In this paper we investigate some properties of the harmonic Bergman spaces Ap(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.
Harmonic Bergman Projectors on Homogeneous Trees
Filippo De Mari;
2024-01-01
Abstract
In this paper we investigate some properties of the harmonic Bergman spaces Ap(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.
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