Moduli of p-continuity provide a measure of fractional smoothness of functions via p-variation. We prove a sharp estimate of the modulus of p-continuity in terms of the modulus of q-continuity (1<p<q<∞).
[1] Andrienko, V. A. 1969 Necessary conditions for imbedding the function classes Mat. Sb. (N.S.) 78 120 280–300 English transl.: Math. USSR Sb., 7 (1969), 273–292.
[2] Bourdaud, G. Lanza de Cristoforis, M. Sickel, W. 2005 Superposition operators and functions of bounded p-variation II Nonlinear Anal. 62 483–517 .
[3] DeVore, R. A. Lorentz, G. G. 1993 Constructive Approximation Springer-Verlag Berlin, Heidelberg.
[4] Golubov, B. I. 1968 Criteria for the compactness of sets in spaces of functions of bounded generalized variation Izv. Akad. Nauk Armjan. SSR Ser. Mat. 3 409–416 in Russian.
[5] Hardy, G. H. Littlewood, J. E. 1928 A convergence criterion for Fourier series Math. Z. 28 612–634 .
[6] Kolyada, V. I. 1988 On relations between moduli of continuity in different metrics Trudy Mat. Inst. Steklov 181 117–136 English transl. in Proc. Steklov Inst. Math. (1989), 127–147.
[7] Kolyada, V. I. Lind, M. 2009 On functions of bounded p-variation J. Math. Anal. Appl. 356 582–604 .
[8] Love, E. R. 1951 A generalization of absolute continuity J. Lond. Math. Soc. 26 1–13 .
[9] Musielak, J. Orlicz, W. 1959 On generalized variation (I) Studia Math. 18 11–41.
[10] Oskolkov, K. I. 1977 Approximation properties of integrable functions on sets of full measure Mat. Sb. 103 563–589 English transl. in Math. USSR Sb., 32 (1977), 489–514.
[11] Peetre, J. 1966 Espaces d'interpolation et théorème de Soboleff Ann. Inst. Fourier (Grenoble) 16 279–317 .
[12] Terehin, A. P., Approximation of functions of bounded p-variation, Izv. Vyssh. Uchebn. Zaved. Mat. (1965), 171–187 (in Russian).
[13] Terehin, A. P. 1967 Integral smoothness properties of periodic functions of bounded p-variation Mat. Zametki 2 289–300 in Russian.
[14] Terehin, A. P. 1972 Functions of bounded p-variation with a given modulus of continuity Mat. Zametki 53 523–530 English transl. in Math. Notes, 12 (1972), 751–755.
[15] Timan, A. F. 1994 Theory of Approximation of Functions of a Real Variable Dover Publications, Inc. New York.
[16] Ul'yanov, P. L. 1967 Absolute and uniform convergence of Fourier series Mat. Sb. 72 193–225.
[17] Ul'yanov, P. L. 1968 Imbeddings of certain function classes Izv. Akad. Nauk SSSR, Ser. Mat. 32 649–686 English transl. in Math. USSR Izv., 2 (1968).
[18] Ul'yanov, P. L. 1970 Imbedding theorems and relations between best approximation in different metrics Mat. Sb. 81 123 104–131 English transl. in Math. USSR Sb., 70 (1970).
[19] Wiener, N. 1924 The quadratic variation of a function and its Fourier coefficients J. Math. Phys. 3 72–94.