Exponential Orlicz Spaces: New Norms and Applications. E.I.Ostrovsky. Department of Mathematics and Computer Science, Ben Gurion University, Israel, Beer - Sheva, 84105, Ben Gurion street, 2, P.O. BOX 61, E - mail: [email protected] Abstract The aim of this paper is investigating of Orlicz spaces with exponential N function and correspondence Orlicz norm: we introduce some new equivalent norms, obtain the tail characterization, study the product of functions in Orlicz spaces etc. We consider some applications: estimation of operators in Orlicz spaces and problem of martingales convergence and divergence. Key words: Orlicz spaces, ∆2 condition, martingale, slowly varying func- tion, absolute continuous norm. Math. Sub. Classification (2000): 47A45, 47A60, 47B10, 18D05. 60F10, 60G10, 60G50. 1. INTRODUCTION. Let (Ω, F, P) be a probability space. Introduce the following set of N Orlicz functions: − LW = N = N(u)= N(W, u) = exp(W (log u)) , u e2 { } ≥ where W is a continuous strictly increasing convex function in domain [2, ) arXiv:math/0406534v1 [math.FA] 25 Jun 2004 / / ∞ such that u limu W (u) = ; here W (u) denotes the left →∞ ⇒ →∞ − ∞ − derivative of the function W. We define the function N(W (u)) arbitrary for the values u [0, e2) but so that N(W (u)) will be continuous convex strictly increasing and∈ such that u 0+ N(W (u)) C(W )u2, → ⇒ ∼ for some C(W ) = const (0, ). For u < 0 we define as usually ∈ ∞ N(W (u)) = N(W ( u )). | | 1 We denote the set of all those N functions as ENF : ENF = N(W ( )) (Exponential N - Functions)− and denote also the correspondence { · } Orlicz space as EOS(W ) = Exponential Orlicz Space or simple EW with Orlicz (or, equally, Luxemburg) norm 1 η L(N)= η L(N(W )) = inf v− (1 + EN(W ( ), vη) , || || || || v>0{ · } where E, D denote the expectation and variance with respect to the prob- ability measure P : Eη = η(ω) P(dω). ZΩ Let us introduce the following important function: ψ(p)= ψ(W, p) = exp(W ∗(p)/p), p 2, (1) ≥ where def W ∗(p) = sup(pz W (z)) z 2 − − ≥ is the Young - Fenchel transform of W. The function p p log ψ( )= W ∗( ) → · · is continuous convex increasing and such that p p log ψ(p) . ↑∞ ⇒ ↑∞ We denote the set of all those functions ψ = ψ(W, p) = exp(W ∗(p)/p) by the symbol Ψ : { } { } { } Ψ= ψ(W, p) = exp(W ∗(p)/p) . ∪W { } { } Inversely, W ( ) may be constructed by means of ψ : · W (p)=(p log ψ(p))∗ (theorem of Fenchel - Moraux). Definition. We introduce the so - called G(ψ) norm for arbitrary func- tion ψ( ) Ψ and the correspondent G(ψ) space: η G(ψ) iff · ∈ ∈ def η G(ψ) = sup η p/ψ(p) < , || || p 2 | | ∞ ≥ where denotes the classical L norm: |·|p p η = E1/p η p, p 1. | |p | | ≥ 2 2 2 In particular, η 2 = Dη + E η. It is easy to prove (see, for example, [3], p. 373; [12], p. 67)| | that G(ψ) is some (full) Banach space. Note that if there exists a family of measurable functions ηα , α so that { } ∈ A sup ηα p < , α | | ∞ then there exists a G(ψ) space, ψ Ψ such that α η G(ψ). For − ∈ ∀ ⇒ α ∈ example, put ψ(p) = sup ηα p. α | | Remark 1. In this paper the letters C,Cj will denote positive finite various constants which may differ from one formula to another and which do not depend on the essential parameters: x, u, p, z, λ. 2. MAIN RESULTS. Theorem 1. The norms η L(N(W )) and η G(ψ) are equivalent. Further, η =0, η G(ψ) iff C,C|| || (0, ) u>|| ||2C 6 ∈ ∃ 1 ∈ ∞ ⇒ ∀ P( η >u) C exp( W (log(u/C))). (2) | | ≤ 1 − Remark 2. This result is a little generalization of [3]; see also [16], p.305, as long as we do not suppose that the function η = η(ω), ω Ω to be exponential integrable: ∈ λ> 0 E exp(λ η ) < ∃ ⇒ | | ∞ (so - called Kramer condition). Proof. a). Suppose η EW, η = 0. Then for some C (0, ) E exp(W (log C η )) < . Proposition∈ 6 (2) follows from Chebyshev∈ inequal-∞ ⇒ ity. | | ∞ b). Inversely, let us assume that η(ω) is a measurable function such that P( η >u) exp( W (log u)), u e2. | | ≤ − ≥ Then, by virtue of properties W ( ) we have: · EN(W ( η /e2)) = N(W, η(ω) /e2) P(dω) | | ZΩ | | ≤ ∞ 2 C1 + exp((W ( η /e )) P(dω) C1+ ek< η ek+1 | | ≤ kX=2 Z | |≤ 3 ∞ ∞ exp((W (k 1)) P( η >k) C + exp(W (k) W (k + 1)) < . − | | ≤ 2 − ∞ kX=2 kX=2 Hence η L(N(W )) < . c) Let|| || η G(ψ);∞ without loss of generality we can assume that ∈ η G(ψ)=1. We deduce: E η p ψp(p), || || | | ≤ p p 2 P( η > x) x− ψ (p) = exp( p log x + p log ψ(p)), x>e , | | ≤ − and, after the minimization over p : x e2 ≥ ⇒ P( η > x) exp sup(p log x p log ψ(p)) = exp( W (log x)). | | ≤ − p 2 − ! − ≥ d) Let us next assume that P( η > x) exp( W (log x)),x>e2. We have: | | ≤ − p p ∞ p 1 E η C + px − exp( W (log x) dx = 2 | | ≤ Ze − Cp + p ∞ exp(py W (y)) dy, p 2. Z2 − ≥ Using Laplace’s method and theorem of Fenchel - Moraux we get: p p p p E η C + C1 exp sup(py W (y)) = C + | | ≤ y 2 − ! ≥ p p p p p C exp (W ∗(p)) = C + C exp(p log ψ(p)) C ψ (p). (3) 2 2 ≤ 3 Finally, η G(ψ) C < . || || ≤ 3 ∞ Remark 3. If conversely P( η > x) exp( W (log x)), x e2, | | ≥ − ≥ then for sufficiently large values of p; p p = p (W ) 2 ≥ 0 0 ≥ η C (W )ψ(p), C (0, ). | |p ≥ 0 0 ∈ ∞ For arbitrary Orlicz spaces L(N),EOS(W )= EW,G(ψ) we denote cor- respondently L(N)0, EW 0,G0(ψ) a closure in L(N),EW,G(ψ) norm the set 4 of all bounded measurable functions. It is known (see, for example, [15], p.75, [10], p.138) that in our conditions (P(Ω) = 1, N(W (u)) Cu2 etc.) ∼ LN(W )0 = η : k > 0 EN(W, η /k) < . { ∀ ⇒ | | ∞ } Theorem 2. Let ψ Ψ. We assert that η EW 0, or, equally, η ∈ ∈ ∈ G0(ψ) if and only if lim η p/ψ(p)=0. (4) p →∞ | | Proof. It is sufficient by virtue of theorem 1 to consider only the case of the G(ψ) spaces. 0 0 1. Denote GB (ψ)= η : limp η p/ψ(p)=0. Let η G (ψ), η =0. →∞ Then for arbitrary δ = const{ > 0 there| | exists a constant} K∈ (0, ) such6 that ∈ ∞ η ηI( η K) G(ψ) δ/2, || − | | ≤ || ≤ where for any event A F I(A)=1,ω A, I(A) = 0 if ω / A. Since ηI( η K) K, we deduce∈ ∈ ∈ | | | ≤ | ≤ ηI( η K) /ψ(p) K/ψ(p). | | | ≤ |p ≤ Using the triangular inequality we obtain for sufficiently large values p : p>p0(δ)= p0(δ, K): η /ψ(p) δ/2+ K/ψ(p) < δ, | |p ≤ as long as ψ(p) as p . Therefore, G0(ψ) GB0(ψ). (The set GB→∞0(ψ) is a closed→∞ subspace of G(ψ) with⊂ respect to the G(ψ) norm and contains all bounded random variables). 2. Conversely, assume η GB0(ψ). Let us denote η(K) = ηI( η > ∈ | | K), K (0, ). We deduce: ∈ ∞ Q 2 lim η(K) Q =0. K ∀ ≥ ⇒ →∞ | | Further, η(K) G(ψ) = sup η(K) p/ψ(p) max η(K) p/ψ(p)+ || || p 2 | | ≤ p [2,Q] | | ≥ ∈ def sup η(K) p/ψ(p) = σ1 + σ2; p>Q | | 5 σ2 = sup η(K) p/ψ(p) sup ( η p/ψ(p)) δ/2 p>Q | | ≤ p Q | | ≤ ≥ for sufficiently large Q as long as η GB0(ψ). Further, ∈ σ1 max η(K) p/ψ(2) η(K) Q/ψ(2) δ/2 p [2,Q] ≤ ∈ | | ≤| | ≤ for sufficiently large K = K(Q). Therefore, lim η(K) G(ψ)=0, η G0(ψ). K →∞ || || ⇒ ∈ Hence GB0(ψ) G0(ψ). ⊂ Let now η , a be some family of functions from the G0(ψ) space. { a} ∈A Theorem 3. Let ψ Ψ. In order to a family ηa of a function belonging to the LG(ψ) space∈ has the uniform absolute continuous{ } norm in this space, briefly: η UCN(G(ψ)), it is necessary and sufficient: { a} ∈ lim sup ηa p/ψ(p)=0. (5) p a | | →∞ ∈A Proof. Recall that by definition η UCN(G(ψ)) if { a} ∈ lim sup sup ηaI(V ) G(ψ)=0. δ 0+ P a → V : (V )<δ || || 1. Let the condition (5) be satisfied, then there exists a function ǫ = ǫ(p) 0 as p such that a and p 2 ηa p ǫ(p)ψ(p). It follows→ that for→∞ all Q 2 the family∀ ∈A of functions∀ ≥ η⇒|Q is| uniform≤ inte- ≥ | a| grable. Let V be an arbitrary measurable set: V F with sufficiently small measure: P(V ) δ, δ (0, 1/2). We have: ∈ ≤ ∈ sup ηaI(V ) G(ψ) sup max ηa p/ψ(p)+ a a p Q || || ≤ ≤ | | def sup sup ηa p/ψ(p) = s1 + s2; a p>Q | | s2 sup ǫ(p) 0, Q ; ≤ p Q → →∞ ≥ s1 sup ηI(V ) Q/ψ(2) 0, δ 0+ . ≤ a | | → → Hence the family η has the uniform absolute continuous norms in our { a} Orlicz space G(ψ).