The Poisson kernel and the Fourier transform of the slice monogenic Cauchy kernels

The Fueter-Sce-Qian (FSQ for short) mapping theorem is a two-steps procedure to extend holomorphic functions of one complex variable to slice monogenic functions and to monogenic functions. Using the Cauchy formula of slice monogenic functions the FSQ-theorem admits an integral representation for n odd. In this paper we show that the relation ∆ (n−1)/2 n+1 S −1 L = F L n between the slice monogenic Cauchy kernel S L and the F-kernel F L n , that appear in the integral form of the FSQ-theorem for n odd, holds also in the case we consider the fractional powers of the Laplace operator ∆n+1 in dimension n + 1, i.e., for n even. Moreover, this relation is proven computing explicitly Fourier transform of the kernels S L and F L n as functions of the Poisson kernel. Similar results hold for the right kernels S R and of F R n . AMS Classification .

[1]  F. Sommen,et al.  Generalizations of Fueter’s theorem , 2002 .

[2]  F. Sommen,et al.  The inverse Fueter mapping theorem in integral form using spherical monogenics , 2013 .

[3]  F. Sommen,et al.  The Fueter primitive of biaxially monogenic functions , 2013 .

[4]  M. Mitrea.  Clifford Wavelets, Singular Integrals, and Hardy Spaces , 1994 .

[5]  Run Fueter.  Die Funktionentheorie der DifferentialgleichungenΔu=0 undΔΔu=0 mit vier reellen Variablen , 1934 .

[6]  F. Colombo,et al.  Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes , 2019, Operator Theory: Advances and Applications.

[7]  D. Struppa,et al.  Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions , 2011 .

[8]  V. Souček,et al.  The Radon transform between monogenic and generalized slice monogenic functions , 2014, Mathematische Annalen.

[9]  F. Sommen,et al.  The inverse Fueter mapping theorem , 2011 .

[10]  F. Colombo,et al.  Spectral Theory on the S-Spectrum for Quaternionic Operators , 2019 .

[11]  Algebraic Approach to Slice Monogenic Functions , 2015, 1511.03902.

[12]  B. Jefferies.  Spectral Properties of Noncommuting Operators , 2004 .

[13]  F. Sommen,et al.  Clifford Algebra and Spinor-Valued Functions , 1992 .

[14]  D. Eelbode.  The biaxial Fueter theorem , 2014 .

[15]  D. Struppa,et al.  Entire slice regular functions , 2015, 1512.04215.

[16]  J. Gilbert,et al.  Clifford Algebras and Dirac Operators in Harmonic Analysis , 1991 .

[17]  D. Struppa,et al.  Duality theorems for slice hyperholomorphic functions , 2010 .

[18]  The Growth and Distortion Theorems for Slice Monogenic Functions , 2014, 1410.4369.

[19]  D. Alpay,et al.  Slice Hyperholomorphic Schur Analysis , 2016 .

[20]  K. Gürlebeck,et al.  Holomorphic Functions in the Plane and n-dimensional Space , 2007 .

[21]  D. Eelbode,et al.  Higher spin generalisation of Fueter's theorem , 2018 .

[22]  Tao Qian,et al.  Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces. , 1994 .

[23]  F. Sommen,et al.  Fueter’s Theorem for Monogenic Functions in Biaxial Symmetric Domains , 2016, 1611.01324.

[24]  F. Colombo,et al.  A nonconstant coefficients differential operator associated to slice monogenic functions , 2012 .

[25]  F. Sommen,et al.  Clifford Algebra and Spinor-Valued Functions: A Function Theory For The Dirac Operator , 2012 .

[26]  F. Sommen,et al.  An alternative proof of Fueter's theorem , 2006 .

[27]  F. Colombo,et al.  A Structure Formula for Slice Monogenic Functions and Some of its Consequences , 2008 .

[28]  G. Fitzgerald,et al.  'I. , 2019, Australian journal of primary health.

[29]  Tao Qian,et al.  On the inversion of Fueter's theorem , 2016 .

[30]  An extension theorem for slice monogenic functions and some of its consequences , 2010 .

[31]  The F-functional calculus for unbounded operators , 2014 .

[32]  F. Colombo,et al.  The Fueter mapping theorem in integral form and the ℱ‐functional calculus , 2010 .

[33]  F. Sommen,et al.  Analysis of Dirac Systems and Computational Algebra , 2004 .

[34]  P. Eberlein,et al.  Clifford Algebras , 2022 .

[35]  F. Sommen,et al.  A new integral formula for the inverse Fueter mapping theorem , 2013, 1302.0685.

[36]  D. Struppa,et al.  Slice monogenic functions , 2007, 0708.3595.

[37]  Tao Qian,et al.  The inverse Fueter mapping theorem for axially monogenic functions of degree k , 2019, Journal of Mathematical Analysis and Applications.

[38]  A theorem on Fourier transforms of radial functions , 1979 .

[39]  L. Grafakos.  Classical Fourier Analysis , 2010 .

[40]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[41]  D. Struppa,et al.  Noncommutative Functional Calculus , 2011 .

[42]  Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space , 1998 .