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.
Publisher
Springer Science and Business Media LLC
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