Abstract
We prove the embedding theorems of the Sobolev–Morrey spaces into the space of uniformly continuous functions so extending the classical Sobolev Theorems.
Similar content being viewed by others
References
Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics, Amer. Math. Soc., Providence (1991).
Morrey C. B., “On the solution of quasi-linear elliptic partial differential equations,” Trans. Amer. Math. Soc., 43, 126–166 (1938).
Besov O. V., Il’in V. P., and Nikol’skiĭ S. M., Integral Representations of Functions and Embedding Theorems, John Wiley and Sons, New York etc. (1978).
Dzhumakaeva G. T., “A criterion for the imbedding of the Sobolev–Morrey class Wl p,F in the space C,” Math. Notes, 37, No. 3, 224–228 (1985).
Dzhumakaeva G. T. and Nauryzbaev K. Zh., “On Lebesgue–Morrey spaces,” Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat., No. 5, 7–12 (1982).
Dzhumakaeva G. T., “On the continuity of smooth functions in mixed norm,” in: Abstracts of the VII Interuniversity Scientific Conference on Mathematics and Mechanics, Karaganda, 1981, pp. 19–20.
Dzhumakaeva G. T., “On a combination of embedding theorems by S. M. Nikol’skiĭ and C. Morrey,” in: Methods for Studying Operator Equations [Russian], Yaroslavl’, 1982, pp. 53–66.
Nauryzbaev K. Zh., Temirgaliev N., and Dzhumakaeva G. T., “A criterion for the embedding of Lebesgue–Morrey spaces in Lorentz spaces and related problems,” Herald of Gumilyov ENU, No. 6, 6–28 (2012).
Yuan W., Sickel W., and Yang D., Morrey and Campanato Meet Besov, Lizorkin and Triebel, Springer-Verlag, Berlin and Heidelberg (2010) (Lect. Notes Math.; V. 2005).
8th Intern. Conf. on Function Spaces, Differential Operators, Nonlinear Analysis (FSDONA-2011), September 18–24, 2011, Tabarz/Thur. http://fsdona2011.uni-jena.de/.
Haroske D. D. and Skrzypczak L., “On Sobolev and Franke–Jawerth embeddings of smoothness Morrey spaces,” Rev. Math. Comput., 27, No. 2, 541–573 (2014).
Rosenthal M. and Triebel H., “Calderón–Zygmund operators in Morrey spaces,” Rev. Math. Comput., 27, No. 1, 1–11 (2014).
Sawano Y., Hakim D. I., and Gunawan H., “Non-smooth atomic decomposition for generalized Orlicz–Morrey spaces,” Math. Nachr., 288, No. 14–15, 1741–1775 (2015).
Yuan W., Haroske D. D, Moura S. D., Skrzypczak L., and Yang D. C., “Limiting embeddings in smoothness Morrey spaces, continuity envelopes and applications,” J. Approx. Theory, 192, 306–335 (2015).
Zhu Y. P., Yang Q. X., and Li P. T., “Stability and Morrey spaces related to multipliers,” Taiwanese J. Math., 19, No. 3, 819–848 (2015).
Ho K. P., “Atomic decomposition of Hardy–Morrey Spaces with variable exponents,” Ann. Acad. Sci. Fenn., Math., 40, No. 1, 31–62 (2015).
Yuan W., Sickel W., and Yang D. C., “Compact embeddings of radial and subradial subspaces of some Besov-type spaces related to Morrey spaces,” J. Approx. Theory, 174, 121–139 (2013).
Haroske D. D. and Skrzypczak L., “Embeddings of Besov–Morrey spaces on bounded domains,” Stud. Math., 218, No. 2, 119–144 (2013).
Rosenthal M., “Local means, wavelet bases and wavelet isomorphisms in Besov–Morrey and Triebel–Lizorkin–Morrey spaces,” Math. Nachr., 286, No. 1, 59–87 (2013).
Persson L.-E., Samko N., and Wall P., “Calderón–Zygmund type singular operators in weighted generalized Morrey spaces,” J. Fourier Anal. Appl., 22, No. 2, 1–14 (2015).
Deringoz F., Guliev V., and Samko S., “Boundedness of maximal, potential and singular operators on generalized Orlicz–Morrey spaces,” Oper. Theory, Adv. Appl., 242, 135–158 (2014).
Chen Y., Ding Y., and Wang X., “Compactness of commutators of Riesz potential on Morrey spaces,” Potential Anal., 30, No. 4, 301–313 (2009).
Lu Y., Yang D., and Yuan W., “Interpolation of Morrey spaces on metric measure spaces,” Canad. Math. Bull., 57, No. 3, 598–608 (2014).
Sobolev S. L., “On one theorem of functional analysis,” Mat. Sb., 4, No. 3, 471–497 (1938).
Temirgaliev N., Zhainibekova M. A., and Dzhumakaeva G. T., “Criteria for embedding of classes of Morrey type,” Russian Math. (Iz. VUZ), 59, No. 5, 69–73 (2015).
Hardy G. H., Littlewood J. E., and Pó lya G., Inequalities, Cambridge University Press, Cambridge (UK) etc. (1988).
Author information
Authors and Affiliations
Corresponding author
Additional information
Astana. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 5, pp. 1156–1170, September–October, 2016; DOI: 10.17377/smzh.2016.57.520.
Rights and permissions
About this article
Cite this article
Temirgaliev, N., Zhainibekova, M.A. & Dzhumakaeva, G.T. A criterion for embedding of anisotropic Sobolev–Morrey spaces into the space of uniformly continuous functions. Sib Math J 57, 905–917 (2016). https://doi.org/10.1134/S0037446616050207
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446616050207