Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T06:24:03.575Z Has data issue: false hasContentIssue false
  • English
  • Français

EMBEDDING THEOREM OF THE WEIGHTED SOBOLEV–LORENTZ SPACES

Part of: Linear function spaces and their duals Inequalities

Published online by Cambridge University Press:  10 June 2021

HONGLIANG LI ,
JIANMIAO RUAN  and
QINXIU SUN
HONGLIANG LI
Affiliation:
Department of Mathematics, Zhejiang International Studies University, Hangzhou 310023, China e-mail: honglli@126.com
JIANMIAO RUAN
Affiliation:
Department of Mathematics, Zhejiang International Studies University, Hangzhou 310023, China e-mail: rjmath@163.com
QINXIU SUN
Affiliation:
Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou 310023, China e-mail: qxsun@126.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Weight criteria for embedding of the weighted Sobolev–Lorentz spaces to the weighted Besov–Lorentz spaces built upon certain mixed norms and iterated rearrangement are investigated. This gives an improvement of some known Sobolev embedding. We achieve the result based on different norm inequalities for the weighted Besov–Lorentz spaces defined in some mixed norms.

MSC classification

Primary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26D10: Inequalities involving derivatives and differential and integral operators
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 2 , May 2022 , pp. 358 - 375
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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

*

Project supported by the Zhejiang Provincial Natural Science Foundation of China (LY19A010001 and LY18A010015) and the National Natural Science Foundation of China (11961056 and 11771358).

References

Alvino, A., Sulla diseguaglianza di Sobolev in spazi di Lorentz, Bull. Un. Mat. Ital. A (5) 14(1) (1977), 148–156.Google Scholar
Ariño, M. A. and Muckenhoupt, B., Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for non-increasing functions, Trans. Amer. Math. Soc. 320 (1990), 727–735.Google Scholar
Barza, S., Persson, L. E., Kamińska, A. and Soria, J. Mixed norm and multidimensional Lorentz spaces, Positivity 10 (2006), 539–554.CrossRefGoogle Scholar
Barza, S., Persson, L. E. and Soria, J., Multidimensional rearrangement and Lorentz spaces, Acta Math. Hungar. 104 (2004), 203–228.CrossRefGoogle Scholar
Bennett, C. and Sharpley, R., Interpolation of operators (Academic Press, Boston, 1988).Google Scholar
Blozinski, A. P., Multivariate rearrangements and Banach function spaces with mixed norms, Trans. Amer. Math. Soc. 263(1) (1981), 149–167.CrossRefGoogle Scholar
Carro, M. J. and Soria, J., Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal. 112(2) (1993), 480–494.CrossRefGoogle Scholar
Carro, M. J. and Soria, J., The Hardy-Littlewood maximal function and weighted Lorentz spaces, J. London Math. Soc. 55(1) (1997), 146–158.CrossRefGoogle Scholar
Carro, M. J., Raposo, J. A. and Soria, J., Recent developements in the theory of Lorentz spaces and weighted inequalities, Mem. Amer. Math. Soc. 187 (2007).Google Scholar
Cwikel, M., On (Lp0 (A0), Lp1 (A1)) θ,q, Proc. Amer. Math. Soc. 44 (1974), 286–292 Google Scholar
Edmunds, D. E., Gurka, P. and Pick, L., Compactness of Hardy-type integral operators in weighted Banach function spaces, Studia Math. 109(1) (1994), 73–90.Google Scholar
Edmunds, D. E., Kokilashvili, V. and Meskhi, A., Boundedness and compactness integral operators (Springer, Kluwer Academic Publishers, Dordrecht, Boston London, 2002).CrossRefGoogle Scholar
Herz, C. S., Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms, J. Math. Mech. 18 (1968), 283–324.Google Scholar
Hunt, R. A., On L(p,q) spaces, Enseignement Math. 12(2) (1966), 249–276.Google Scholar
Kolyada, V. I., On relations between moduli of continuity in different metrics, Trudy Mat. Inst. Steklov 181 (1988), 117–136 (in Russian); English transl. in Proc. Steklov Inst. Math. (4) (1989), 127–148.Google Scholar
Kolyada, V. I., On embedding of Sobolev spaces, Mat. Zametki 54(3) (1993), 48–71; English transl.: Math. Notes 54(3) (1993), 908–922.Google Scholar
Kolyada, V. I., Rearrangement of functions and embedding of anisotropic spaces of Sobolev type, East J. Approx. 4 (1998), 111–199.Google Scholar
Kolyada, V. I., On embedding theorems, in: Nonlinear analysis, function spaces and applications: Proceedings of the Spring School held in Prague, 2006, vol. 8, Prague (2007), 35–94.Google Scholar
Kolyada, V. I., Embedding theorems for Sobolev and Hardy-Sobolev spaces and estimates of Fourier transforms, Ann. Mat. Pur. Appl. 198 (2019), 615–637.CrossRefGoogle Scholar
Kolyada, V. I and Lerner, A. K., On limiting embeddings of Besov spaces, Studia Math. 171(1) (2005), 1–13.CrossRefGoogle Scholar
Kudryavtsev, L. D. and Nikol’skij, S. M., Spaces of differentiable functions of several variables and embedding theorems, current problems in mathematics, fundamental directions, Itogi nauki i Techniki, Akad. Nauk SSSR 26 (1988), 5–157. Moscow (In Russian).Google Scholar
Li, H. L. and Kamińska, A., Boundedness and compactness of Hardy operator on Lorentz-type spaces, Math. Nachr. 290(5) (2017), 852–866.CrossRefGoogle Scholar
Li, H. L. and Sun, Q. X., Some notes on embedding for anisotropic Sobolev spaces, Czechoslovak Math. J. 61 (2010), 97–111.CrossRefGoogle Scholar
Nikol’skij, S. M., Approximation of functions of several variables and imbedding theorems (Springer, Berlin-Heidelberg-New York, 1975).CrossRefGoogle Scholar
O’Neil, R., Convolution operators and L(p.q) spaces, Duke Math. J. 30 (1963), 129–142.CrossRefGoogle Scholar
Peetre, J., Espaces d’interpolation et espaces de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966), 279–317.CrossRefGoogle Scholar
Pérez Lázaro, F. J., A note on extreme cases of Sobolev embeddings, J. Math. Anal. Appl. 320 (2006), 973–982.CrossRefGoogle Scholar
Pérez Lázaro, F. J., Embeddings for anisotropic Besov spaces, Acta Math. Hung. 119 (2008), 25–40.CrossRefGoogle Scholar
Poornima, S., An embedding theorem for the Sobolev space W 1,1, Bull. Sci. Math. 107(2) (1983), 253–259.Google Scholar
Schep, A. R., Minkowskis integral inequality for function norms, in Operator theory in function spaces and Banach lattices, Operator Theory: Advances and Applications, vol. 75 (Birkhäuser, Basel, 1995), 299–308.CrossRefGoogle Scholar
Stein, E. M., Singular integrals and differentiability properties of functions (Princeton University Press, Princeton, New Jersey, 1970).Google Scholar
Storozhenko, E. A., Necessary and sufficient conditions for the embedding of certain classes of functions, Izv. Akad. Nauk SSSR, Ser. Mat. 37 (1973), 386–398 (Russian); English transl. Math USSR-Izv. 7 (1973), 388–400.Google Scholar
Triebel, H., Theory of function spaces (Birkhäuser, Basel, 1983).CrossRefGoogle Scholar
Triebel, H., Theory of function spaces II (Birkhäuser, Basel, 1992).CrossRefGoogle Scholar
Ulyanov, P. L., Embedding of certain function classes $$H_p^\omega $$ , Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 649–686; English transl. in Math. USSR Izv. 2 (1968), 601–637.Google Scholar
Yatsenko, A. A., Iterative rearrangements of functions and the Lorentz spaces, Izv. VUZ. Mat. (5) (1998), 73–77; English transl.: Russian Math. (Iz. VUZ) 42(5) (1998), 71–75.Google Scholar
Yosida, K., Functional analysis (Springer-Verlag, Berlin, 1965).Google Scholar