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Register a new userQuasianalytic functionals and ultradistributions as boundary values of harmonic functions
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We study boundary values of harmonic functions in spaces of quasianalytic functionals and spaces of ultradistributions of non-quasianalytic type. As an application, we provide a new approach to Hormanders support theorem for quasianalytic functionals. Our main technical tool is a description of ultradifferentiable functions by almost harmonic functions, a concept that we introduce in this article. We work in the setting of ultradifferentiable classes defined via weight matrices. In particular, our results simultaneously apply to the two standard classes defined via weight sequences and via weight functions.
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On a countable tree $T$, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with eigenvalue $lambda in mathbb{C}$. This is possible whenever $lambda$ is in the resolvent set of $P$ as a self-adjoint operator on a suitable $ell^2$-space and the on-diagonal elements of the resolvent (Green function) do not vanish at $lambda$. We show that when $P$ is invariant under a transitive (not necessarily fixed-point-free) group action, the latter condition holds for all $lambda e 0$ in the resolvent set. These results extend and complete previous results by Cartier, by Fig`a-Talamanca and Steger, and by Woess. For those eigenvalues, we also provide an integral representation of $lambda$-polyharmonic functions of any order $n$, that is, functions $f: T to mathbb{C}$ for which $(lambda cdot I - P)^n f=0$. This is a far-reaching extension of work of Cohen et al., who provided such a representation for simple random walk on a homogeneous tree and eigenvalue $lambda =1$. Finally, we explain the (much simpler) analogous results for forward only transition operators, sometimes also called martingales on trees.
In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.
We consider a countable tree $T$, possibly having vertices with infinite degree, and an arbitrary stochastic nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with eigenvalue $lambda in mathbb{C}$, under the condition that the oriented edges can be equipped with complex-valued weights satisfying three natural axioms. These axioms guarantee that one can construct a $lambda$-Poisson kernel. The boundary integral is with respect to distributions, that is, elements in the dual of the space of locally constant functions. Distributions are interpreted as finitely additive complex measures. In general, they do not extend to $sigma$-additive measures: for this extension, a summability condition over disjoint boundary arcs is required. Whenever $lambda$ is in the resolvent of $P$ as a self-adjoint operator on a naturally associated $ell^2$-space and the diagonal elements of the resolvent (`Green function) do not vanish at $lambda$, one can use the ordinary edge weights corresponding to the Green function and obtain the ordinary $lambda$-Martin kernel. We then consider the case when $P$ is invariant under a transitive group action. In this situation, we study the phenomenon that in addition to the $lambda$-Martin kernel, there may be further choices for the edge weights which give rise to another $lambda$-Poisson kernel with associated integral representations. In particular, we compare the resulting distributions on the boundary. The material presented here is closely related to the contents of our `companion paper arXiv:1802.01976
This work is dedicated to the development of the theory of Fourier hyperfunctions in one variable with values in a complex non-necessarily metrisable locally convex Hausdorff space $E$. Moreover, necessary and sufficient conditions are described such that a reasonable theory of $E$-valued Fourier hyperfunctions exists. In particular, if $E$ is an ultrabornological PLS-space, such a theory is possible if and only if E satisfies the so-called property $(PA)$. Furthermore, many examples of such spaces having $(PA)$ resp. not having $(PA)$ are provided. We also prove that the vector-valued Fourier hyperfunctions can be realized as the sheaf generated by equivalence classes of certain compactly supported $E$-valued functionals and interpreted as boundary values of slowly increasing holomorphic functions.
In 2000 V. Lomonosov suggested a counterexample to the complex version of the Bishop-Phelps theorem on modulus support functionals. We discuss the $c_0$-analog of that example and demonstrate that the set of sup-attaining functionals is non-trivial, thus answering an open question, asked in cite{KLMW}.
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