Illinois Journal of Mathematics

On weighted inequalities for fractional integrals of radial functions

Pablo L. De Nápoli, Irene Drelichman, and Ricardo G. Durán

Full-text: Open access

Abstract

We prove a weighted version of the Hardy–Littlewood–Sobolev inequality for radially symmetric functions, and show that the range of admissible power weights appearing in the classical inequality due to Stein and Weiss can be improved in this particular case.

Article information

Source
Illinois J. Math., Volume 55, Number 2 (2011), 575-587.

Dates
First available in Project Euclid: 1 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1359762403

Digital Object Identifier
doi:10.1215/ijm/1359762403

Mathematical Reviews number (MathSciNet)
MR3020697

Zentralblatt MATH identifier
1277.26026

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators 47G10: Integral operators [See also 45P05]
Secondary: 31B10: Integral representations, integral operators, integral equations methods

Citation

De Nápoli, Pablo L.; Drelichman, Irene; Durán, Ricardo G. On weighted inequalities for fractional integrals of radial functions. Illinois J. Math. 55 (2011), no. 2, 575--587. doi:10.1215/ijm/1359762403. https://projecteuclid.org/euclid.ijm/1359762403


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References

  • P. L. De Nápoli, I. Drelichman and R. G. Durán, Radial solutions for Hamiltonian elliptic systems with weights, Adv. Nonlinear Stud. 9 (2009), 579–593.
  • G. Gasper, K. Stempak and W. Trebels, Fractional integration for Laguerre expansions, Mathods Appl. Anal. 2 (1995), 67–75.
  • L. Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004.
  • K. Hidano and Y. Kurokawa, Weighted HLS inequalities for radial functions and Strichartz estimates for Wave and Schrödinger equations, Illinois J. Math. 52 (2008), 365–388.
  • G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals, I, Math. Z. 27 (1928), 565–606.
  • P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315–334.
  • W. Rother, Some existence theorems for the equation $-\Delta u + K(x) u^p = 0$. Comm. Partial Differential Equations 15 (1990), 1461–1473.
  • E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J Math. 114 (1992), 813–874.
  • E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, 1970.
  • E. M. Stein and G. Weiss, Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514.
  • M. C. Vilela, Regularity solutions to the free Schrödinger equation with radial initial data, Illinois J. Math. 45 (2001), 361–370.