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H p (p ≥ 1) Martingales

In: Martingale Spaces and Inequalities

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  • Ruilin Long

Abstract

In Chapter 1, we have introduced the concept of martingale and one kind of martingale spaces, i.e. L P , and just mentioned two important operators defined on martingales, i.e. maximal operator M and square function operator S. In this chapter, we will study several other martingale spaces, among which Hardy space H p is the most important one. Among other things, in the chapter, we will establish the L P (1 ≤ p ≤ ∝) equivalence between M and S (i.e. Davis’ inequality and Burkholder-Gundy’s inequality); establish the Fefferman’s H 1 BMO duality by two different proofs (one of which is via atomic decomposition); discuss the weak compactness of subsets and the convergence of sequences in H 1; and as a comparison, we will introduce two versions of H p , i.e. h p and P p .

Suggested Citation

  • Ruilin Long, 1993. "H p (p ≥ 1) Martingales," Springer Books, in: Martingale Spaces and Inequalities, chapter 2, pages 33-79, Springer.
    • Handle: RePEc:spr:sprchp:978-3-322-99266-6_2
    DOI: 10.1007/978-3-322-99266-6_2
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