∼ p^{−2}2^{(p+pβ+2)/2}x^{p}(log^{(p+pβ−2)/2} x)p log x
= p^{−1}2^{(p+pβ+2)/2}x^{p} log^{(p+pβ)/2} x.

Hence they have the same asymptotic behavior.

(b) β < 0 and large k.

The asymptotic behavior at x = ∞ can be obtained similarly as in the case β ≥ 0.

As x → 0, LHS ∼ x^{p} is clear. Further we have

θ(x) ∼ x^{(p−2)/2}

Θ(x) ∼ \frac{2}{p}x^{p/2}

Θ(x)^{2} log(e + Θ(x)^{2}) ∼ \frac{4}{p^{2}} x^{p}

Thus we have the desired result. Q.E.D.

We recall the following fact. Let U and V be a non-negative functions on a measure space (M, m). Assume that

∫_{M} Uϕ(U)dm < ∞,
∫_{M} Uϕ(U)dm ≤ ∫_{M} V ϕ(U)dm + C.

Then it follows that

∫_{M} Φ(U)dm ≤ ∫_{M} Φ(V)dm + C. (33)

For the proof, see [4, Lemma VI.98]. Now we have the following theorem. In the sequel, we denote by Φ_{p,β} in place of Φ_{p,β,k} because the index k is not essential.

Orlicz norm equivalence 15

Proposition 5.2. For p > 2, β ∈ ℝ, there exists a positive constant C such that

E[Φ_{p,(β+1)/2}(|f|)] ≤ CE[Φ_{p,(1+β)/2−(1/p)}(|f|)] + CE[Φ_{p,β/2}(|Df|)]. (34)
Proof. Set g = \sqrt{Θ(|f|)^{2} + e}. Then

Dg = \frac{2Θ(|f|)Θ′(|f|)D|f|}{2\sqrt{Θ(|f|)^{2} + e}}

and hence |Dg| ≤ θ(|f|)|Df|. Now, by using the logarithmic Sobolev inequality

E [g^{2} log(g^{2}/∥g∥_{2}^{2})] ≤ 2E [|Dg|^{2}] ,

we have E [{Θ(|f|)^{2} + e} log(e + Θ(|f|)^{2})] ≤ E [Θ(|f|)^{2} + e] log E [e + Θ(|f|)^{2}]

+ E [|Df|^{2}{|f|^{2} log(e + |f|^{2})}^{(p−2)/2} log^{pβ/2}(k + |f|^{2} log(e + |f|^{2}))] , and E [Θ(|f|)^{2} log(e + Θ(|f|)^{2})] ≤ E [Θ(|f|)^{2}] log E [e + Θ(|f|)^{2}]

+ E [|Df|^{2}{|f|^{2} log(e + |f|^{2})}^{(p−2)/2} log^{pβ/2}(k + |f|^{2} log(e + |f|^{2}))] . We set

ϕ(x) = ϕ_{p/2,β,k}(x) = x^{(p/2)−1} log^{pβ/2}(k + x),
U = |f|^{2} log(e + |f|^{2}).

Then

Uϕ(U) = |f|^{2} log(e + |f|^{2})
× {|f|^{2} log(e + |f|^{2})}^{(p/2)−1} log^{pβ/2}(k + |f|^{2} log(e + |f|^{2}))
= |f|^{p} log^{p/2}(e + |f|^{2}) log^{pβ/2}(k + |f|^{2} log(e + |f|^{2}))
≤ KΘ(|f|)^{2} log(e + Θ(|f|)^{2}). (∵ (32))

Combining this with the previous result, we have K^{−1}E[Uϕ(U)] ≤ E[e + Θ(|f|)^{2}] log E[e + Θ(|f|)^{2}] + E[|Df|^{2}ϕ(U)].

16 I. Shigekawa

Now, by (33), it follows that

E[Φ(U)] ≤ KE[e + Θ(|f|)^{2}] log E[e + Θ(|f|)^{2}] + KE[Φ(|Df|^{2})]. Here Φ is the integral of ϕ. Since Φ = Φ_{p/2,β},

Φ(x^{2}) ≤ c_{1}x^{2}ϕ(x^{2})
≤ c_{1}x^{2}(x^{2})^{(p/2)−1} log^{pβ/2}(k + x^{2})
≤ c_{1}x^{p} log^{pβ/2}(k + x^{2})
≤ c_{2}Φ_{p,β/2}(x).

Further

Φ(x^{2} log(e + x^{2})) ≥ c_{3}x^{2} log(e + x^{2})ϕ(x^{2} log(e + x^{2}))
= c_{3}x^{2} log(e + x^{2}){x^{2} log(e + x^{2})}^{(p/2)−1}
× log^{pβ/2}(k + x^{2} log(e + x^{2}))
= c_{3}x^{p} log^{p/2}(e + x^{2}) log^{pβ/2}(k + x^{2} log(e + x^{2}))
≥ c_{4}x^{p} log^{p/2}(e + x) log^{pβ/2}(k + x)
≥ c_{5}x^{p} log^{p(1+β)/2}(k + x)
≥ c_{6}Φ_{p,(β+1)/2}(x)

and

Θ(x)^{2} ≤ x^{2}θ(x)^{2}
≤ x^{2}{x^{2} log(e + x^{2})}^{(p−2)/2} log^{pβ/2}(k + x^{2} log(e + x))
≤ c_{7}x^{p} log^{(p−2)/2}(e + x) log^{pβ/2}(k + x)
≤ c_{7}x^{p} log^{(p+pβ−2)/2}(k + x)
≤ c_{8}Φ_{p,(1+β)/2−(1/p)}(x).

Thus we have eventually obtained

E[Φ_{p,(β+1)/2}(|f|)] ≤ CE[Φ_{p,(1+β)/2−(1/p)}(|f|)] + CE[Φ_{p,β/2}(|Df|)]. This completes the proof. Q.E.D.

If p = 2 and β ≥ 0, the above proof works as well in this case. We only state the result.

Proposition 5.3. For p = 2, β ≥ 0, there exists a positive constant C such that E[Φ_{2,(β+1)/2}(|f|)] ≤ CE[Φ_{2,β/2}(|f|)] + CE[Φ_{p,β/2}(|Df|)]. (35)

In Section 3, we showed that the right hand side of (34) is equivalent to E [Φ_{p,β/2}(\sqrt{1 − L}f)]. Therefore we easily get the following theorem.

Theorem 5.4. For p > 1, β ≥ 0, the following map is continuous:

\sqrt{1 − L}^{−1} : L^{p} log^{pβ} L → L^{p} log^{p(β+1/2)} L. (36)

Recall that the dual space of L^{p} log^{pβ} is L^{q} log^{−qβ} L (see Proposition 4.4). Hence, when 1 < p < 2, the above equation (36) is shown by the duality. By iterating the map \sqrt{1 − L}^{−1}, we can have the continuity of (1 − L)^{−1} from L^{p} log^{pβ} L to L^{p} log^{p(β+1)} L.

References

[ 1 ] R. A. Adams, “Sobolev spaces,” Academic press, New York, 1975. [ 2 ] R. A. Adams, General logarithmic Sobolev inequalities and Orlicz imbed-

dings, J. Funct. Anal., 34 (1979), 292–303. [ 3 ] D. Bakry and P. A. Meyer, Sur les inégalités de Sobolev logarithmiques,

I, II, Lecture Notes in Math., 920 (1982), 138–145, 146–150, Springer-Verlag, Berlin.

[ 4 ] C. Dellacherie and P.-A. Meyer, “Probabilities and potential. B. Theory

of martingales,” North-Holland Publishing Co., Amsterdam, 1982. [ 5 ] G. F. Feissner, Hypercontractive semigroups and Sobolev’s inequality,

Trans. Amer. Math. Soc., 210 (1975), 51–62. [ 6 ] 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.

[ 7 ] 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.

[ 8 ] M. M. Rao and Z. D. Ren, “Theory of Orlicz spaces,” Marcel Dekker,

New York, 1991. [ 9 ] I. Shigekawa, Littlewood-Paley inequality for a diffusion satisfying the

logarithmic Sobolev inequality and for the Brownian motion on a Riemannian manifold with boundary, Osaka J. Math., 39 (2002), 897–930.

[10] I. Shigekawa and N. Yoshida, Littlewood-Paley-Stein inequality for a symmetric diffusion, J. Math. Soc. Japan, 44 (1992), 251–280.

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