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Keywords:
integral modulus of continuity; variation of a function
integral modulus of continuity; variation of a function
Summary:
Given a modulus of continuity $\omega$ and $q \in[1, \infty[ $ then $H_q^\omega$ denotes the space of all functions $f$ with the period $1$ on $\R$ that are locally integrable in power $q$ and whose integral modulus of continuity of power $q$ (see(1)) is majorized by a multiple of $ \omega$. The moduli of continuity $ \omega$ are characterized for which $H_q^\omega$ contains "many" functions with infinite "essential" variation on an interval of length $1$.
References:
[1] O. Kováčik: A necessary condition of embedding of $H_p^{\omega}$ into the space of functions with bounded variations. Izvestija vysšich učebnych zaveděnij Matematika 10 (1983), 26-28. (In Russian.)
[2] W. Orlicz: Application of Baire's category method to certain problems of mathematical analysis. Wiadomości Matematyczne XXIV (1982), 1-15. (In Polish.) MR 0705608
[3] J. C. Oxtoby: Mass und Kategorie. Springeг-Verlag, 1971. MR 0393404 | Zbl 0217.09202
[4] A. F. Timan: Theory of Approximation of function of Real Variable. Moskva, 1960. (In Russian.)
[5] G. H. Hardy J. E. Littlewood: Some properties of fractional integrals I, II. Math. Z. 27 (1928), 565-606; З4 (1932), 403-439. MR 1544927
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