Abstract
We obtain pointwise and integral type estimates of higher-order partial moduli of continuity in C via partial derivatives. Also, a Gagliardo–Nirenberg type inequality for partial derivatives in a fixed direction is proved. Our methods enable us to study the case when different partial derivatives belong to different spaces, including the space L 1.
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References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, New York (1988)
Besov, O.V., Il’in, V.P., Nikol’skiĭ, S.M.: Integral Representation of Functions and Imbedding Theorems, vols. 1–2. Winston, Washington (1978)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)
DeVore, R., Riemenschneider, S., Sharpley, R.: Weak interpolation in Banach spaces. J. Funct. Anal. 33, 58–94 (1979)
Hardy, G.H., Littlewood, J.E.: A convergence criterion for Fourier series. Math. Z. 28, 612–634 (1928)
Herz, C.S.: Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms. J. Math. Mech. 18, 283–324 (1968)
Kolyada, V.I.: On relations between moduli of continuity in different metrics. Trudy Mat. Inst. Steklov. 181, 117–136 (1988) (in Russian); English transl. in Proc. Steklov Inst. Math. 4, 127–148 (1989)
Kolyada, V.I.: On an embedding of Sobolev spaces. Mat. Zametki 54, 48–71 (1993); English transl. in Math. Notes 54, 908–922 (1993)
Kolyada, V.I.: Inequalities of Gagliardo–Nirenberg type and estimates for the moduli of continuity. Russ. Math. Surv. 60(6), 1147–1164 (2005)
Kolyada, V.I., Pérez, F.J.: Estimates of difference norms for functions in anisotropic Sobolev spaces. Math. Nachr. 267, 46–64 (2004)
Leindler, L.: Generalization of inequalities of Hardy and Littlewood. Acta Sci. Math. (Szeged) 31, 279–285 (1970)
Nikol’skiĭ, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems. Springer, Berlin (1975)
Pełczyński, A., Wojciechowski, M.: Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm. Stud. Math. 107(1), 61–100 (1993)
Poornima, S.: An embedding theorem for the Sobolev space W 1,1. Bull. Sci. Math. (2) 107(3), 253–259 (1983)
Tartar, L.: Imbedding theorems of Sobolev spaces into Lorentz spaces. Bolletino U.M.I. 8(1-B), 479–500 (1998)
Timan, A.F.: Theory of Approximation of Functions of a Real Variable. Pergamon, Oxford (1963)
Triebel, H.: The Structure of Functions. Birkhäuser, Basel (2001)
Ziemer, W.P.: Weakly Differentiable Functions. Springer, New York (1989)
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Communicated by Ronald A. DeVore.
Research of the second author was partially supported by Grant MTM2009-12740-C03-03 of the D.G.I. of Spain.
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Kolyada, V.I., Pérez Lázaro, F.J. Inequalities for Partial Moduli of Continuity and Partial Derivatives. Constr Approx 34, 23–59 (2011). https://doi.org/10.1007/s00365-010-9088-5
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DOI: https://doi.org/10.1007/s00365-010-9088-5
Keywords
- Moduli of continuity
- Gagliardo–Nirenberg type inequalities
- Sobolev spaces
- Besov norms
- Embeddings
- Rearrangements