We introduce the notion of strip complex. A strip complex is a special type of complex obtained by gluing "strips" along their natural boundaries according to a given graph structure. The most familiar example is the one dimensional complex classically associated with a graph, in which case the strips are simply copies of the unit interval (our setup actually allows for variable edge length). A leading key example is treebolic space, a geometric object studied in a number of recent articles, which arises as a horocyclic product of a metric tree with the hyperbolic plane. In this case, the graph is a regular tree, the strips are [0 , 1] × R, and each strip is equipped with the hyperbolic geometry of a specific strip in upper half plane. We consider natural families of Dirichlet forms on a general strip complex and show that the associated heat kernels and harmonic functions have very strong smoothness properties. We study questions such as essential selfadjointness of the underlying differential operator acting on a suitable space of smooth functions satisfying a Kirchoff type condition at points where the strip complex bifurcates. Compatibility with projections that arise from proper group actions is also considered.

In this paper we study spaces of holomorphic functions on the Siegel upper halfspace U and prove Paley-Wiener type theorems for such spaces. The boundary of U can be identified with the Heisenberg group Hn. Using the group Fourier transform on Hn, Ogden-Vagi [OV79] proved a Paley-Wiener theorem for the Hardy space H 2 (U).We consider a scale of Hilbert spaces on U that includes the Hardy space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in particular the Drury-Arveson space, and the Dirichlet space D. For each of these spaces, we prove a Paley-Wiener theorem, some structure theorems, and provide some applications. In particular we prove that the norm of the Dirichlet space modulo constantsḊ is the unique Hilbert space norm that is invariant under the action of the group of automorphisms of U.

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We study the fractional Laplacian and the homogeneous Sobolev spaces on R d , by considering two definitions that are both considered classical. We compare these different definitions, and show how they are related by providing an explicit correspondence between these two spaces, and show that they admit the same representation. Along the way we also prove some properties of the fractional Laplacian.

We introduce and study some new spaces of holomorphic functions on the right halfplane R. In a previous work, S. Krantz, C. Stoppato and the first named author formulated the Müntz-Szász problem for the Bergman space, that is, the problem to characterize the sets of complex powers {ζ λ j −1 } with Re λj > 0 that form a complete set in the Bergman space A 2 (∆), where ∆ = {ζ : |ζ − 1| < 1}.In this paper, we construct a space of holomorphic functions on the right half-plane, that we denote by M 2 ω (R), whose sets of uniqueness {λj} correspond exactly to the sets of powers {ζ λ j −1 } that are a complete set in A 2 (∆).We show that M 2 ω (R) is a reproducing kernel Hilbert space and we prove a Paley-Wiener type theorem among several other structural properties.We introduce a transform M∆ modelled on the classical Mellin transform and show thatWe determine a sufficient condition on a set {λj } to be a set of uniqueness for M 2 ω (R), thus providing a sufficient condition for the solution of the Müntz-Szász for the Bergman space. Introduction and statement of the main resultsIn this paper we introduce and begin the analysis of a space of holomorphic functions on the right half-plane. The initial motivation for such a study arose in the work on Bergman spaces of worm domains in C 2 by S. Krantz and the first named author. In collaboration also with C. Stoppato [7] we stated the Müntz-Szász problem for the Bergman space and proved a preliminary result.We denote by ∆ the disk {ζ : |ζ − 1| < 1}, by dA the Lebesgue measure in C and consider the (unweighted) Bergman space A 2 (∆). Then the complex powers {ζ λ−1 } with Re λ > 0 are well defined and in A 2 (∆). We denote by R the right half-plane and by R its closure.Following [7], the Müntz-Szász problem for the Bergman space is the question of characterizing the sequencesThe classical Müntz-Szász theorem concerns with the completeness of a set of powers {t λ j − 1 2 } in L 2 [0, 1] , where Re λ j > 0. The solution was provided in two papers separate by C. Müntz [12] and by O. Szász [16] where they show that the set {t λ j − 1 2 } is complete L 2 [0, 1] if and only