Measure, Integration & Real Analysis · Problem 8C.25

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · dirichlet-versus-bergman-space

Problem

(a) Prove that the Dirichlet space

\mathcal{D}

is contained in the Bergman space

L^2_a(\mathbf{D})

. (b) Prove that there exists a function

f \in L^2_a(\mathbf{D})

such that

f

is uniformly continuous on

\mathbf{D}

and

f \notin \mathcal{D}