  The first two authors and Kohnen have recently introduced a new class of
modular objects called locally harmonic Maass forms, which are annihilated
almost everywhere by the hyperbolic Laplacian operator. In this paper, we
realize these locally harmonic Maass forms as theta lifts of harmonic weak
Maass forms. Using the theory of theta lifts, we then construct examples of
(non-harmonic) local Maass forms, which are instead eigenfunctions of the
hyperbolic Laplacian almost everywhere.
