Meyer’s inequality in the framework of Orlicz space^{∗}

Ichiro Shigekawa^{†} (Kyoto University)

Let (B, H, μ) be an abstract Wiener space and L be the Ornstein-Uhlenbeck operator. The following norm equivalence is well-known as Meyer’s equivalence [4]:

∥\sqrt{1 − L}f∥_{p} ∼ ∥∇f∥_{p} + ∥f∥_{p}.  (1)

Here, ∇ denotes the H-derivative in the sense of Malliavin, ∥ · ∥_{p} denotes the L^{p} norm (1 < p < ∞) and the notation A ∼ B means that cA ≤ B ≤ CA for some constants c > 0 and C > 0 which is independent of f. We extends this equivalence to the Orlicz spaces.

Orlicz spaces

Let ϕ be a non-negative non-decreasing right continuous function on [0, ∞) with ϕ(0) = 0. Integrating it, we define the function Φ:

Φ(x) = ∫_{0}^{x} ϕ(t)dt, x ≥ 0.  (2)

This type of function is called an N-function (nice Young function). For measurable functions f on a measure space (M, m), define a norm ∥ ∥_{Φ} by

∥f∥_{Φ} := inf{λ > 0 ; ∫_{M} Φ(|f|/λ) ≤ 1}.  (3)

We denote the set of all functions satisfying ∥f∥_{Φ} < ∞ by L^{Φ} and call it a Orlicz space.

Let us introduce some classes of N-functions. Definition 1. For a constant α > 0, define L(α), U(α) as follows:

(i) Φ ∈ L(α) ⟺^{def} αΦ(x) ≤ xΦ′(x), ∀x > 0.

(ii) Φ ∈ U(α) ⟺^{def} xΦ′(x) ≤ αΦ(x), ∀x > 0.

\underline{Main result}

Our main result is the following.

[figure]

Theorem 2. We assume that there exist constants 1 < α < β such that Φ ∈ L(α) ∩ U(β). Further we assume that ϕ is either convex or concave. Then it holds that

∥f∥_{Φ} + ∥∇f∥_{Φ} ∼ ∥\sqrt{1 − L}f∥_{Φ}.  (4) [figure]

To show this theorem, we adopt a probabilistic method. To do this, we recall that many inequalities, e.g., Doob’s inequality, Burkholder’s inequality, etc., are all established in the framework of Orlicz space (see e.g, [3]). Our methods relies on Littelwood-Paley inequalities.

∗November 6–8, 2002, Workshop on “Stochastic analysis in infinite dimensional spaces” at RIMS.

†E-mail: ichiro@kusm.kyoto-u.ac.jp URL: http://www.kusm.kyoto-u.ac.jp/~ichiro

Littlewood-Paley inequalities

Let {T_{t}} be an Ornstein-Uhlenbeck semigroup. For α ≥ 0, define

T_{t}^{(α)} = e^{−αt}T_{t}.  (5)

Then the generator of \{T_{t}^{(α)}\} is L − α． We denote the semigroup generated by −\sqrt{α − L} by {Q_{t}^{(α)}}. {Q_{t}^{(α)}} can be obtained as a subordination of {T_{t}^{(α)}}. Using {Q_{t}^{(α)}}, we define the Littlewood-Paley G-functions as follows:

G^{→}f(x) = {∫_{0}^{∞} t|∂_{t}Q_{t}^{(α)}(x, f)|^{2}dt}^{1/2},  (6)
G^{↑}f(x) = {∫_{0}^{∞} t|∇Q_{t}^{(α)}(x, f)|_{H}^{2}dt}^{1/2},  (7)
Gf(x) = {∫_{0}^{∞} t(|∂_{t}Q_{t}^{(α)}(x, f)|^{2} + |∇Q_{t}^{(α)}(x, f)|_{H}^{2})dt}^{1/2}.  (8)

We can show the following theorem.

[figure]

Theorem 3. Assume that there exist constants 1 < α < β such that Φ ∈ L(α)∩U(β). Further we assume that ϕ is either convex or concave. Then we have

∥Φ(Gf)∥_{1} ≲ ∥Φ(f)∥_{1},  (9)
∥Φ(f)∥_{1} ≲ ∥Φ(G^{→}f)∥_{1}.  (10) [figure]

Using this theorem, we cans show Theorem 2 by a standard argument.

References

[1] R. A. Adams, “Sobolev spaces.” Academic press, New York, 1975.

[2] 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.

[3] C. Dellacherie and P.-A. Meyer, “Probabilities and potential. B. Theory of martingales,” North-Holland Publishing Co., Amsterdam, 1982.

[4] 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.

[5] E. M. Stein, “Topics in harmonic analysis, related to Littlewood-Paley theory,” Annals of Math. Studies, 63, Princeton Univ. Press, 1974.