Skip to content


  • Research Article
  • Open Access

Inequalities for differentiable reproducing kernels and an application to positive integral operators

Journal of Inequalities and Applications20062006:53743

  • Received: 18 October 2005
  • Accepted: 13 November 2005
  • Published:


Let be an interval and let be a reproducing kernel on . We show that if is in the appropriate differentiability class, it satisfies a 2-parameter family of inequalities of which the diagonal dominance inequality for reproducing kernels is the 0th order case. We provide an application to integral operators: if is a positive definite kernel on (possibly unbounded) with differentiability class and satisfies an extra integrability condition, we show that eigenfunctions are and provide a bound for its Sobolev norm. This bound is shown to be optimal.


  • Integral Operator
  • Integrability Condition
  • Sobolev Norm
  • Order Case
  • Diagonal Dominance


Authors’ Affiliations

Departamento de Matemática, Instituto Superior Técnico, Lisbon, 1049-001, Portugal
Departamento de Engenharia Mecânica, ISEL, Lisbon, 1949-014, Portugal


  1. Aronszajn N: Theory of reproducing kernels. Transactions of the American Mathematical Society 1950,68(3):337–404. 10.1090/S0002-9947-1950-0051437-7MathSciNetView ArticleMATHGoogle Scholar
  2. Buescu J: Positive integral operators in unbounded domains. Journal of Mathematical Analysis and Applications 2004,296(1):244–255. 10.1016/j.jmaa.2004.04.007MathSciNetView ArticleMATHGoogle Scholar
  3. Buescu J, Garcia F, Lourtie I, Paixão AC: Positive-definiteness, integral equations and Fourier transforms. Journal of Integral Equations and Applications 2004,16(1):33–52. 10.1216/jiea/1181075257MathSciNetView ArticleMATHGoogle Scholar
  4. Buescu J, Paixão AC: Positive definite matrices and integral equations on unbounded domains. Differential and Integral Equations 2006,19(2):189–210.MathSciNetMATHGoogle Scholar
  5. Moore EH: General Analysis. Pt. I, Memoirs of Amer. Philos. Soc.. American Philosophical Society, Pennsylvania; 1935.Google Scholar
  6. Moore EH: General Analysis. Pt. II, Memoirs of Amer. Philos. Soc.. American Philosophical Society, Pennsylvania; 1939.Google Scholar
  7. Riesz F, Nagy B: Functional Analysis. Ungar, New York; 1952.MATHGoogle Scholar
  8. Saitoh S: Theory of Reproducing Kernels and Its Applications, Pitman Research Notes in Mathematics Series. Volume 189. Longman Scientific & Technical, Harlow; 1988.Google Scholar