Korean J. Math.  Vol 26, No 1 (2018)  pp.87-101
DOI: https://doi.org/10.11568/kjm.2018.26.1.87

On hyperholomorphic $F^{\alpha}_{\omega,G}(p,q,s)$ spaces of quaternion valued functions

Alaa Kamal, Taha Ibrahim Yassen


The purpose of this paper is to define a new class of hyperholomorphic functions spaces, which will be called $F^{\alpha}_{\omega,G}(p,q,s)$ type spaces. For this class, we characterize hyperholomorphic weighted  $\alpha$-Bloch functions by functions belonging to $F^{\alpha}_{\omega,G}(p,q,s)$ spaces under some mild conditions. Moreover, we give some essential properties for the extended weighted little $\alpha$-Bloch spaces. Also, we give the characterization for the hyperholomorphic weighted Bloch space by the integral norms of $F^{\alpha}_{\omega,G}(p,q,s)$ spaces of hyperholomorphic functions. Finally, we will give the relation between the hyperholomorphic  ${\mathcal{B}}^{\alpha}_{\omega,0}$ type spaces and the hyperholomorphic  valued-functions space $F^{\alpha}_{\omega,G}(p,q,s)$.


Quaternionic analysis, $F^{\alpha}_{\omega,G}(p,q,s)$ spaces, hyperholomorphic functions, Clifford analysis

Subject classification

$30G35$, $46E15$


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R. Ablamowicz and B. Fauser, Mathematics of Clifford - a Maple package for Clifford and Graßmann algebras, Adv.Appl. Clifford Algebr. 15 (2) (2005), 157–181. (Google Scholar)

R. Ablamowicz, Computations with Clifford and Graßmann algebras, Adv. Appl. Clifford Algebr. 19 (3-4) (2009), 499–545. (Google Scholar)

S. Bernstein, Harmonic Qp spaces, Comput. Meth. Fun. Theory. 9 (1) (2009), 285–304. (Google Scholar)

K.M. Dyakonov, Weighted Bloch spaces, Hp and BMOA, J. Lond. Math. Soc. II. Ser. 65 (2) (2002), 411–417. (Google Scholar)

A. El-Sayed Ahmed, On weighted α-Besov spaces and α-Bloch spaces of quaternion-valued functions, N. Fun. Anal. Opt., 29(9-10)(2008), 1064-1081 (Google Scholar)

A. El-Sayed Ahmed and S. Omran, Weighted classes of quaternion-valued functions, Banach J. Math. Anal. 6 (2) (2012), 180–191. (Google Scholar)

A.El-SayedAhmed,A.KamalandT.I.Yassen,CharacterizationsforQK,ω(p,q) type functions by series expansions with Hadamard gaps, CUBO. A Math. J. 01 (2014), 81–93. (Google Scholar)

A. El-Sayed Ahmed and Fatima Asiri, Characterizations of weighted Bloch Space by Qp,ω -type Spaces of quaternion-valued functions, J. Comput. Theor. Nanosc. 12 (2015), 4250–4255. (Google Scholar)

A. Kamal, A. El-Sayed Ahmed and T. Yassen, Carleson measures and Hadamard products in some general analytic function spaces, J. Comput. Theor. Nsc. 12 (2015), 2227–2236 (Google Scholar)

K. Gu ̈rlebeck, U. K ̈ahler, M. Shapiro, and L. M. Tovar, On Qp spaces of quaternion-valued functions, Complex Variables, 39 (1999), 115–135. (Google Scholar)

K. Gu ̈rlebeck and A. El-Sayed Ahmed, Integral norms for hyperholomorphic Bloch-functions in the unit ball of IR3, Progress in Analysis (Begehr et al., eds.) Kluwer Academic, (2003), 253–263. (Google Scholar)

K. Gu ̈rlebeck and A. El-Sayed Ahmed, On Bq Spaces of Hyperholomorphic Functions and the Bloch Space in R3, Finite or Infinite Dim. Com. Anal. Appl. (2004), 269–286. (Google Scholar)

K. Gu ̈rlebeck, K. Habetha, and W. Spr ̈oßig, Holomorphic functions in the plane and n-dimensional space, Birkhau ̈ser Verlag, Basel (2008). (Google Scholar)

H.R. Malonek, Quaternions in applied sciences, a historical perspective of a mathematical concept, 17th Inter.Conf. on the Appl. of Computer Science and Mathematics on Architecture and Civil Engineering, Weimar (2003). (Google Scholar)

R. A. Rashwan, A. El-Sayed Ahmed and Alaa Kamal, Integral characterizations of weighted Bloch spaces and QK,ω(p,q) spaces, Math. tome. 51 (74) (2) (2009), 63–76. (Google Scholar)

L. F. Res ́endis and L. M. Tovar, Besov-type characterizations for Quaternionic Bloch functions,In: Le Hung Son et al (Eds) finite or infinite complex Analysis and its applications, Adv. Complex Analysis and applications, (Boston MA: Kluwer Academic Publishers) (2004), 207–220. (Google Scholar)

R. Zhao, On a general family of function spaces, Annal. Acad. Scien. Fennicae. Series A I. Math. Diss. 105. Helsinki: Suomalainen Tiedeakatemia, (1996), 1–56. (Google Scholar)


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