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Register a new userSpherical $Pi$-type Operators in Clifford Analysis and Applications
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The $Pi$-operator (Ahlfors-Beurling transform) plays an important role in solving the Beltrami equation. In this paper we define two $Pi$-operators on the n-sphere. The first spherical $Pi$-operator is shown to be an $L^2$ isometry up to isomorphism. To improve this, with the help of the spectrum of the spherical Dirac operator, the second spherical $Pi$ operator is constructed as an isometric $L^2$ operator over the sphere. Some analogous properties for both $Pi$-operators are also developed. We also study the applications of both spherical $Pi$-operators to the solution of the spherical Beltrami equations.
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In this paper, we study a specific system of Clifford-Appell polynomials and in particular their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows to obtain various function spaces, by specifying a suitable sequence of real numbers. We focus on the Fock and Hardy cases in this setting, and we study the action of the Fueter mapping and its range.
We prove that the Gram--Schmidt orthogonalization process can be carried out in Hilbert modules over Clifford algebras, in spite of the un-invertibility and the un-commutativity of general Clifford numbers. Then we give two crucial applications of the orthogonalization method. One is to give a constructive proof of existence of an orthonormal basis of the inner spherical monogenics of order $k$ for each $kinmathbb{N}.$ The second is to formulate the Clifford Takenaka--Malmquist systems, or in other words, the Clifford rational orthogonal systems, as well as define Clifford Blaschke product functions, in both the unit ball and the half space contexts. The Clifford TM systems then are further used to establish an adaptive rational approximation theory for $L^2$ functions on the sphere and in $mathbb{R}^m.$
We completely characterize the boundedness of the Volterra type integration operators $J_b$ acting from the weighted Bergman spaces $A^p_alpha$ to the Hardy spaces $H^q$ of the unit ball of $mathbb{C}^n$ for all $0<p,q<infty$. A partial solution to the case $n=1$ was previously obtained by Z. Wu in cite{Wu}. We solve the cases left open there and extend all the results to the setting of arbitrary complex dimension $n$. Our tools involve area methods from harmonic analysis, Carleson measures and Kahane-Khinchine type inequalities, factorization tricks for tent spaces of sequences, as well as techniques and integral estimates related to Hardy and Bergman spaces.
Let $Omega$ be a domain in $mathbb{C}$ with hyperbolic metric $lambda_Omega(z)|dz|$ of Gaussian curvature $-4.$ Mejia and Minda proved in their 1990 paper that $Omega$ is (Euclidean) convex if and only if $d(z,partialOmega)lambda_Omega(z)ge1/2$ for $zinOmega,$ where $d(z,partialOmega)$ denotes the Euclidean distance from $z$ to the boundary $partialOmega.$ In the present note, we will provide similar characterizations of spherically convex domains in terms of the spherical density of the hyperbolic metric.
Classically, theorems of Fatou and Julia describe the boundary regularity of functions in one complex variable. The former says that a complex analytic function on the disk has non-tangential boundary values almost everywhere, and the latter describes when a function takes an extreme value at a boundary point and is differentiable there non-tangentially. We describe a class of intermediate theorems in terms of averaged Julia-Fatou quotients. Boundary regularity is related to integrability of certain quantities against a special measure, the so-called Nevanlinna measure. Applications are given to spectral theory.
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