zbMATH — the first resource for mathematics

Characterizations of logarithmic Besov spaces in terms of differences, Fourier-analytical decompositions, wavelets and semi-groups. (English) Zbl 1356.46026
Summary: We work with Besov spaces \(\mathbf{B}_{p, q}^{0, b}\) defined by means of differences, with zero classical smoothness and logarithmic smoothness with exponent \(b\). We characterize \(\mathbf{B}_{p, q}^{0, b}\) by means of Fourier-analytical decompositions, wavelets and semi-groups. We also compare those results with the well-known characterizations for classical Besov spaces \(B_{p, q}^s\).

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46M35 Abstract interpolation of topological vector spaces
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42B25 Maximal functions, Littlewood-Paley theory
Full Text: DOI
[1] Almeida, A., Wavelet bases in generalized Besov spaces, J. Math. Anal. Appl., 304, 198-211, (2005) · Zbl 1108.42007
[2] Bennett, C.; Rudnick, K., On Lorentz-Zygmund spaces, Dissertationes Math., 175, 1-72, (1980)
[3] Bennett, C.; Sharpley, R., Interpolation of operators, (1988), Academic Press Boston · Zbl 0647.46057
[4] Besov, O. V., On spaces of functions of smoothness zero, Sb. Math., 203, 1077-1090, (2012) · Zbl 1259.46027
[5] Butzer, P. L.; Berens, H., Semi-groups of operators and approximation, (1967), Springer New York · Zbl 0164.43702
[6] Caetano, A. M.; Gogatishvili, A.; Opic, B., Sharp embeddings of Besov spaces involving only logarithmic smoothness, J. Approx. Theory, 152, 188-214, (2008) · Zbl 1161.46017
[7] Caetano, A. M.; Gogatishvili, A.; Opic, B., Embeddings and the growth envelope of Besov spaces involving only slowly varying smoothness, J. Approx. Theory, 163, 1373-1399, (2011) · Zbl 1238.46024
[8] Caetano, A. M.; Leopold, H.-G., On generalized Besov and Triebel-Lizorkin spaces of regular distributions, J. Funct. Anal., 264, 2676-2703, (2013) · Zbl 1291.46031
[9] Cobos, F.; Domínguez, O., Embeddings of Besov spaces of logarithmic smoothness, Studia Math., 223, 193-204, (2014) · Zbl 1325.46039
[10] Cobos, F.; Domínguez, O., Approximation spaces, limiting interpolation and Besov spaces, J. Approx. Theory, 189, 43-66, (2015) · Zbl 1326.46020
[11] Cobos, F.; Domínguez, O., On Besov spaces of logarithmic smoothness and Lipschitz spaces, J. Math. Anal. Appl., 425, 71-84, (2015) · Zbl 1320.46028
[12] F. Cobos, O. Domínguez, On the relationship between two kinds of Besov spaces with smoothness near zero and some other applications of limiting interpolation, J. Fourier Anal. Appl. http://dx.doi.org/10.1007/s00041-015-9456-6.
[13] Cobos, F.; Fernandez, D. L., Hardy-Sobolev spaces and Besov spaces with a function parameter, (Function Spaces and Applications, Lecture Notes in Math., vol. 1302, (1988), Springer Berlin), 158-170
[14] Cobos, F.; Segurado, A., Description of logarithmic interpolation spaces by means of the J-functional and applications, J. Funct. Anal., 268, 2906-2945, (2015) · Zbl 1335.46018
[15] DeVore, R. A.; Riemenschneider, S. D.; Sharpley, R. C., Weak interpolation in Banach spaces, J. Funct. Anal., 33, 58-94, (1979) · Zbl 0433.46062
[16] Ditzian, Z.; Ivanov, K. G., Strong converse inequalities, J. Anal. Math., 61, 61-111, (1993) · Zbl 0798.41009
[17] Domínguez, O., Tractable embeddings of Besov spaces into small Lebesgue spaces, Math. Nachr., (2016) · Zbl 1361.46026
[18] Evans, W. D.; Opic, B., Real interpolation with logarithmic functors and reiteration, Canad. J. Math., 52, 920-960, (2000) · Zbl 0981.46058
[19] Farkas, W.; Leopold, H.-G., Characterizations of function spaces of generalized smoothness, Ann. Mat. Pura Appl., 185, 1-62, (2006)
[20] Fehér, F.; Grässler, G., On an extremal scale of approximation spaces, J. Comput. Anal. Appl., 3, 95-108, (2001) · Zbl 1033.46021
[21] Haroske, D. D.; Moura, S. D., Continuity envelopes of spaces of generalized smoothness, entropy and approximation numbers, J. Approx. Theory, 128, 151-174, (2004) · Zbl 1055.46020
[22] Kalyabin, G. A.; Lizorkin, P. I., Spaces of functions of generalized smoothness, Math. Nachr., 133, 7-32, (1987) · Zbl 0636.46033
[23] Karadzhov, G. E.; Milman, M.; Xiao, J., Limits of higher-order Besov spaces and sharp reiteration theorems, J. Funct. Anal., 221, 323-339, (2005) · Zbl 1102.46022
[24] Moura, S. D., Function spaces of generalized smoothness, Dissertationes Math., 398, 1-88, (2001)
[25] Nikolskiĭ, S. M., Approximation of functions of several variables and imbedding theorems, (1975), Springer Berlin
[26] Petrushev, P. P.; Popov, V. A., Rational approximation of real functions, Encyclopedia of Mathematics and Its Applications, vol. 28, (1987), Cambridge Univ. Press Cambridge · Zbl 0644.41010
[27] Pietsch, A., Approximation spaces, J. Approx. Theory, 32, 115-134, (1981) · Zbl 0489.47008
[28] Stein, E. M., Singular integrals and differentiability properties of functions, (1970), Princeton University Press Princeton · Zbl 0207.13501
[29] Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, (1993), Princeton University Press Princeton · Zbl 0821.42001
[30] Trebels, W.; Westphal, U., Characterizations of K-functionals built from fractional powers of infinitesimal generators of semigroups, Constr. Approx., 19, 355-371, (2003) · Zbl 1030.41021
[31] Triebel, H., Interpolation theory, function spaces, differential operators, (1978), North-Holland Amsterdam · Zbl 0387.46033
[32] Triebel, H., Theory of function spaces, (1983), Birkhäuser Basel · Zbl 0546.46028
[33] Triebel, H., Theory of function spaces II, (1992), Birkhäuser Basel · Zbl 0778.46022
[34] Triebel, H., Theory of function spaces III, (2006), Birkhäuser Basel · Zbl 1104.46001
[35] Triebel, H., Function spaces and wavelets on domains, (2008), European Math. Soc. Publishing House Zürich · Zbl 1158.46002
[36] Triebel, H., Bases in function spaces, sampling, discrepancy, numerical integration, (2010), European Math. Soc. Publishing House Zürich · Zbl 1202.46002
[37] Triebel, H., Hybrid function spaces, heat and Navier-Stokes equations, (2014), European Math. Soc. Publishing House Zürich
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.