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Sobolev Embedding Theorems and Their Generalizations for Maps Defined on Topological Spaces with Measures

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Abstract

For mappings of a topological space \((X,\mu)\) to a Banach space \((Y,|\cdot|_{Y})\), we define analogs of Sobolev classes \(W_{p}^{r}(X;Y)\), \(r=1,2,\dots,\) as well as Sobolev–Slobodetsky classes \(W_{p}^{r}\), \(r\in[1,\infty)\), and their generalizations. We prove theorems on the embedding into the scale of Lebesgue spaces \(L_{q}\) and into Orlizc spaces corresponding to rapidly increasing generating functions; other properties of Sobolev functions are studied as well.

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REFERENCES

  1. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Nauka, Moscow, 1988).

    MATH  Google Scholar 

  2. V. G. Maz’ya, Spaces of S. L. Sobolev (Leningrad. Gos. Univ., Leningrad, 1985).

    Book  Google Scholar 

  3. S. M. Nikol’skii, Approximation of Multivariable Functions and Embedding Theorems (Nauka, Moscow, 1977).

    Google Scholar 

  4. O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems (Nauka, Moscow, 1996).

    MATH  Google Scholar 

  5. H. Triebel, ‘‘Limits of Besov norms,’’ Arch. Math. 96, 169–175 (2011). https://doi.org/10.1007/s00013-010-0214-1

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Bojarski and P. Hajlasz, ‘‘Pointwise inequalities for Sobolev functions and some applications,’’ Stud. Math. 106, 77–92 (1993).

    MathSciNet  MATH  Google Scholar 

  7. P. Hajlasz, ‘‘Sobolev spaces on an arbitrary metric spaces,’’ Potential Anal. 5, 403–415 (1996). https://doi.org/10.1007/BF00275475

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Franchi, P. Hajlasz, and P. Koskela, ‘‘Definitions of Sobolev classes on metric spaces,’’ Ann. Inst. Fourier 49, 1903–1924 (1999). https://doi.org/10.5802/aif.1742

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Hajłasz and P. Koskela, ‘‘Sobolev met Poincaré,’’ Mem. Am. Math. Soc. 145 (688), 1–101 (2000).

    MATH  Google Scholar 

  10. J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext (Springer, New York, 2001). https://doi.org/10.1007/978-1-4613-0131-8

  11. V. M. Gol’dshtein and M. Troyanov, ‘‘Axiomatic theory of Sobolev spaces,’’ Expositiones Math. 19, 289–336 (2001). https://doi.org/10.1016/S0723-0869(01)80018-9

    Article  MathSciNet  MATH  Google Scholar 

  12. B. Bojarski, ‘‘Pointwise characterization of Sobolev classes,’’ Proc. Steklov Inst. Math. 255, 65–81 (2006). https://doi.org/10.1134/S0081543806040067

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Johnsson, ‘‘Brownian motion on fractals and function spaces,’’ Math. Z. 222, 495–504 (1996). https://doi.org/10.1007/BF02621879

    Article  MathSciNet  Google Scholar 

  14. M. T. Barlow, ‘‘Diffusions on fractals,’’ in Lectures on Probability Theory and Statistics, Ed. by P. Bernard, Lecture Notes in Mathematics, vol. 1690 (Springer, Berlin, 1998). https://doi.org/10.1007/BFb0092537

  15. J. Lott and C. Villani, ‘‘Ricci curvature for metric-measure spaces via optimal transport,’’ Ann. Math. 169, 903–991 (2009).

    Article  MathSciNet  Google Scholar 

  16. L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Lecture in Mathematics ETH Zürich (Birkhäuser, Basel, 2008). https://doi.org/10.1007/978-3-7643-8722-8

  17. K. Kuwada, ‘‘Duality on gradient estimates and Wasserstein controls,’’ J. Funct. Anal. 258, 3758–3774 (2010). https://doi.org/10.1016/j.jfa.2010.01.010

    Article  MathSciNet  MATH  Google Scholar 

  18. S. K. Vodop’yanov and N. N. Romanovskiĭ, ‘‘Sobolev classes of mappings on a Carnot–Carathéodory space: Various norms and variational problems,’’ Sib. Math. J. 49, 814 (2008). https://doi.org/10.1007/s11202-008-0080-2

    Article  MathSciNet  Google Scholar 

  19. Yu. G. Reshetnyak, ‘‘Sobolev-type classes of functions with values in a metric space,’’ Sib. Math. J. 38, 567–583 (1997). https://doi.org/10.1007/BF02683844

    Article  MATH  Google Scholar 

  20. Yu. G. Reshetnyak, ‘‘Sobolev-type classes of functions with values in a metric space. II,’’ Sib. Math. J. 45, 709–721 (2004). https://doi.org/10.1023/B:SIMJ.0000035834.03736.b6

    Article  Google Scholar 

  21. Yu. G. Reshetnyak, ‘‘To the theory of Sobolev-type classes of functions with values in a metric space,’’ Sib. Math. J. 47, 117–134 (2006). https://doi.org/10.1007/s11202-006-0013-x

    Article  Google Scholar 

  22. A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics (Springer, Berlin, 2007). https://doi.org/10.1007/978-3-540-71897-0

  23. D. Jerison, ‘‘The Poincare inequality for vector fields satisfying Hörmander’s condition,’’ Duke Math. J. 53, 503–523 (1986). https://doi.org/10.1215/S0012-7094-86-05329-9

    Article  MathSciNet  MATH  Google Scholar 

  24. C. Villani, Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften, vol. 338 (Springer, Berlin, 2009). https://doi.org/10.1007/978-3-540-71050-9

  25. M.-K. von Renesse and K.-T. Sturm, ‘‘Transport inequalities, gradient estimates, entropy, and Ricci curvature,’’ Commun. Pure Appl. Math. 58, 923–940 (2005). https://doi.org/10.1002/cpa.20060

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Ambrosio, N. Gigli, and G. Savaré, ‘‘Calculus and heat flows in metric measure spaces and applications to spaces with Ricci bounds from below,’’ Inventiones Math. 195, 289–391 (2014). https://doi.org/10.1007/s00222-013-0456-1

    Article  MathSciNet  MATH  Google Scholar 

  27. N. N. Romanovskiĭ, ‘‘Sobolev spaces on an arbitrary metric measure space: Compactness of embeddings,’’ Sib. Math. J. 54, 353–367 (2013). https://doi.org/10.1134/S0037446613020171

    Article  MathSciNet  MATH  Google Scholar 

  28. N. N. Romanovskiĭ, ‘‘Embedding theorems and a variational problem for functions on a metric measure space,’’ Sib. Math. J. 55, 511–529 (2014). https://doi.org/10.1134/S0037446614030136

    Article  MathSciNet  MATH  Google Scholar 

  29. N. N. Romanovskiĭ, ‘‘Sobolev embedding theorems and generalizations for functions on a metric measure space,’’ Sib. Math. J. 59, 126–135 (2018). https://doi.org/10.1134/S0037446618010147

    Article  MathSciNet  MATH  Google Scholar 

  30. A. A. Dezin, ‘‘To the embedding theorems and problem of generalization of functions,’’ Dokl. Akad. Nauk SSSR 88, 741–743 (1953).

    MATH  Google Scholar 

  31. Yu. A. Brudnyĭ, ‘‘Criteria for the existence of derivatives in \(L_{p}\),’’ Math. USSR Sb. 2, 35–55 (1967). https://doi.org/10.1070/sm1967v002n01abeh002323

    Article  Google Scholar 

  32. I. A. Ivanishko and V. G. Krotov, ‘‘Compactness of embeddings of Sobolev type on metric measure spaces,’’ Math. Notes 86, 775–788 (2009). https://doi.org/10.1134/S0001434609110200

    Article  MathSciNet  MATH  Google Scholar 

  33. V. G. Krotov, ‘‘Criteria for compactness in \(L_{p}\) spaces, \(p>0\),’’ Sb. Math. 203, 1045–1064 (2012). https://doi.org/10.1070/sm2012v203n07abeh004253

    Article  MathSciNet  MATH  Google Scholar 

  34. S. I. Pokhozhaev, ‘‘On the theorem of Sobolev embedding in the case \(pl=n\),’’ in Proc. Sci.-Tech. Conf. Mathematical Section (Mosk. Energ. Inst., Moscow, 1965), pp. 158–170.

  35. N. S. Trudinger, ‘‘On imbeddings into Orlicz spaces and some applications,’’ J. Math. Mech. 17, 473–483 (1967).

    MathSciNet  MATH  Google Scholar 

  36. A. A. Cianchi, ‘‘A sharp embedding theorem for Orlicz–Sobolev spaces,’’ Indiana Univ. Math. J. 45, 39–65 (1996).

    Article  MathSciNet  Google Scholar 

  37. B. V. Trushin, ‘‘Embedding of Sobolev space in Orlicz space for a domain with irregular boundary,’’ Math. Notes 79, 707–718 (2006). https://doi.org/10.1007/s11006-006-0080-0

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This work was supported by the Russian Foundation for Basic Research, project no. 20-01-00661.

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  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia

    N. N. Romanovski

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  1. N. N. Romanovski
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Correspondence to N. N. Romanovski.

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Translated by A. Muravnik

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Romanovski, N.N. Sobolev Embedding Theorems and Their Generalizations for Maps Defined on Topological Spaces with Measures. Moscow Univ. Math. Bull. 77, 27–40 (2022). https://doi.org/10.3103/S0027132222010053

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