TY - JOUR
T1 - A Hilbert Transform for Matrix Functions on Fractal Domains
AU - Abreu-Blaya, R.
AU - Bory-Reyes, J.
AU - Brackx, F.
AU - de Schepper, H.
AU - Sommen, F.
N1 - Funding Information:
Acknowledgments This paper was written during a scientific stay of the first author at the Clifford Research Group of Ghent University, supported by a “Visiting Postdoctoral Fellowship” of the Flemish Research Foundation. He wishes to thank the members of the Clifford Research Group for their kind hospitality during this stay.
PY - 2012/4
Y1 - 2012/4
N2 - We consider Hölder continuous circulant (2 × 2) matrix functions G 2 1 defined on the fractal boundary Γ of a Jordan domain Ω in ℝ 2n. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitian Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitian Dirac operators. In Brackx et al. (Bull Braz Math Soc 40(3): 395-416, 2009) a Hermitian Cauchy integral was constructed by means of a matrix approach using circulant (2 × 2) matrix functions, from which a Hilbert transform was derived in Brackx et al. (J Math Anal Appl 344: 1068-1078, 2008) for the case of domains with smooth boundary. However, crucial parts of the method are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitian Hilbert transform in that case. As a consequence, we give necessary and sufficient conditions for the Hermitian monogenicity of a circulant matrix function G 2 1 in the interior and exterior of Ω, in terms of its boundary value g 2 1 = G 2 1{pipe}Γ, extending in this way also results of Abreu Blaya et al. (Bound. Value Probl. 2008: 2008) (article ID 425256), (article ID 385874), where Γ is required to be Ahlfors-David regular.
AB - We consider Hölder continuous circulant (2 × 2) matrix functions G 2 1 defined on the fractal boundary Γ of a Jordan domain Ω in ℝ 2n. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitian Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitian Dirac operators. In Brackx et al. (Bull Braz Math Soc 40(3): 395-416, 2009) a Hermitian Cauchy integral was constructed by means of a matrix approach using circulant (2 × 2) matrix functions, from which a Hilbert transform was derived in Brackx et al. (J Math Anal Appl 344: 1068-1078, 2008) for the case of domains with smooth boundary. However, crucial parts of the method are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitian Hilbert transform in that case. As a consequence, we give necessary and sufficient conditions for the Hermitian monogenicity of a circulant matrix function G 2 1 in the interior and exterior of Ω, in terms of its boundary value g 2 1 = G 2 1{pipe}Γ, extending in this way also results of Abreu Blaya et al. (Bound. Value Probl. 2008: 2008) (article ID 425256), (article ID 385874), where Γ is required to be Ahlfors-David regular.
KW - Cauchy integral
KW - Fractal geometry
KW - Hermitian Clifford analysis
KW - Hilbert transform
UR - http://www.scopus.com/inward/record.url?scp=84858795340&partnerID=8YFLogxK
U2 - 10.1007/s11785-010-0121-2
DO - 10.1007/s11785-010-0121-2
M3 - Artículo
SN - 1661-8254
VL - 6
SP - 359
EP - 372
JO - Complex Analysis and Operator Theory
JF - Complex Analysis and Operator Theory
IS - 2
ER -