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Embeddings of Besov spaces of logarithmic smoothness. (English) Zbl 1325.46039
Summary: This paper deals with Besov spaces of logarithmic smoothness $$B_{p,r}^{0,b}$$ formed by periodic functions. We study embeddings of $$B_{p,r}^{0,b}$$ into Lorentz-Zygmund spaces $$L_{p,q}(\log L)_{\beta }$$. Our techniques rely on the approximation structure of $$B_{p,r}^{0,b}$$, Nikol’skiĭ type inequalities, extrapolation properties of $$L_{p,q}(\log L)_{\beta }$$ and interpolation.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 46B70 Interpolation between normed linear spaces
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##### References:
 [1] [1]V. V. Arestov, Inequality of various metrics for trigonometric polynomials, Math. Notes 27 (1980), 265–268. · Zbl 0508.42001 [2] [2]C. Bennett and K. Rudnick, On Lorentz–Zygmund spaces, Dissertationes Math. 175 (1980), 72 pp. · Zbl 0456.46028 [3] [3]J. Bergh and J. L\"ofstr\"om, Interpolation Spaces. An Introduction, Springer, Berlin, 1976. [4] [4]P. Borwein and T. Erd\'elyi, Polynomials and Polynomial Inequalities, Springer, New York, 1995. [5] [5]P. L. Butzer and K. Scherer, Approximationsprozesse und Interpolationsmethoden, Bibliographisches Institut, Mannheim–Z\"urich, 1968. · Zbl 0177.08501 [6] [6]A. M. Caetano, A. Gogatishvili and B. Opic, Sharp embeddings of Besov spaces involving only logarithmic smoothness, J. Approx. Theory 152 (2008), 188–214. · Zbl 1161.46017 [7] [7]A. M. Caetano, A. Gogatishvili and B. Opic, Compact embeddings of Besov spaces involving only slowly varying smoothness, Czechoslovak Math. J. 61 (2011), 923–940. · Zbl 1249.46026 [8] [8]F. Cobos and \'O. Dom\'ınguez, Approximation spaces, limiting interpolation and Besov spaces, preprint, Madrid, 2014. [9] [9]F. Cobos, \'O. Dom\'ınguez and A. Mart\'ınez, Compact operators and approximation spaces, Colloq. Math. 136 (2014), 1–11. [10] [10]F. Cobos and D. L. Fernandez, Hardy-Sobolev spaces and Besov spaces with a function parameter, in: Function Spaces and Applications (Lund, 1986), Lecture Notes in Math. 1302, Springer, Berlin, 1988, 158–170. [11] [11]F. Cobos, L. M. Fern\'andez-Cabrera, A. Manzano and A. Mart\'ınez, Logarithmic interpolation spaces between quasi-Banach spaces, Z. Anal. Anwendungen 26 (2007), 65–86. [12] [12]F. Cobos and M. Milman, On a limit class of approximation spaces, Numer. Funct. Anal. Optim. 11 (1990), 11–31. · Zbl 0729.41033 [13] [13]F. Cobos and I. Resina, Representation theorems for some operator ideals, J. London Math. Soc. 39 (1989), 324–334. [14] [14]R. A. DeVore, S. D. Riemenschneider and R. C. Sharpley, Weak interpolation in Banach spaces, J. Funct. Anal. 33 (1979), 58–94. · Zbl 0433.46062 [15] [15]Z. Ditzian and A. Prymak, Nikol’skii inequalities for Lorentz spaces, Rocky Mountain J. Math. 40 (2010), 209–223. [16] [16]D. E. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer, Berlin, 2004. 204F. Cobos and \'O. Dom\'ınguez [17] [17]D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, Cambridge, 1996. · Zbl 0865.46020 [18] [18]W. Farkas and H.-G. Leopold, Characterisations of function spaces of generalised smoothness, Ann. Mat. Pura Appl. 185 (2006), 1–62. · Zbl 1116.46024 [19] [19]F. Feh\'er and G. Gr\"assler, On an extremal scale of approximation spaces, J. Comput. Anal. Appl. 3 (2001), 95–108. · Zbl 1033.46021 [20] [20]V. I. Ivanov, Direct and inverse theorems of the theory of approximation in Lp spaces, 0 < p < 1, Mat. Zametki 18 (1975), 641–658 (in Russian). · Zbl 0337.42001 [21] [21]G. E. Karadzhov and M. Milman, Extrapolation theory: new results and applications, J. Approx. Theory 133 (2005), 38–99. · Zbl 1081.46018 [22] [22]C. Merucci, Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces, in: Interpolation Spaces and Allied Topics in Analysis, Lecture Notes in Math. 1070, Springer, Berlin, 1984, 183–201. [23] [23]S. M. Nikol’ski\breveı, Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Berlin, 1975. [24] [24]J. Peetre and G. Sparr, Interpolation of normed abelian groups, Ann. Mat. Pura Appl. 92 (1972), 217–262. · Zbl 0237.46039 [25] [25]L. E. Persson, Interpolation with a parameter function, Math. Scand. 59 (1986), 199–222. · Zbl 0619.46064 [26] [26]P. P. Petrushev and V. A. Popov, Rational Approximation of Real Functions, Cambridge Univ. Press, Cambridge, 1988. · Zbl 1209.41002 [27] [27]A. Pietsch, Approximation spaces, J. Approx. Theory 32 (1981), 115–134. [28] [28]A. Pietsch, Tensor products of sequences, functions, and operators, Arch. Math. (Basel) 38 (1982), 335–344. · Zbl 0464.47006 [29] [29]A. Pietsch, History of Banach Spaces and Linear Operators, Birkh\"auser, Boston, 2007. [30] [30]L. A. Sherstneva, Nikol’ski\breveı inequalities for trigonometric polynomials in Lorentz spaces, Moscow Univ. Math. Bull. 39 (1984), 75–81. · Zbl 0563.42001 [31] [31]E. A. Storozhenko, V. G. Krotov and P. Osval’d, Direct and inverse theorems of Jackson type in Lpspaces, 0 < p < 1, Mat. Sb. 98 (1975), 395–415 (in Russian). [32] [32]H. Triebel, The Structure of Functions, Birkh\"auser, Basel, 2001. [33] [33]H. Triebel, Comments on tractable embeddings and function spaces of smoothness near zero, report, Jena, 2012.
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