Embeddings between weak Orlicz martingale spaces

作者：Jiao Yong^*

来源：Journal of Mathematical Analysis and Applications, 2011, 378(1): 220-229.

DOI：10.1016/j.jmaa.2010.12.001

• Weak Orlicz spaces
• Martingale inequalities
• Uniformly convex (smooth) spaces

Let Phi be an increasing and convex function on [0, infinity) with Phi(0) = 0 satisfying that for any alpha > 0, there exists a positive constant C(alpha) such that Phi(alpha t) <= C(alpha)Phi(t), t > 0. Let wL(Phi) denote the corresponding weak Orlicz space. We obtain some embeddings between vector-valued weak Orlicz martingale spaces by establishing the wL(Phi)-inequalities for martingale transform operators with operator-valued multiplying sequences. These embeddings are closely related to the geometric properties of the underlying Banach space. In particular, for any scalar valued martingale f = (f(n))(n >= 1). we claim that
parallel to sup(n)vertical bar f(n)vertical bar parallel to(wL Phi) approximate to parallel to (Sigma(infinity)(n = 1) vertical bar df(n)vertical bar(2))(1/2)parallel to wL(Phi)
where df(n) = f(n) - f(n-1) and "similar to" only depends on Phi.

• 出版日期2011-6-1
• 单位中南大学