Operators with Continuous Kernels

  • W. Arendt
  • A. F. M. ter ElstEmail author


Let \(\Omega \subset \mathbb {R}^d\) be open. We investigate conditions under which an operator T on \(L_2(\Omega )\) has a continuous kernel \(K \in C({{\overline{\Omega }}} \times \overline{\Omega })\). In the centre of our interest is the condition \(T L_2(\Omega ) \subset C({{\overline{\Omega }}})\), which one knows for many semigroups generated by elliptic operators. This condition implies that \(T^3\) has a kernel in \(C({{\overline{\Omega }}} \times \overline{\Omega })\) if T is self-adjoint and \(\Omega \) is bounded, and the power 3 is best possible. We also analyse Mercer’s theorem in our context.


Intergral operator Continuous kernel Mercer’s theorem 

Mathematics Subject Classification

Primary 47G10 Secondary 47B10 



The first-named author is most grateful for the hospitality extended to him during a fruitful stay at the University of Auckland and the second-named author for a wonderful stay at the University of Ulm. Both authors wish to thank the referee for the comments and corrections which improved significantly this paper.


  1. 1.
    Arendt, W., Daners, D.: The Dirichlet problem by variational methods. Bull. Lond. Math. Soc. 40, 51–56 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arendt, W., ter Elst, A.F.M.: Gaussian estimates for second order elliptic operators with boundary conditions. J. Oper. Theory 38, 87–130 (1997)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arendt, W., ter Elst, A.F.M.: The Dirichlet-to-Neumann operator on \(C(\partial \Omega )\). Ann. Sci. Norm. Super. Pisa Cl. Sci. (2019).
  4. 4.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 3rd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  5. 5.
    Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. Graduate Texts in Mathematics, vol. 100. Springer, Berlin (1984)CrossRefGoogle Scholar
  6. 6.
    Daners, D.: Heat kernel estimates for operators with boundary conditions. Math. Nachr. 217, 13–41 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1989)CrossRefGoogle Scholar
  8. 8.
    ter Elst, A.F.M., Rehberg, J.: Hölder estimates for second-order operators on domains with rough boundary. Adv. Differ. Equ. 20, 299–360 (2015)zbMATHGoogle Scholar
  9. 9.
    Ferreira, J.C., Menegatto, V.A., Oliveira, C.P.: On the nuclearity of integral operators. Positivity 13, 519–541 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Keller, M., Lenz, D., Vogt, H., Wojciechowski, R.: Note on basic features of large time behaviour of heat kernels. J. Reine Angew. Math. 708, 73–95 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ouhabaz, E.-M.: Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series, vol. 31. Princeton University Press, Princeton (2005)zbMATHGoogle Scholar
  12. 12.
    Pedersen, G.K.: Analysis Now. Graduate Text in Mathematics, vol. 118. Springer, New York (1989)Google Scholar
  13. 13.
    Sun, H.: Mercer theorem for RKHS on noncompact sets. J. Complex. 21, 337–349 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence, RI (1975)zbMATHGoogle Scholar
  15. 15.
    Werner, D.: Funktional Analysis, 7th edn. Springer, Berlin (2011)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Applied AnalysisUniversity of UlmUlmGermany
  2. 2.Department of MathematicsUniversity of AucklandAuckland 1142New Zealand

Personalised recommendations