Harmonic Bergman Projectors on Homogeneous Trees

• Published in: Potential analysis Vol. 61; no. 1; pp. 153 - 182
• Main Authors: De Mari, Filippo, Monti, Matteo, Vallarino, Maria
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 2024; Springer Nature B.V
• Subjects: Functional Analysis; Geometry; Hilbert space; Mathematics; Mathematics and Statistics; Potential Theory; Probability Theory and Stochastic Processes; Projectors; 46E22; Bergman projectors; Homogeneous trees; Bergman spaces; 43A85; Calderón-Zygmund theory; 05C05
• ISSN: 0926-2601; 1572-929X
• DOI: 10.1007/s11118-023-10106-4

In this paper we investigate some properties of the harmonic Bergman spaces
A
p
(
σ
)
on a
q
-homogeneous tree, where
q
≥
2
,
1
≤
p
<
∞
, and
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
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
(
σ
)
for
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.