We consider trees with root at infinity endowed with flow measures, which are nondoubling measures of at least exponential growth and which do not satisfy the isoperimetric inequality. In this setting, we develop a Calderón–Zygmund theory and we define BMO and Hardy spaces, proving a number of desired results extending the corresponding theory as known in more classical settings.
We consider the Dirichlet problem on infinite and locally finite rooted trees, and we prove that the set of irregular points for continuous data has zero capacity. We also give some uniqueness results for solutions in Sobolev W 1,p of the tree.
We obtain weak type (1, 1) and L p boundedness, for p ∈ (1, ∞), of the first order Riesz transform and its adjoint operator on a homogeneous tree endowed with the canonical flow measure. This is a model case of measure metric space which is nondoubling, of exponential growth, does not satisfy the Cheeger isoperimetric inequality, and where the Laplacian does not have spectral gap. This complements a previous work by W. Hebisch and T. Steger.
In the Drury-Arveson space, we consider the subspace of functions whose Taylor coefficients are supported in the complement of a set Y ⊂ N d with the property that Y +ej ⊂ Y for all j = 1, . . . , d. This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitely calculated. Every such a space can be seen as an intersection of kernels of Hankel operators with explicit symbols. Finally, this is the right space on which Drury's inequality can be optimally adapted to a sub-family of the commuting and contractive operators originally considered by Drury.Following Drury, we will relate A to an operator acting on a Hilbert space of holomorphic functions of several variables on the unit ball. We write B d for the open unit ball {z = (z 1 , . . . , z d ) ∈ C d : |z| < 1}, where |z| 2 := d j=1 |z j | 2 . Assuming that multiplication by z j defines a bounded linear operator (and it does on the spaces we are dealing with), on such a space we can consider a very natural d-tuple of operators, namely the d-shift
We give a survey of results and proofs on two-weight Hardy's inequalities on infinite trees. Most of the results are already known but some results are new. Among the new results that we prove there is the characterization of the compactness of the Hardy operator, a reverse Hölder inequality for trace measures and a simple proof of the characterization of trace measures based on a monotonicity argument. Furthermore we give a probabilistic proof of an inequality due to Wolff. We also provide a list of open problems and suggest some possible lines of future research.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.