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Note on functions satisfying the integral Hölder condition. (English) Zbl 0863.26006
For \(q\geq 1\) and a 1-periodic function \(f:\mathbb R \to \mathbb R\cup \{-\infty,\infty\}\) define \(|f|_q=[\int_0^1|f(x)|^q\text{d}x]^{1/q}\) and in the case \(|f|_q<\infty\) its modulus of continuity \(\omega(f,t)_q=\sup_{|h|\leq t}[\int_0^1|f(x+h) - f(x)|^q\text{d}x]^{1/q}\). Two such functions are equivalent if they differ almost everywhere by a constant. Given a modulus of continuity \(\omega\), \(H^\omega_q\) is the set of classes of equivalent functions \(f\) for which there is a nonnegative constant \(c\) such that \(\omega(f,t)_q\leq c\omega(t)\). It is shown that if \(\liminf_{t\to 0}\omega(t)/t=\infty\) then the set \(\{f\in H^\omega_q:\text{var}(f) <\infty\}\) is of the first category in \(H^\omega_q\) and that if \(\liminf_{t\to 0}\omega(t)/t<\infty\) then \(\{f\in H^\omega_q:\text{var}(f) <\infty\}=H^\omega_q\).
MSC:
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A45 Functions of bounded variation, generalizations
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