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This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.

Let a , b , c ∈ ℂ 2 be three non-collinear points such that their mutual joining complex lines do not intersect the unit ball B 2 and such that the line through a and b is tangent to B 2 . Then the set of lines concurrent to a , b or c is a testing family for continuous functions on S 3 . This improves a result by the authors and solves a case left open in the literature as described by Globevnik.

The aim of this paper is to introduce the H^ınfty -functional calculus for harmonic functions over the quaternions. More precisely, we give meanring to Df(T) for unbounded sectorial operators T and polynomially growing functions of the form Df , where f is a slice hyperholomorphic function and D=_q_0+e_1_q_1+e_2_q_2+e_3_q_3 is the Cauchy–Fueter operator. The harmonic functional calculus can be viewed as a modification of the well-known S -functional calculus f(T) , with a different resolvent operator. The harmonic H^ınfty -functional calculus is defined in two steps. First, for functions with a certain decay property, one can make sense of the bounded operator Df(T) directly via a Cauchy-type formula. In a second step, a regularization procedure is used to extend the functional calculus to polynomially growing functions and consequently unbounded operators Df(T) . The harmonic functional calculus is an important functional calculus of the quaternionic fine structures on the S -spectrum, which arise also in the Clifford setting and they encompass a variety of function spaces and the corresponding functional calculi. These function spaces emerge through all possible factorizations of the second map of the Fueter–Sce extension theorem. This field represents an emerging and expanding research area that serves as a bridge connecting operator theory, harmonic analysis, and hypercomplex analysis.

The spectral theory on the S -spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S -spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmonic functional calculus is the commutative version of the pseudo S -resolvent operator. This new calculus also appears, in a natural way, in the product rule for the F -functional calculus.

Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define the functions of operators. The Fueter–Sce–Qian extension theorem is a two-step procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called S -functional calculus for noncommuting operators based on the S -spectrum. In the second step this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, contains the Weyl functional calculus as a particular case. In this paper we show that the extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces. The integral representations in such spaces allow to define the associated functional calculi based on the S -spectrum. The function spaces and the associated functional calculi define the so-called fine structure of the spectral theories on the S-spectrum . Among the possible fine structures there are the harmonic and polyharmonic functions and the associated harmonic and polyharmonic functional calculi. The study of the fine structures depends on the dimension considered and in this paper we study in detail the case of dimension five, and we describe all of them. The five-dimensional case is of crucial importance because it allows to determine almost all the function spaces will also appear in dimension greater than five, but with different orders.

Infinite order differential operators appear in different fields of mathematics and physics. In the past decade they turned out to play a crucial role in the theory of superoscillations and provided new insight in the study of the evolution as initial data for the Schrödinger equation. Inspired by the infinite order differential operators arising in quantum mechanics, in this paper we investigate the continuity of a class of infinite order differential operators acting on spaces of entire hyperholomorphic functions. Precisely, we consider homomorphisms acting on functions in the kernel of the Dirac operator. For this class of functions, often called monogenic functions, we introduce the proximate order and prove some fundamental properties. As an important application, we are able to characterize infinite order differential operators that act continuously on spaces of monogenic entire functions.

The aim of this paper is to introduce the [H.sup.[infinity] -functional calculus for harmonic functions over the quaternions. More precisely, we give meanring to Df(T) for unbounded sectorial operators T and polynomially growing functions of the form Df, where f is a slice hyperholo-morphic function and [Please download the PDF to view the mathematical expression] is the Cauchy-Fueter operator. The harmonic functional calculus can be viewed as a modification of the well-known S-functional calculus f(T), with a different resolvent operator. The harmonic [H.sup.[infinity]]-functional calculus is defined in two steps. First, for functions with a certain decay property, one can make sense of the bounded operator Df(T) directly via a Cauchy-type formula. In a second step, a regularization procedure is used to extend the functional calculus to polynomially growing functions and consequently unbounded operators Df(T). The harmonic functional calculus is an important functional calculus of the quaternionic fine structures on the S-spectrum, which arise also in the Clifford setting and they encompass a variety of function spaces and the corresponding functional calculi. These function spaces emerge through all possible factorizations of the second map of the Fueter-Sce extension theorem. This field represents an emerging and expanding research area that serves as a bridge connecting operator theory, harmonic analysis, and hyper-complex analysis. Keywords: H-infinity functional calculus, harmonic functional calculus, Cauchy-Fueter operator, S-spectrum, fine structures.

Using the$H^{\\infty }$-functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order$m\\geq 1$, acting on the right linear quaternionic Hilbert space$L^{2}(\\Omega,\\mathbb {C}\\otimes \\mathbb {H})$. The operators that we consider are of the type align* T=i^m-1(a_1(x) e_1_x_1^m+a_2(x) e_2_x_2^m+a_3(x) e_3_x_3^m), x=(x_1,\\, x_2,\\, x_3)ın align* where$\\overline {\\Omega }$is the closure of either a bounded domain$\\Omega$with$C^{1}$boundary, or an unbounded domain$\\Omega$in$\\mathbb {R}^{3}$with a sufficiently regular boundary, which satisfy the so-called property$(R)$(see Definition 1.3),$e_1,\\, e_2,\\, e_3\\in \\mathbb {H}$which are pairwise anticommuting imaginary units,$a_1,\\,a_2,\\, a_3: \\overline {\\Omega } \\subset \\mathbb {R}^{3}\\to \\mathbb {R}$are the coefficients of$T$. In particular, it will be given sufficient conditions on the coefficients of$T$in order to generate the fractional powers of$T$, denoted by$P_{\\alpha }(T)$for$\\alpha \\in (0,1)$, when the components of$T$, i.e. the operators$T_l:=a_l\\partial _{x_l}^{m}$, do not commute among themselves. This kind of result is to be understood in the more general setting of the fractional diffusion problems. The method used to construct the fractional power of a quaternionic linear operator is a generalization of the method developed by Balakrishnan.

This paper aims to state compactness estimates for the Kohn-Laplacian on an abstract CR manifold in full generality. The approach consists of a tangential basic estimate in the formulation given by the first author in his thesis, which refines former work by Nicoara. It has been proved by Raich that on a CR manifold of dimension 2 n − 1 2n-1 which is compact pseudoconvex of hypersurface type embedded in the complex Euclidean space and orientable, the property named “ ( C R − P q ) (CR-P_q) ” for 1 ≤ q ≤ n − 1 2 1 q n-12 , a generalization of the one introduced by Catlin, implies compactness estimates for the Kohn-Laplacian ◻ b _b in any degree k k satisfying q ≤ k ≤ n − 1 − q q k n-1-q . The same result is stated by Straube without the assumption of orientability. We regain these results by a simplified method and extend the conclusions to CR manifolds which are not necessarily embedded nor orientable. In this general setting, we also prove compactness estimates in degree k = 0 k=0 and k = n − 1 k=n-1 under the assumption of ( C R − P 1 ) (CR-P_1) and, when n = 2 n=2 , of closed range for ∂ ¯ b _b . For n ≥ 3 n 3 , this refines former work by Raich and Straube and separately by Straube.

This paper is inspired by a class of infinite order differential operators arising in the time evolution of superoscillations. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e., functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra. We introduce the concept of proximate order and establish some fundamental properties of entire hyperholomorphic functions that are crucial for this characterization.