Abstract
The so-called sharp Marchaud inequality and some converse of it, as well as the Ulyanov and Kolyada inequalities are equivalent to some embeddings between Besov and potential spaces. Peetre’s (modified) K-functional, its characterization via moduli of smoothness (also of fractional order), and limit cases of the Holmstedt formula are essentially used.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bennett C., Sharpley R.: Interpolation of Operators. Academic Press, Boston (1988)
Bergh J., Löfström J.: Interpolation Spaces. An Introduction. Springer-Verlag, Berlin (1976)
ButzerP.L. Berens H.: Semi-Groups of Operators and Approximation. Springer-Verlag, Berlin (1967)
Dai F., Ditzian Z.: Littlewood-Paley theory and a sharp Marchaud inequality. Acta Sci. Math. (Szeged) 71, 65–90 (2005)
Dai F., Ditzian Z., Tikhonov S.: Sharp Jackson inequalities. J. Approx. Theory 151, 86–112 (2008)
DitzianZ. Tikhonov S.: Ulyanov and Nikolskiĭ-type inequalities. J. Approx. Theory 133, 100–133 (2005)
Ditzian Z., Tikhonov S.: Moduli of smoothness of functions and their derivatives. Studia Math. 180, 143–160 (2007)
Goldman M.L.: A criterion for the embedding of different metrics for isotropic Besov spaces with arbitrary moduli of continuity. Trudy Mat. Inst. Steklov. 201, 186–218 (1992) Proc. Steklov Inst. Math. 1994, no. 2, 155–181 (201)
Herz C.S.: Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms. J. Math. Mech. 18, 283–323 (1968/69)
Hörmander L.: Estimates for translation invariant operators in L p spaces. Acta Math. 104, 93–139 (1960)
Kolyada V. I.: On relations between moduli of continuity in different metrics. Trudy Mat. Inst. Steklov. 181, 117–136 (1988) 270; Proc. Steklov Inst. Math. 1989, Issue 4, 127–148
P. Nilsson, Reiteration theorems for real interpolation and approximation spaces, Ann. Mat. Pura Appl. (4) 132 (1982), 291–330 (1983).
Peetre J.: Espaces d’interpolation et théorème de Soboleff. Ann. Inst. Fourier (Grenoble) 16, 279–317 (1966)
B. Simonov and S. Tikhonov, Weak inequalities of smoothness, K-functionals and embedding theorems, manuscript 23 pp.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.
Trebels W., Westphal U.: K-functionals related to semigroups of operators. Rend. Circ. Mat. Palermo (2) Suppl.(76), 603–620 (2005)
Triebel H.: Theory of Function Spaces. Birkhäuser, Basel (1983)
Wilmes G.: On Riesz-type inequalities and K-functionals related to Riesz potentials in \({{\mathbb R}^N}\). Numer. Funct. Anal. Optim. 1, 57–77 (1979)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Trebels, W. Inequalities for moduli of smoothness versus embeddings of function spaces. Arch. Math. 94, 155–164 (2010). https://doi.org/10.1007/s00013-009-0078-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-009-0078-4