DOUBLE Φ-INEQUALITIES FOR BANACH-SPACE-VALUED MARTINGALES

doi:10.1016/S0252-9602(12)60129-5

Abstract

Abstract:

Let B be a Banach space, Φ₁, Φ₂ be two generalized convex Φ-functions and ψ₁, ψ₂ the Young complementary functions of Φ₁, Φ₂respectively with
∫^t_t₀ψ₂(s)/s ds ≤ c₀ ψ₁(c₀t) (t > t0)
for some constants c₀ > 0 and t₀ > 0, where ψ₁ and ψ₂ are the left-continuous derivative functions of ψ₁ and ψ₂, respectively. We claim that: (i) If B is isomorphic to a p-uniformly smooth space (or q-uniformly convex space, respectively), then there exists a constant c > 0 such that for any B-valued martingale f = (f_n)_n≥0,

||f *||_Φ1 ≤ c||S^(p)(f)||_Φ2 (or ||S^(q)(f)||_Φ1 ≤ c||f *||_Φ2 , respectively),

where f * and S^(p)(f) are the maximal function and the p-variation function of f respec-tively; (ii) If B is a UMD space, Tvf is the martingale transform of f with respect to v = (v_n)_n≥0 (v* ≤ 1), then ||(T_vf )||_Φ1 ≤ c||f *||_Φ2 .

Key words: martingale, convex Φ-inequality, martingale transform, weighted average

CLC Number:

60G42

WANG Ying-Zhan, ZHANG Chao, HOU You-Liang. DOUBLE Φ-INEQUALITIES FOR BANACH-SPACE-VALUED MARTINGALES[J].Acta mathematica scientia,Series B, 2012, 32(4): 1627-1636.

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DOUBLE Φ-INEQUALITIES FOR BANACH-SPACE-VALUED MARTINGALES

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