On the Martingale Transforms in expL~p and Its Application

摘要

Let 1 < a < β < +∞ and 1 < β < γ < +∞. Let {v_n} be an adapted process of {F_n}, and {f_n} be a martingale about {F_n}. {v_n} is multiplier of martingale transform to be of type (exp L~a, exp L~β ). Let Φ(t) be continuous nonnegative strictly increasing convex function defined on [0, + ∞), satisfying the △~2 -condition such that there is a constant c_1 >1 such that Φ~(_1)(t)[ln(l + t)]~(1/α) is non-increasing in [c_1, + ∞), and ψ(t) be continuous nonnegative strictly increasing function defined on [0, + ∞), satisfying Φ~(-1) (t)[ln(1 + u)]δ ≤ Kψ~(-1)(t) for all t>c_2, where K>0 and c_2 > 1 are constants and δ = max {1/ β-1/α, 1/γ), then transform of multiplier {v_n} is to be type (L~ψ,L~Φ): ||T_vf||_(LΦ) ≤ C||f||_(LΦ), where C is a constant.