Real-Variable Theory of Musielak-Orlicz Hardy Spaces by Dachun Yang Yiyu Liang & Luong Dang Ky

Author:Dachun Yang, Yiyu Liang & Luong Dang Ky
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

It is easy to see that S( f)(x) ≤ g λ ∗( f)(x) for all and , which, together with Theorem 7.3.5, immediately implies the following conclusion.

Proposition 7.3.8

Let φ be as in Definition 1.1.4 and λ ∈ (1,∞). If vanishes weakly at infinity and , then and

with C being a positive constant independent of f.

Next we consider the boundedness of g λ ∗ on . To this end, we need to introduce the following variant of the Lusin area function S. For all α ∈ (0, ∞), and , let

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