Comparison of Morrey spaces and Nikol'skii spaces

V. I. Burenkov ; V. S. Guliyev ; T. V. Tararykova

Eurasian mathematical journal, Tome 12 (2021) no. 1, pp. 9-20.

Résumé

We consider two popular function spaces: the Morrey spaces and the Nikol'skii spaces and investigate the relationship between them in the one-dimensional case. In particular, we prove that, under the appropriate assumptions on the numerical parameters, their restrictions to the class of functions f of the form $f(x) = g(|x|)$, where g is a non-negative non-increasing function on $[0,\infty)$, coincide.

Bibliographie

[1] D. R. Adams, Morrey spaces, Birkhäuser, 2015 | MR | Zbl

[2] O. V. Besov, V. P. Il'in, S. M. Nikol'skii, Integral Representations of functions and embedding theorems, 2nd Edition, Fizmatlit, Nauka, M., 1996 (in Russian) | MR

[3] V. I. Burenkov, Sobolev spaces on domains, B.G. Teubner, Stuttgart–Leipzig, 1998 | MR | Zbl

[4] V. I. Burenkov, “Sharp estimates for integrals over small intervals for functions possessing some smoothness”, Progress in Analysis, Proceedings of the 3rd ISAAC Congress, World Scientific, New Yersey-London-Singapore–Hong Kong, 2003, 45–56 | DOI | MR

[5] V. I. Burenkov, “Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. I”, Eurasian Math. J., 3:3 (2012), 11–32 | MR | Zbl

[6] V. I. Burenkov, “Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. II”, Eurasian Math. J., 4:1 (2013), 21–45 | MR | Zbl

[7] V. S. Guliyev, “Generalized weighted Morrey spaces and higher order commutators of sublinear operators”, Eurasian Math. J., 3:3 (2012), 33–61 | MR | Zbl

[8] B. Kuttner, “Some theorems on fractional derivatives”, Proc. London Math. Soc., 3:2 (1953), 480–497 | DOI | MR | Zbl

[9] P. G. Lemari'e-Rieusset, “The role of Morrey spaces in the study of Navier-Stokes and Euler equations”, Eurasian Math. J., 3:3 (2012), 62–93 | MR | Zbl

[10] C. B. Morrey, “On the solutions of quasi-linear elliptic partial differential equations”, Trans. Amer. Math. Soc., 43 (1938), 126–166 | DOI | MR

[11] S. M. Nikol'skii, “Inequalities for entire functions of exponential type and applications”, Doklady Mathematics. Academy of Sciences of the USSR, 38 (1951), 244–278 (in Russian)

[12] S. M. Nikol'skii, “Inequalities for entire functions of exponential type and applications to the theory of differentiable functions of several variables”, Proc. Steklov Math. Inst., 38 (1951), 244–278 (in Russian)

[13] S. M. Nikol'skii, “On a property of the $H_p^r$ classes”, Ann. Univ. Sci. Budapest, Sect. Math., 3–4 (1960/1961), 205–216 | MR

[14] Springer-Verlag, 1975 | MR | Zbl

[15] J. Peetre, “On the theory of $\mathcal{L}_{p,\lambda}$-spaces”, J. Func. Anal., 4 (1969), 71–87 | DOI | MR | Zbl

[16] M. A. Ragusa, “Partial differential equations involving Morrey spaces as initial conditions”, Eurasian Math. J., 3:3 (2012), 94–109 | MR | Zbl

[17] W. Sickel, “Some generalizations of the spaces $F_{1,q}^s(\mathbb{R}^d)$ and relations to Lizorkin-Triebel spaces built on Morrey spaces. I”, Eurasian Math. J., 3:3 (2012), 110–149 | MR | Zbl

[18] W. Sickel, “Some generalizations of the spaces $F_{1,q}^s(\mathbb{R}^d)$ and relations to Lizorkin-Triebel space built on Morrey spaces. II”, Eurasian Math. J., 4:1 (2013), 82–124 | MR | Zbl

[19] G. Stampacchia, “$\mathcal{L}_{p,\lambda}$-spaces and interpolation”, Comm. Pure Appl. Math., 17 (1964), 293–306 | DOI | MR | Zbl

[20] H. Triebel, Theory of function spaces, Birkhäuser, 1983 | MR | Zbl

[21] C. T. Zorko, “Morrey space”, Proc. Amer. Math. Soc., 98:4 (1986), 586–592 | DOI | MR | Zbl