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A Primer on the Dirichlet Space by Thomas Ransford, Javad Mashreghi, Karim Kellay, Omar El-Fallah

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Appendix D

Regularization and the rising-sun lemma

In this appendix we prove a regularization lemma, Lemma 9.4.5, used in establishing Theorem 9.4.1. The principal tool is the notion of the increasing regularization of a function, and we begin by examining this separately.

D.1 Increasing regularization

Definition D.1.1   Given a function image we define its increasing regularization image by

image

Clearly ũ is increasing and Also, ũ is maximal with these two properties, ...

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