Skip to main content

Generalized Fractional Integrals on Central Morrey Spaces and Generalized σ-Lipschitz Spaces

  • Conference paper
Current Trends in Analysis and Its Applications

Part of the book series: Trends in Mathematics ((RESPERSP))

Abstract

For the generalized fractional integrals \({\tilde{I}}_{\alpha,d}\), which were defined in Function Spaces X, to appear, when n/αp<∞, we will consider their boundedness from the central Morrey spaces \(B^{p,\lambda}(\mathbb{R}^{n})\) to the generalized σ-Lipschitz spaces \(\mathrm{Lip}^{(d)}_{\beta,\sigma }(\mathbb{R}^{n})\).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Alvarez, M. Guzmán-Partida, J. Lakey, Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures. Collect. Math. 51, 1–47 (2000)

    MATH  MathSciNet  Google Scholar 

  2. O.V. Besov, V.P. Il’in, S.M. Nikol’skiı̌, in Integral Representations of Functions and Imbedding Theorems, vol. I, II, ed. by M.H. Teibleson Scripta Series in Mathematics (Winston, Washington, 1978). Translated from Russian, 1979

    Google Scholar 

  3. A. Beurling, Construction and analysis of some convolution algebras. Ann. Inst. Fourier 14(2), 1–32 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  4. 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), 11–32 (2012)

    MATH  MathSciNet  Google Scholar 

  5. 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), 21–45 (2013)

    MATH  MathSciNet  Google Scholar 

  6. Y. Chen, K. Lau, Some new classes of Hardy spaces. J. Funct. Anal. 84, 255–278 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Feichtinger, An elementary approach to Wiener’s third Tauberian theorem on Euclidean n-space, in Proceedings, Conference at Cortona 1984. Symposia Mathematica, vol. 29 (Academic Press, New York, 1987), pp. 267–301

    Google Scholar 

  8. Z. Fu, Y. Lin, S. Lu, λ-Central BMO estimates for commutators of singular integral operators with rough kernels. Acta Math. Sin. Engl. Ser. 24(3), 373–386 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. J. García-Cuerva, Hardy spaces and Beurling algebras. J. Lond. Math. Soc. 39, 499–513 (1989)

    Article  MATH  Google Scholar 

  10. J. García-Cuerva, M.J.L. Herrero, A theory of Hardy spaces associated to the Herz spaces. Proc. Lond. Math. Soc. 69, 605–628 (1994)

    Article  MATH  Google Scholar 

  11. G.H. Hardy, J.E. Littlewood, Some properties of fractional integrals. I. Math. Z. 27, 565–606 (1928)

    Article  MATH  MathSciNet  Google Scholar 

  12. G.H. Hardy, J.E. Littlewood, Some properties of fractional integrals. II. Math. Z. 34, 403–439 (1932)

    Article  MathSciNet  Google Scholar 

  13. C. Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms. J. Math. Mech. 18, 283–324 (1968)

    MATH  MathSciNet  Google Scholar 

  14. Y. Komori-Furuya, K. Matsuoka, Some weak-type estimates for singular integral operators on CMO spaces. Hokkaido Math. J. 39, 115–126 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Y. Komori-Furuya, K. Matsuoka, Strong and weak estimates for fractional integral operators on some Herz-type function spaces, in Proceedings of the Maratea Conference FAAT 2009, Rendiconti del Circolo Mathematico di Palermo, Serie II, vol. 82 (2010), pp. 375–385

    Google Scholar 

  16. Y. Komori-Furuya, K. Matsuoka, E. Nakai, Y. Sawano, Integral operators on B σ -Morrey–Campanato spaces. Rev. Mat. Complut. 26, 1–32 (2013). doi:10.1007/s13163-011-0091-6

    Article  MATH  MathSciNet  Google Scholar 

  17. Y. Komori-Furuya, K. Matsuoka, E. Nakai, Y. Sawano, Applications of Littlewood-Paley theory for \(\dot{B}_{\sigma}\)-Morrey spaces to the boundedness of integral operators. J. Funct. Spaces Appl. 2013 (2013). Article ID 859402, 21 pages

    Google Scholar 

  18. K. Matsuoka, B σ -Morrey–Campanato estimates and some estimates for singular integrals on central Morrey spaces and λ-CMO spaces, in Banach and Function Spaces IV (Kitakyushu 2012) (Yokohama Publishers, Yokohama, 2014), pp. 325–335

    Google Scholar 

  19. K. Matsuoka, Generalized fractional integrals on central Morrey spaces and generalized λ-CMO spaces, in Function Spaces X. Banach Center Publ. (Inst. Math., Polish Sci., Warsawa), to appear

    Google Scholar 

  20. K. Matsuoka, E. Nakai, Fractional integral operators on B p,λ with Morrey–Campanato norms, in Function Spaces IX. Banach Center Publ., vol. 92 (Inst. Math., Polish Sci., Warsawa, 2011), pp. 249–264

    Google Scholar 

  21. Y. Mizuta, Potential Theory in Euclidean Spaces (Gakkōtosho Co., Tokyo, 1996)

    Google Scholar 

  22. E. Nakai, Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665–3748 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. S.M. Nikol’skiı̌, Approximation of Functions of Several Variables and Imbedding Theorems (Springer, Berlin, 1975)

    Book  Google Scholar 

  24. J. Peetre, On the theory of \(\mathcal{L}_{p,\lambda}\) spaces. J. Funct. Anal. 4, 71–87 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  25. T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations (de Gruyter, Berlin, 1996)

    Book  MATH  Google Scholar 

  26. W. Sickel, Smoothness spaces related to Morrey spaces----a survey. I. Eurasian Math. J. 3(3), 110–149 (2012)

    MATH  MathSciNet  Google Scholar 

  27. W. Sickel, Smoothness spaces related to Morrey spaces—a survey. II. Eurasian Math. J. 4(1), 82–124 (2013)

    MATH  MathSciNet  Google Scholar 

  28. S.L. Sobolev, On a theorem in functional analysis. Mat. Sb. 4, 471–497 (1938) (Russian)

    MATH  Google Scholar 

  29. A. Zygmund, On a theorem of Marcinkiewicz concerning interpolation of operations. J. Math. Pures Appl. 35, 223–248 (1956)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgement

The author would like to express to the anonymous referee his gratitude for reading his paper carefully and for suggesting the valuable remarks, i.e., Remarks 2.2, 2.4 and 2.9.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Katsuo Matsuoka .

Editor information

Editors and Affiliations

Additional information

Dedicated to Professor Kichi-Suke Saito in celebration of his 65th birthday.

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Matsuoka, K. (2015). Generalized Fractional Integrals on Central Morrey Spaces and Generalized σ-Lipschitz Spaces. In: Mityushev, V., Ruzhansky, M. (eds) Current Trends in Analysis and Its Applications. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-12577-0_22

Download citation

Publish with us

Policies and ethics