Abstract
We study necessary and sufficient conditions for embeddings of Besov and Triebel-Lizorkin spaces of generalized smoothness \(B^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n})\) and \(F^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n})\), respectively, into generalized Hölder spaces \(\Lambda_{\infty,r}^{\mu(\cdot)}(\ensuremath {\ensuremath {\mathbb {R}}^{n}})\). In particular, we are able to characterize optimal embeddings for this class of spaces provided q>1. These results improve the embedding assertions given by the continuity envelopes of \(B^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n})\) and \(F^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n})\), which were obtained recently solving an open problem of D.D. Haroske in the classical setting.
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Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics, vol. 140, 2nd edn. Elsevier/Academic Press, Amsterdam (2003)
Bennett, G.: Some elementary inequalities. III. Q. J. Math. Oxf. Ser. (2) 42(166), 149–174 (1991)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1989)
Bricchi, M.: Tailored Besov spaces and h-sets. Math. Nachr. 263/264, 36–52 (2004)
Caetano, A.M., Farkas, W.: Local growth envelopes of Besov spaces of generalized smoothness. Z. Anal. Anwend. 25(3), 265–298 (2006)
Caetano, A.M., Haroske, D.D.: Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers. J. Funct. Spaces Appl. 3(1), 33–71 (2005)
Caetano, A.M., Leopold, H.-G.: Local growth envelopes of Triebel-Lizorkin spaces of generalized smoothness. J. Fourier Anal. Appl. 12(4), 427–445 (2006)
Caetano, A.M., Moura, S.D.: Local growth envelopes of spaces of generalized smoothness: the critical case. Math. Inequal. Appl. 7(4), 573–606 (2004)
Caetano, A.M., Moura, S.D.: Local growth envelopes of spaces of generalized smoothness: the subcritical case. Math. Nachr. 273, 43–57 (2004)
Edmunds, D.E., Haroske, D.D.: Spaces of Lipschitz type, embeddings and entropy numbers. Diss. Math. (Rozprawy Mat.) 380, 43 (1999)
Edmunds, D.E., Haroske, D.D.: Embeddings in spaces of Lipschitz type, entropy and approximation numbers, and applications. J. Approx. Theory 104(2), 226–271 (2000)
Edmunds, D.E., Triebel, H.: Spectral theory for isotropic fractal drums. C. R. Acad. Sci. Paris Sér. I Math. 326(11), 1269–1274 (1998)
Edmunds, D.E., Triebel, H.: Eigenfrequencies of isotropic fractal drums. In: The Maz’ya Anniversary Collection, vol. 2, Rostock, 1998. Oper. Theory Adv. Appl., vol. 110, pp. 81–102. Birkhäuser, Basel (1999)
Farkas, W., Leopold, H.-G.: Characterisations of function spaces of generalised smoothness. Ann. Mat. Pura Appl. (4) 185(1), 1–62 (2006)
Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34(4), 777–799 (1985)
Gogatishvili, A., Neves, J.S., Opic, B.: Sharpness and non-compactness of embeddings of Bessel-potential-type spaces. Math. Nachr. 280(9–10), 1083–1093 (2007)
Gogatishvili, A., Neves, J.S., Opic, B.: Optimal embeddings of Bessel-potential-type spaces into generalized Hölder spaces involving k-modulus of smoothness. Potential Anal. 32, 201–228 (2010)
Gol’dman, M.L.: Hardy type inequalities on the cone of quasimonotone functions. Research Report 98/31, Computing Centre FEB Russian Academy of Sciences, Khabarovsk (1998)
Haroske, D.D.: On more general Lipschitz spaces. Z. Anal. Anwend. 19(3), 781–799 (2000)
Haroske, D.D.: Envelopes and Sharp Embeddings of Functions Spaces. Research Notes in Mathematics, vol. 437. Chapman & Hall/CRC, Boca Raton (2007)
Heinig, H.P., Stepanov, V.D.: Weighted Hardy inequalities for increasing functions. Can. J. Math. 45(1), 104–116 (1993)
Leopold, H.-G.: Limiting embeddings and entropy numbers. Preprint Math/Inf/98/05, Univ. Jena, Germany (1998)
Leopold, H.-G.: Embeddings and entropy numbers in Besov spaces of generalized smoothness. In: Hudzig, H., Skrzypczak, L. (eds.) Function Spaces: The Fifth Conference. Lecture Notes in Pure and Appl. Math., vol. 213, pp. 323–336. Marcel Dekker, New York (2000)
Moura, S.D., Neves, J.S., Piotrowski, M.: Continuity envelopes of spaces of generalized smoothness in the critical case. J. Fourier Anal. Appl. 15, 775–795 (2009)
Moura, S.D.: Function spaces of generalised smoothness. Diss. Math. (Rozprawy Mat.) 398, 88 (2001)
Moura, S.D.: Function spaces of generalised smoothness, entropy numbers, applications. PhD thesis, University of Coimbra, Portugal (2001)
Neves, J.S.: Extrapolation results on general Besov-Hölder-Lipschitz spaces. Math. Nachr. 230, 117–141 (2001)
Neves, J.S.: Fractional Sobolev-type spaces and embeddings. PhD thesis, University of Sussex, UK (2001)
Neves, J.S.: Lorentz-Karamata spaces, Bessel and Riesz potentials and embeddings. Diss. Math. (Rozprawy Mat.) 405, 46 (2002)
Opic, B., Kufner, A.: Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series, vol. 219. Longman Scientific & Technical, Harlow (1990)
Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser, Basel (1983)
Triebel, H.: Fractals and Spectra, Related Fourier Analysis and Function Spaces. Monographs in Mathematics, vol. 91. Birkhäuser, Basel (1997)
Triebel, H.: The Structure of Functions. Monographs in Mathematics, vol. 97. Birkhäuser, Basel (2001)
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Communicated by Hans Triebel.
Research partially supported by Centre of Mathematics of the University of Coimbra and FCT Project PTDC/MAT/098060/2008.
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Moura, S.D., Neves, J.S. & Schneider, C. Optimal Embeddings of Spaces of Generalized Smoothness in the Critical Case. J Fourier Anal Appl 17, 777–800 (2011). https://doi.org/10.1007/s00041-010-9155-0
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DOI: https://doi.org/10.1007/s00041-010-9155-0