Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-18T02:34:56.187Z Has data issue: false hasContentIssue false

On the smoothness of slowly varying functions

Published online by Cambridge University Press:  16 May 2024

Dalimil Peša*
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Praha 8, Czech Republic

Abstract

In this paper, we consider the question of smoothness of slowly varying functions satisfying the modern definition that, in the last two decades, gained prevalence in the applications concerning function spaces and interpolation. We show that every slowly varying function of this type is equivalent to a slowly varying function that has continuous classical derivatives of all orders.

Type
Research Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research was supported by the Grant schemes at Charles University, reg. No. CZ.02.2.69/0.0/0.0/19_073/0016935, the grants no. P201/21-01976S and P202/23-04720S of the Czech Science Foundation and Charles University Research program No. UNCE/SCI/023.

References

Baena-Miret, S., Gogatishvili, A., Mihula, Z. and Pick, L.. Reduction principle for Gaussian K-inequality. J. Math. Anal. Appl., 516(2) (2022), .CrossRefGoogle Scholar
Bathory, M.. Joint weak type interpolation on Lorentz-Karamata spaces. Math. Inequal. Appl., 21(2) (2018) 385419.Google Scholar
Bennett, C. and Rudnick, K.. On Lorentz-Zygmund spaces. Dissertationes Math. (Rozprawy Mat.), 175 (67) (1980).Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L.. Regular Variation, Encyclopedia of Mathematics and its Applications, Vol. 27, Cambridge University Press, Cambridge, 1987.CrossRefGoogle Scholar
Brézis, H. and Wainger, S.. A note on limiting cases of Sobolev embeddings and convolution inequalities. Comm. Partial Differential Equations., 5 (7) (1980), 773789.CrossRefGoogle Scholar
Caetano, A. M., Gogatishvili, A. and Opic, B.. Embeddings and the growth envelope of Besov spaces involving only slowly varying smoothness. J. Approx. Theory, 163(10) (2011), 13731399.CrossRefGoogle Scholar
Cianchi, A. and Pick, L.. Optimal Gaussian Sobolev embeddings. J. Funct. Anal., 256(11) (2009), 35883642.CrossRefGoogle Scholar
de Haan, L.. On regular variation and its application to the weak convergence of sample extremes, Mathematical Centre Tracts, Vol. 32, Mathematisch Centrum, Amsterdam, 1970.Google Scholar
de Haan, L. and Ferreira, A.. Extreme Value Theory: An introduction. Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.CrossRefGoogle Scholar
di Blasio, G., Feo, F. and Posteraro, M. R.. Regularity results for degenerate elliptic equations related to Gauss measure. Math. Inequal. Appl., 10(4) (2007), 771797.Google Scholar
Edmunds, D. E., Gurka, P. and Opic, B.. Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces. Indiana Univ. Math. J., 44(1) (1995), 1943.CrossRefGoogle Scholar
Edmunds, D. E., Kerman, R. and Pick, L.. Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms. J. Funct. Anal., 170(2) (2000), 307355.CrossRefGoogle Scholar
Edmunds, D. E. and Opic, B.. Alternative characterisations of Lorentz-Karamata spaces. Czechoslovak Math. J., 58(133)(2) (2008), 517540.CrossRefGoogle Scholar
Feller, W.. An introduction to Probability Theory and Its Applications, Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, Second edition, 1971.Google Scholar
Gogatishvili, A., Neves, J. S. and Opic, B.. Optimal embeddings of Bessel-potential-type spaces into generalized Hölder spaces involving k-modulus of smoothness. Potential Anal., 32(3) (2010), 201228.CrossRefGoogle Scholar
Gogatishvili, A., Opic, B. and Neves, J. S.. Optimality of embeddings of Bessel-potential-type spaces into Lorentz-Karamata spaces. Proc. Roy. Soc. Edinburgh Sect. A, 134(6) (2004), 11271147.CrossRefGoogle Scholar
Gogatishvili, A., Opic, B. and Trebels, W.. Limiting reiteration for real interpolation with slowly varying functions. Math. Nachr., 278(1–2) (2005) 86107.CrossRefGoogle Scholar
Gurka, P. and Opic, B.. Sharp embeddings of Besov spaces with logarithmic smoothness. Rev. Mat. Complut., 18(1) (2005), 81110.CrossRefGoogle Scholar
Gurka, P. and Opic, B.. Sharp embeddings of Besov-type spaces. J. Comput. Appl. Math., 208(1) (2007), 235269.CrossRefGoogle Scholar
Jessen, A. H. and Mikosch, T.. Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.), 80(94) (2006), 171192.CrossRefGoogle Scholar
Karamata, J.. Sur un mode de croissance reâguilieáre des fonctions. Mathematica (Cluj), 4 (1930), 3853.Google Scholar
Karamata, J.. Sur un mode de croissance régulière. Théorèmes fondamentaux. Bull. Soc. Math. France, 61 (1933), 5562.CrossRefGoogle Scholar
Marić, V.. Regular Variation and Differential Equations, volume 1726 of Lecture Notes in Mathematics, Vol. 1726, Springer-Verlag, Berlin, 2000.Google Scholar
Moura, S. D., Neves, J. S. and Piotrowski, M.. Continuity envelopes of spaces of generalized smoothness in the critical case. J. Fourier Anal. Appl., 15(6) (2009), 775795.CrossRefGoogle Scholar
Neves, J. S.. Lorentz-Karamata spaces, Bessel and Riesz potentials and embeddings. Dissertationes Math. (Rozprawy Mat.), 405 (46) (2002).Google Scholar
Neves, J. S. and Opic, B.. Optimal local embeddings of Besov spaces involving only slowly varying smoothness. J. Approx. Theory, 254 (105393) (2020), .CrossRefGoogle Scholar
Nikolić, A.. Karamata functions and differential equations: achievements from the 20th century. Historia Math., 45(3) (2018), 277299.CrossRefGoogle Scholar
Opic, B. and Grover, M.. Description of K-spaces by means of J-spaces and the reverse problem in the limiting real interpolation. Math. Nachr., 296 (2023), 40024031.CrossRefGoogle Scholar
Opic, B. and Pick, L.. On generalized Lorentz-Zygmund spaces. Math. Inequal. Appl., 2(3) (1999), 391467.Google Scholar
Peša, D.. Lorentz-Karamata spaces. 2023, arXiv:2006.14455 http://arxiv.org/abs/2006.14455.Google Scholar
Rădulescu, V. D.. Singular phenomena in nonlinear elliptic problems: from blow-up boundary solutions to equations with singular nonlinearities. In Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. IV, 485593. Elsevier/North-Holland, Amsterdam, 2007.CrossRefGoogle Scholar
Zygmund, A.. Trigonometric Series. Vol. I, II. Cambridge Mathematical Library. Cambridge University Press, Cambridge, Third edition, 2002. With a foreword by Robert A. Fefferman.Google Scholar