arXiv subjects
Publications and source records attributed to Elena Rizzo.
We prove vector-valued boundedness of (suitable) Calderon-Zygmund operators and of the (truncated) Hardy-Littlewood maximal function on a connected locally doubling metric measure space.
We prove endpoint and sparse-like bounds for Bergman projectors on nonhomogeneous, radial trees $X$ that model manifolds with possibly unbounded geometry. The natural Bergman measures on $X$ may fail to be doubling, and even locally doubling, with respect to the right metric in our setting. Weighted consequences of our sparse domination results are also considered, and are in line with the known results in the disk. Our endpoint results are partly a consequence of a new Calder\'on-Zygmund theory for discrete, non-locally doubling metric spaces.
The main focus of this contribution is on the harmonic Bergman spaces $\mathcal{B}_α^{p}$ on the $q$-homogeneous tree $\mathfrak{X}_q$ endowed with a family of measures $σ_α$ that are constant on the horocycles tangent to a fixed boundary point and turn out to be doubling with respect to the corresponding horocyclic Gromov distance. A central role is played by the reproducing kernel Hilbert space $\mathcal{B}_α^{2}$ for which we find a natural orthonormal basis and formulae for the kernel. We also consider the atomic Hardy space and the bounded mean oscillation space. Appealing to an adaptation of Calderón-Zygmund theory and to standard boundedness results for integral operators on $L^p_α$ spaces with Hörmander-type kernels, we determine the boundedness properties of the Bergman projection.