A new Hilbert transform on the unit sphere in ‡

Given a harmonic function U in a domain Ω in Euclidean space, the problem of finding a harmonic conjugate V, generalizing the well-known case of the complex plane, was considered in [4] in the framework of Clifford analysis. By the nature of the given construction, which is genuinely cartesian, this approach lead to geometric constraints on the domain Ω. In this paper we consider the problem in a larger class of domains, by a spherical approach. Starting from a real-valued function u, and singling out the radial direction, we explicitly construct a harmonic function of the form w = erv, with v ∈ span(eθ1,…, eθm−1), such that u+w is monogenic, i.e. a null solution of the Dirac operator. As an illustration, the construction is applied to important classes of homogeneous monogenic polynomials and functions. Finally, it is investigated to which extent the approach also applies to the complex plane case.

In this paper we study the rigidity theorem for smooth Bakry-Emery-Kohn harmonic function u u in the unit ball B n B_n in C n \textit {C}^n , which satisfies ◻ ψ u = ∑ i , j = 1 n ( δ i j − z i z ¯ j ) ∂ 2 u ∂ z i ∂ z ¯ j + ( 1 − | z | 2 ) ψ ′ ( | z | 2 ) ∑ j = 1 n z ¯ j ∂ u ∂ z ¯ j = 0 \begin{equation*} \Box ^{\psi } u=\sum _{i, j=1}^n (\delta _{ij}-z_i \overline {z}_j){\partial ^2 u \over \partial z_i \partial \overline {z}_j}+(1-|z|^2) \psi ’(|z|^2) \sum _{j=1}^n \overline {z}_j {\partial u\over \partial \overline {z}_j}=0 \end{equation*} with some restriction of the coefficients of Taylor expansion for ψ \psi at 1 1 . We prove that any smooth B-E-K harmonic function on B ¯ n \overline {B}_n must be holomorphic in B n B_n . We study the regularity problem for the solution of the Dirichlet boundary value problem: { ◻ ψ u = 0 , if z ∈ B n , u = f , if z ∈ ∂ B n . \begin{equation*} \begin {cases} \Box ^\psi u=0, \hbox { if } z\in B_n,\cr \quad \ u=f, \hbox { if } z\in \partial B_n.\cr \end{cases} \end{equation*}

Let D be a simply-connected domain in the complex plane ℂ, and let f = u + iv be a univalent, orientation-preserving, harmonic mapping from D into ℂ. Then f can be written in the form f = h + \(\bar g\) where h and g belong to the linear space H(D) of analytic functions on D. In addition, f can be viewed as a solution of the elliptic partial differential equation $$ {\bar f_{\bar z}} = a{f_z} $$ (1.1) where the function a = g′/h′ belongs to H(D) and satisfies |a(z)| < 1 for all z ∈ D. Hence the mapping f is locally quasiconformal. Conversely, any univalent solution of (1.1) with a analytic and |a| < 1 is an orientation-preserving harmonic mapping of D (see [4]). Observe that if φ is a conformal mapping from D 1 onto D, then f 1 = f o φ is a harmonic mapping defined on D 1 with D 1 with a 1 = a o φ.

In this paper we derive for the even dimensional case a closed form of the Fourier–Borel kernel in the Clifford analysis setting. This kernel is obtained as the monogenic component in the Fischer decomposition of the exponential function \({e^{\langle \underline{x}, \underline{u} \rangle}}\) where \({\langle . , . \rangle}\) denotes the standard inner product on the m-dimensional Euclidean space. A first approach based on Clifford analysis techniques leads to a conceptual formula containing the Gamma operator and the so-called Clifford–Bessel function, two fundamental objects in the theory of Clifford analysis. To obtain an explicit expression for the Fourier–Borel kernel in terms of a finite sum of Bessel functions, this formula remains however hard to work with. To that end we have also elaborated a more direct approach based on special functions leading to recurrence formulas for a closed form of the Fourier–Borel kernel.

We provide new sharp decomposition theorems for multifunctional Bergman spaces in the unit ball and bounded pseudoconvex domains with smooth boundary expanding known results from the unit ball. Namely we prove that mΠ j=1 jjfj jjXj ≍ jjf1 : : : fmjj Ap for various (Xj) spaces of analytic functions in bounded pseudoconvex domains with smooth boundary where f; fj ; j = 1; : : : ;m are analytic functions and where Ap ; 0 &lt; p &lt; 1; &gt; �����1 is a Bergman space. This in particular also extend in various directions a known theorem on atomic decomposition of Bergman Ap spaces.

Strong asymptotics for Bergman polynomials over non-smooth domains

We prove the following. If f is a harmonic quasiconformal mapping between the unit ball in $\mathbb {R}^{n}$ and a spatial domain with C1,α boundary, then f is Lipschitz continuous in B. This generalizes some known results for n = 2 and improves some others in higher dimensional case.

The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected domain in the plane in terms of a single Ahlfors mapping of the domain, which is a proper holomorphic mapping of the domain onto the unit disc that is the analogue of the Riemann map in the multiply connected setting.

In this paper we investigate the relationship among the following integrals $$\\int_B|u(x)|^{p-i} |\\nabla u(x)|^i(1-|x|)^\\alpha dV(x),$$ where $i\\in\\{0,1,2\\}$ , $1&lt;p&lt;\\infty$ , $\\alpha&gt;0$ , and where $u$ is an arbitrary harmonic function on the unit ball $B\\subset\\bm{R}^n$ . Growth of the integral means of harmonic functions is also compared to the integral means of their gradient.

Optimal Cartan-type covers by hyperbolic discs of carriers of Green α-potentials are obtained in a simply connected domain in the complex plane and estimates of the potentials outside the carriers are presented. These results are applied to problems on the separation of singularities of analytic and harmonic functions. For instance, uniform and integral estimates in terms of Green capacities of components of meromorphic functions are obtained.

The local analysis of signals arising on the sphere is a common task in earth sciences. On the real line the analytic signal turned out to be an important representation in local one-dimensional signal processing. Its generalization to two dimensions is the monogenic signal, and the properties of the analytic and the monogenic signal in the Fourier domain are well known. A generalization to the sphere is given by the Hilbert transform on the sphere known from Clifford analysis. To obtain a spectral characterization, the transform has to be decomposed into spherical harmonic functions. In this paper, we derive the spherical harmonic coefficients of the Hilbert transform on the sphere and give a series expansion. This will show that it acts as a differential operator on the spherical harmonic basis functions of the Laplace equation solution, analogously to the Riesz transform in two dimensions. This allows an interpretation of the Hilbert transform suitable for signal processing of signals naturally arising on the two-sphere. We show that the scale space naturally arising is a Poisson scale space in the unit ball. In addition, the obtained interpretation of the Hilbert transform is used for orientation analysis of plane waves. This representation is justified as a novel signal model on the sphere which can be used to construct intensity and rotation-invariant operators for local signal analysis in a scale-space concept.

In recent work a radial deformation of the Fourier transform in the setting of Clifford analysis was introduced. The key idea behind this deformation is a family of new realizations of the Lie superalgebra $${\mathfrak {osp}}(1|2)$$ in terms of a so-called radially deformed Dirac operator $${\mathbf {D}}$$ depending on a deformation parameter c such that for $$c=0$$ the classical Dirac operator is reobtained. In this paper, several versions of the Paley–Wiener theorems for this radially deformed Fourier transform are investigated, which characterize the supports of functions associated to this generalized Fourier transform in Clifford analysis.

We develop a rigorous algebraic–analytic framework for multidual complex numbers DCn within the setting of Clifford analysis and establish a comprehensive theory of hyperholomorphic multidual complex-valued functions. Our main contributions are (i) a fully coupled multidual Cauchy–Riemann system derived from the Dirac operator, yielding precise differentiability criteria; (ii) generalized conjugation laws and the associated norms that clarify metric and geometric structure; and (iii) explicit operator and kernel constructions—including generalized Cauchy kernels and Borel–Pompeiu-type formulas—that produce new representation theorems and regularity results. We further provide matrix–exponential and functional calculus representations tailored to DCn, which unify algebraic and analytic viewpoints and facilitate computation. The theory is illustrated through a portfolio of examples (polynomials, rational maps on invertible sets, exponentials, and compositions) and a solvable multidual boundary value problem. Connections to applications are made explicit via higher-order automatic differentiation (using nilpotent infinitesimals) and links to kinematics and screw theory, highlighting how multidual analysis expands classical holomorphic paradigms to richer, nilpotent-augmented coordinate systems. Our results refine and extend prior work on dual/multidual numbers and situate multidual hyperholomorphicity within modern Clifford analysis. We close with a concise summary of notation and a set of concrete open problems to guide further development.

We study Hardy spaces H p \mathcal {H}^p , 0 &gt; p &gt; ∞ 0&gt;p&gt;\infty for quasiregular mappings on the unit ball B B in R n {\mathbb R}^n which satisfy appropriate growth and multiplicity conditions. Under these conditions we recover several classical results for analytic functions and quasiconformal mappings in H p \mathcal {H}^p . In particular, we characterize H p \mathcal {H}^p in terms of non-tangential limit functions and non-tangential maximal functions of quasiregular mappings. Among applications we show that every quasiregular map in our class belongs to H p \mathcal {H}^p for some p = p ( n , K ) p=p(n,K) . Moreover, we provide characterization of Carleson measures on B B via integral inequalities for quasiregular mappings on B B . We also discuss the Bergman spaces of quasiregular mappings and their relations to H p \mathcal {H}^p spaces and analyze correspondence between results for H p \mathcal {H}^p spaces and A \mathcal {A} -harmonic functions. A key difference between the previously known results for quasiconformal mappings in R n {\mathbb R}^n and our setting is the role of multiplicity conditions and the growth of mappings that need not be injective. Our paper extends results by Astala and Koskela, Jerison and Weitsman, Jones, Nolder, and Zinsmeister.

We adapt Sklyanin’s K-matrix formalism to the sinh–Gordon equation, and prove that all free boundary constant mean curvature annuli in the unit ball in R3 are of finite type.