Skip to main content
SpringerLink
Log in

Some New Hardy-type Integral Inequalities on Cones of Monotone Functions

  • Conference paper
  • First Online:
Advances in Harmonic Analysis and Operator Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 229))

  • 1374 Accesses

Abstract

Some new Hardy-type inequalities with Hardy-Volterra integral operators on the cones of monotone functions are obtained. The case 1 < p ≤ q < ∞ is considered and the involved kernels satisfy conditions which are less restrictive than the classical Oinarov condition.

Mathematics Subject Classification (2010). 47G10, 47B38.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Ari˜no and B. Muckenhoupt, 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 

  2. I. Asekritova, N. Krugljak, L. Maligranda and L.-E. Persson, Distribution and rearrangement estimates of the maximal function and interpolation. Studia Math 124 (1997), no. 2, 107–132.

    MathSciNet  MATH  Google Scholar 

  3. A. Gogatishvili, M. Johansson, C. A. Okpoti and L.-E. Persson, Characterisation of embeddings in Lorentz spaces. Bull. Austral. Math. Soc. 76 (2007), no. 1, 69–92.

    Article  MathSciNet  MATH  Google Scholar 

  4. H.P. Heinig and V.D. Stepanov, Weighted Hardy inequalities for increasing functions. Canad. J. Math. 45 (1993), no. 1, 104–116.

    Article  MathSciNet  MATH  Google Scholar 

  5. V. Kokilashvili, A. Meskhi and L.-E. Persson, Weighted norm inequalities for integral transforms with product kernels. Nova Science Publishers Inc. New York, 2010.

    Google Scholar 

  6. A. Kufner, L. Maligranda and L.-E. Persson, The Hardy inequality. About its history and some related results. Vydavatelsk´y Servis, Plseˇn, 2007.

    Google Scholar 

  7. A. Kufner and L.-E. Persson, Weighted inequalities of Hardy type. World Scientific, New Jersey, 2003.

    Google Scholar 

  8. C.J. Neugebauer, Weights for the Hardy operator on non-decreasing functions. Int. J. Pure Appl. Math. 60 (2010), no. 1, 71–82.

    MathSciNet  MATH  Google Scholar 

  9. R. Oinarov, Boundedness and compactness of Volterra-type integral operators. Sibirsk. Mat. Zh. 48 (2007), no. 5, 1100–1115; English transl. in Siberian Math J. 48 (2007), no. 5, 884–896.

    Google Scholar 

  10. B. Opic and A. Kufner, Hardy-type inequalities. Pitman Research Notes in Mathematics, Series 219, Longman Science and Technology, Harlow, 1990.

    Google Scholar 

  11. L.-E. Persson, O.V. Popova and V.D. Stepanov, Two-sided Hardy-type inequalities for monotone functions. Doklady Mathematics 80 (2009), no. 3, 814–817.

    Article  MathSciNet  MATH  Google Scholar 

  12. L.-E. Persson, O.V. Popova and V.D. Stepanov, Two-sided Hardy-type inequalities for monotone functions. Complex Variables and Elliptic Equations 55 (2010), no. 8-10, 973–989.

    Article  MathSciNet  MATH  Google Scholar 

  13. L.-E. Persson, V.D. Stepanov and E.P. Ushakova, Equivalence of Hardy-type inequalities with general measures on the cones of non-negative respective non-increasing functions. Proc. Amer. Math. Soc. 134 (2006), 2363–2372.

    Article  MathSciNet  MATH  Google Scholar 

  14. E. Sawyer, Boundedness of classical operators on classical Lorentz spaces. Studia Math. 96 (1990), 145–158.

    MathSciNet  MATH  Google Scholar 

  15. G. Sinnamon, Hardy’s inequality and monotonocity. Function Spaces and Nonlinear Analysis (eds.: P. Dr´abec and J. R´akosnik), Mathematical Institute of the Academy of Sciences of the Czech Republic, Prague, 2005, 292–310.

    Google Scholar 

  16. V.D. Stepanov, The weighted Hardy’s inequality for non-increasing functions. Trans. Amer. Math. Soc. 338 (1993), 173–186.

    MathSciNet  MATH  Google Scholar 

  17. V.D. Stepanov, Integral operators on the cone of monotone functions. J. London Math. Soc. 48 (1993), 465–487.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Fundamental and Applied Mathematics, Eurasian National University, Munaitpasov str. 4, Astana, 010008, Kazakhstan

    L. S. Arendarenko & R. Oinarov

  2. Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87, Luleå, Sweden

    L.-E. Persson

  3. Narvik University College, 385, N-8505, Narvik, Norway

    L.-E. Persson

Authors
  1. L. S. Arendarenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Oinarov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L.-E. Persson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. S. Arendarenko .

Editor information

Editors and Affiliations

  1. , Departamento de Matemática, Universidade de Aveiro, Campus Universitário de Santiago, Aveiro, 3810-193, Portugal

    Alexandre Almeida

  2. , Departamento de Matemática, Universidade de Aveiro, Campus Universitário de Santiago, Aveiro, 3810-193, Portugal

    Luís Castro

  3. Instituto Superior Tecnico, Depto. Matematica, Universidade Tecnica Lisboa, Av. Rovisco Pais 1, Lisboa, 1049-001, Portugal

    Frank-Olme Speck

Additional information

Dedicated to Professor Stefan Samko on the occasion of his 70th anniversary

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Basel

About this paper

Cite this paper

Arendarenko, L.S., Oinarov, R., Persson, LE. (2013). Some New Hardy-type Integral Inequalities on Cones of Monotone Functions. In: Almeida, A., Castro, L., Speck, FO. (eds) Advances in Harmonic Analysis and Operator Theory. Operator Theory: Advances and Applications, vol 229. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0516-2_4

Download citation

Publish with us

Policies and ethics

Search

Navigation