Abstract
A general class of energy integrals in ℝn, including standard Riesz and Bessel α-energy integrals, is considered and it is shown that the modulus of continuity of positive measures for which such integrals are finite satisfy certain weighted integrability conditions. These results are deduced from an equivalent formulation of the finite energy condition in terms of related positive harmonic functions in \(R^{n+1}_{+}\). It is also shown that some of the results obtained are sharp.
Similar content being viewed by others
References
H. Aikawa and M. Essén, Potential Theory — Selected Topics, Springer-Verlag, Berlin, 1996.
D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1996.
J. M. Anderson, J. L. Fernandez and A. L. Shields, Inner functions and cyclic vectors in the Bloch space, Trans. Amer. Math. Soc. 323 (1991), 429–448.
F. Holland and J.B. Twomey, Integral means of functions with positive real part, Can. J. Math. 32 (1980), 1008–1020.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to W. K. Hayman on the occasion of his 80th birthday
Rights and permissions
About this article
Cite this article
Twomey, J.B. The Modulus of Continuity of a Measure with Finite Energy. Comput. Methods Funct. Theory 8, 47–56 (2008). https://doi.org/10.1007/BF03321669
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321669