The duals of harmonic Bergman spaces

Abstract:

In this paper we show that for $\Omega$, a starlike Lipschitz domain, the dual of the space of harmonic functions in ${L^p}(\Omega )$ need not be the harmonic functions in ${L^q}(\Omega )$, where $1/p + 1/q = 1$. We show that, as a consequence, the harmonic Bergman projection for $\Omega$ need not extend to a bounded operator on ${L^p}(\Omega )$ for all $1 < p < \infty$. The duality result is a partial answer to a question of Nakai and Sario [9] posed initially in the Proceedings of the London Mathematical Society in 1978. We treat the duality question as a biharmonic problem, and our result follows from the failure of uniqueness for the biharmonic Dirichlet problem in domains with sharp intruding corners.