Harmonic Bergman Projectors on Homogeneous Trees

Abstract

AbstractIn this paper we investigate some properties of the harmonic Bergman spaces $$\mathcal A^p(\sigma )$$ A p ( σ ) on a q-homogeneous tree, where $$q\ge 2$$ q ≥ 2 , $$1\le p<\infty $$ 1 ≤ p < ∞ , and $$\sigma $$ σ is a finite measure on the tree with radial decreasing density, hence nondoubling. These spaces were introduced by J. Cohen, F. Colonna, M. Picardello and D. Singman. When $$p=2$$ p = 2 they are reproducing kernel Hilbert spaces and we compute explicitely their reproducing kernel. We then study the boundedness properties of the Bergman projector on $$L^p(\sigma )$$ L p ( σ ) for $$1<p<\infty $$ 1 < p < ∞ and their weak type (1,1) boundedness for radially exponentially decreasing measures on the tree. The weak type (1,1) boundedness is a consequence of the fact that the Bergman kernel satisfies an appropriate integral Hörmander’s condition.