Real Analysis Exchange

Sufficient conditions for three weight sum inequalities in Lebesgue spaces

Richard C. Brown, Don B. Hinton, and Alois Kufner

Full-text: Open access

Abstract

Conditions (in terms of integrals of the weights) are derived, under which the weighted \(L^q\)-norm of the \(j\)-th order derivative of the function \(u\) can be estimated by the sum of the weighted \(L^r\)-norm of \(u\) and of the weighted \(L^p\)-norm of its \(m\)-th order derivative, \(j\lt m\). All mutual positions of the parameters, \(p,\, q, \, r\) are admissible.

Article information

Source
Real Anal. Exchange, Volume 22, Number 1 (1996), 292-317.

Dates
First available in Project Euclid: 1 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1338515222

Mathematical Reviews number (MathSciNet)
MR1433615

Zentralblatt MATH identifier
0879.26053

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords
weighted inequalities weighted Sobolev space interpolation inequalities

Citation

Brown, Richard C.; Hinton, Don B.; Kufner, Alois. Sufficient conditions for three weight sum inequalities in Lebesgue spaces. Real Anal. Exchange 22 (1996), no. 1, 292--317. https://projecteuclid.org/euclid.rae/1338515222


Export citation

References

  • R. C. Brown and D. B. Hinton, Sufficient conditions for weighted inequalities of sum form, J. Math. Anal. Appl., 112 (1985), 563–578.
  • R. C. Brown and D. B. Hinton, Sufficient conditions for weighted Gabushin inequalities, Časopis P\v est. Mat., 111 (1986), 113–121.
  • R. C. Brown and D. B. Hinton, Necessary and sufficient conditions for one variable weighted interpolation inequalities, Proc. London Math. Soc., 35(2) (1987), 439–453.
  • R. C. Brown and D. B. Hinton, Weighted interpolation inequalities of sum and product form in $R^n$, J. London Math Soc., 56(3) (1988), 261–280.
  • R. C. Brown and D. B. Hinton, Weighted interpolation inequalities and embeddings in $R^n$, Canad. J. Math., 47 (1990), 959–980.
  • B. Curgus and T. Read, Discretness of the spectrum of second-order differential operators and associated embedding theorems, (to appear).
  • E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1969.
  • A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin., 25 (1984), 537–554.
  • P. I. Lizorkin and M. O. Otelbaev, Imbedding and compactness theorems for Sobolev type spaces with weights I, Mat. Sb. (N. S.), 108(150), 358–377, (in Russian).
  • P. I. Lizorkin and M. O. Otelbaev, Imbedding and compactness theorems for Sobolev type spaces with weights II, Mat. Sb. (N. S.), 112(154), no. 1, (1980), 56–85, (in Russian).
  • V. G. Maz'ja, Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1985.
  • R. O\v\i narov, On weighted norm inequalities with three weights, J. London Math. Soc., \bf(2) 48 (1993), 103–116.
  • B. Opic and A. Kufner, Hardy-type Inequalities, Longman Scientific and Technical, Harlow, Essex, UK, 1990.