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Introduction. Let f (x) be a real measurable function defined in the interval (0, 2) we write f (x) L^p (p>0) when |f (x) |^p is integrable in (0, 2), and f (x) L^*k (a>0) when (x) |^k (1+f^ (x) ) is integrable in (0, 2). L^*1 is the function class which was introduced by A. Zygmund
Shigeki Yano (Sat,) studied this question.