Alaa Kamal and Taha Ibrahim Yassen

Transcription

1 Korean J. Math ), No. 1, htts://doi.org/ /kjm ON HYPERHOLOMORPHIC Fω,G α, q, s) SPACES OF QUATERNION VALUED FUNCTIONS Alaa Kamal and Taha Ibrahim Yassen Abstract. The urose of this aer is to define a new class of hyerholomorhic functions saces, which will be called Fω,G α, q, s) tye saces. For this class, we characterize hyerholomorhic weighted α-bloch functions by functions belonging to Fω,G α, q, s) saces under some mild conditions. Moreover, we give some essential roerties for the extended weighted little α-bloch saces. Also, we give the characterization for the hyerholomorhic weighted Bloch sace by the integral norms of Fω,G α, q, s) saces of hyerholomorhic functions. Finally, we will give the relation between the hyerholomorhic,0 α tye saces and the hyerholomorhic valued-functions sace Fω,G α, q, s). 1. Introduction Quaternions were introduced for the first time by William Rowan Hamilton in Quaternion analysis is the generalizations of the theory of holomorhic functions in one comlex variable to Euclidean sace. The concet of the hyerholomorhic functions based on the consideration of functions in the kernel of the generalized Cauchy-Riemann oerator. Quaternions are also recognized as a owerful tool for modeling and solving roblems in theoretical as well as alied mathematics see [14]). The emergence of a large of software ackages to erform Received June 11, Revised March 18, Acceted March 20, Mathematics Subject Classification: 30G35, 46E15. Key words and hrases: Quaternionic analysis, Fω,G α, q, s) saces, hyerholomorhic functions, Clifford analysis. c The Kangwon-Kyungki Mathematical Society, This is an Oen Access article distributed under the terms of the Creative commons Attribution Non-Commercial License htt://creativecommons.org/licenses/by -nc/3.0/) which ermits unrestricted non-commercial use, distribution and reroduction in any medium, rovided the original work is roerly cited.

2 88 A. Kamal and T. I. Yassen comutations in the algebra of the real quaternions see [13]), or more generally, in Clifford Algebra has been enhanced by the increasing interest in using quaternions and their alications in almost all alied sciences see [1, 2]). 2. Preliminaries 2.1. Analytic function saces. Let D = {z C : z < 1} be the comlex unit disk. The well known Bloch sace is defined by: B = {f : f analytic in D and Bf) = su1 z 2 ) f z) < }. z D Comosing the Möbius transform ϕ a z), which mas the unit disk D onto itself, and the fundamental solution of the two-dimensional real Lalacian on D, we have the Green s function gz, a) = ln 1 az with a z logarithmic singularity at a D. Here, ϕ a always stands for the Möbius transformation ϕ a z) = a z. Then, in [17] Zhao gave the following 1 āz definition: Definition 2.1. Let f be an analytic function in D and let 0 < <, 2 < q < and 0 < s <. If f F,q,s) = su f z) 1 z 2 ) q g s z, a)daz) <, a D D then f F, q, s). Moreover, if f z) 1 z 2 ) q g s z, a)daz) = 0, lim a 1 then f F 0, q, s). D Definition 2.2. see [4]) Let a right-continuous and nondecreasing function ω : 0, 1] 0, ), the weighted Bloch sace B ω is defined as the set of all analytic functions f on D satisfying 1 z ) f z) Cω1 z ), z D, for some fixed C = C f > 0. In the secial case where ω 1, B ω reduces to the classical Bloch sace B.

3 On Hyerholomorhic Fω,G α, q, s) Saces 89 Definition 2.3. see [15]) Let 0 < α < and ω : 0, 1] 0, ). For an analytic function f in D, we define the weighted α-bloch sace, α as follows: B α ω = { f : f analytic in D and f B α ω 1 z ) α f z) = su z D ω1 z ) } <. Also, the little weighted α-bloch sace,0 α is a subsace of α consisting of all f, α such that [7, 9] 1 z ) α f z) lim = 0. z 1 ω1 z ) 2.2. Quaternion function saces. To introduce the meaning of hyerholomorhic functions, let IH be the skew field of quaternions. This means we can write each element w IH in the form w = w 0 + w 1 i + w 2 j + w 3 k, w 0, w 1, w 2, w 3 IR, where 1, i, j, k are the basis elements of IH. For these elements we have the multilication rules i 2 = j 2 = k 2 = 1, ij = ji = k, kj = jk = i, ki = ik = j. The conjugate element w is given by w = w 0 w 1 i w 2 j w 3 k. Then, we have the roerty w w = ww = w 2 = w w w w 2 3. Moreover, we can identify each vector x = x 0, x 1, x 2 ) IR 3 with a quaternion x of the form x = x 0 + x 1 i + x 2 j. In follows we will work in B 1 0) IR 3, the unit ball in the real threedimensional sace. We will consider functions f defined on B 1 0) with values in IH. We define a generalized Cauchy-Riemann oerator D by Df = f + i f + j f, x 0 x 1 x 2 and it s conjugate oerator by Df = f i f j f. x 0 x 1 x 2 For these oerators, we have that DD = DD = 3,

4 90 A. Kamal and T. I. Yassen where 3 is the Lalacian for functions defined over domains in IR 3. For a < 1, we will denote by ϕ a x) = a x)1 āx) 1 the Möbius transform, which mas the unit ball onto itself. Furthermore, let gx, a) = 1 ) 1 4π ϕ a x) 1, be the modified fundamental solution of the Lalacian in IR 3 comosed with the Möbius transform ϕ a x). Esecially, we denote for all 0 g x, a) = 1 ) 1 4 π ϕ a x) 1. Let f : B IH be a hyerholomorhic function. Then from [10], we have the seminorms Bf) = su1 x 2 ) 3/2 Dfx), x B Q f) = su B Dfx) 2 g x, a)db x. a B Definition 2.4. Let 0 < α <. Recall that the hyerholomorhic α-bloch sace see [5]) is defined as follows: B α = {f ker D : su1 x 2 ) 3α 2 Dfx) < }, x B the little α-bloch tye sace B0 α is a subsace of B consisting of all f B α such that lim x 1 1 x 2 ) 3α 2 Dfx) = 0. Quite recently, El-Sayed Ahmed and Omran in [6], gave the following definition: Definition 2.5. Let f be quaternion-valued function in B. For 0 < <, 2 < q < and 0 < s <. If ) s f F,q,s) = su Dfx) 1 x 2 ) 3q 2 1 ϕ a x) 2 db x <, a B then f F, q, s). Moreover, if lim a 1 then f F 0, q, s). B B Dfx) 1 x 2 ) 3q 2 1 ϕ a x) 2 ) s db x = 0,

5 On Hyerholomorhic Fω,G α, q, s) Saces 91 Ahmed and asiri in [8], gave the following definition: Definition 2.6. For given right-continuous and nondecreasing function ω : 0, 1] 0, ), and 0 < α <. A quaternion-valued function f on B 1 0) is belong to the weighted α- Bloch sace B α ω, if 1 x 2 ) 3α 2 f B α ω = su x B 1 0) ω1 x ) Dfx) <. Moreover, A quaternion-valued function f B α ω,0, if 1 x 2 ) 3α 2 f B α ω,0 = lim Dfx) <. x 1 ω1 x ) Now, we use the definition of green function in IR 3 see [3]) Gx, a) = 1 ϕ ax) 2. 1 ax Then, we introduce the following new definition of the so called the hyerholomorhic Fω,G α, q, s) saces. Definition 2.7. Let 1 < α, <, 2 < q <, s > 0, and ω : 0, 1] 0, ). Assume that f be hyerholomorhic function in the unit ball B 1 0). Then, f Fω,G α, q, s), if = Fω,G, α q, s) { f kerd : su Dfx) 1 x 2 ) 3αq a B 1 0) B 1 0) ω 1 x ) } ) sdbx Gx, a) <. The sace Fω,G,0 α, q, s) is subsace of F ω,g α, q, s) consisting of all functions f Fω,G α, q, s), such that lim Dfx) 1 x 2 ) 3αq ) sdbx Gx, a) = 0. a 1 ω 1 x ) B 1 0) Our objective in this article is twofold. First, we study the generalized quaternion saces Fω,G α, q, s). Second, we characterize their relations to the quaternion,0 α sace. The following lemma, we will need in the sequel: Lemma 2.1. see [16]). Let f : B 1 0) IH be a hyerholomorhic function. Let 0 < R < 1, 1 < q. Then for every a B 1 0) Dfa) q q Dfx) q dbx. πr 3 1 R 2 ) 2q 1 a 2 ) 3 Ma,R)

6 92 A. Kamal and T. I. Yassen 3. Characterization of Fω,G α, q, s) saces in Clifford Analysis The relations between F α ω,g, q, s) sace and Bα ω saces are given in quaternion sense. The results in this section are extensions and generalize of the results see [10, 11]). Proosition 3.1. Let α, 1, 2 < q <, s > 0, 0 < R < 1, and ω : 0, 1] 0, ). Assume that f be hyerholomorhic function in the unit ball B 1 0). Then 1 a 2 ) 3αq ω 1 a ) Dfa) 482) R) s πr 3 1 R 2 ) 2s+s 1 a 2 ) 3 B 1 0) Dfx) 1 x 2 ) 3αq ω 1 a ) Gx, a) ) sdbx. Proof. Let Ma, R) = {x B 1 0) : ϕ a x) = a x < R} be seudohyerbolic ball with center a and radius R. Then 1 ax Dfx) 1 x 2 ) 3αq ) sdbx Gx, a) B 1 0) ω 1 x ) Dfx) 1 x 2 ) 3αq ) sdbx Gx, a). ω 1 x ) Since and Then, we have Ma,R) ϕ a x) < R, x Ma, R), Gx, a) = 1 ϕ ax) 2. 1 ax Gx, a) = 1 R2, where 1 R 1 ax 1 + R. 1 + R Now, for fixed R 0, 1) and a B 1 0). Let Ea, R) Ma, R), sutch that Then, we deduce that Ea, R) = {x B 1 0) : x a < R 1 a }.

7 On Hyerholomorhic Fω,G α, q, s) Saces 93 Dfx) 1 x 2 ) 3αq ) sdbx Gx, a) B 1 0) ω 1 x ) ) 1 R 2 s Dfx) 1 x 2 ) 3q db 1 + R Ma,R) ω x 1 x ) ) 1 R 2 s Dfx) 1 x 2 ) 3αq db 1 + R Ea,R) ω x 1 x ) ) 1 R 2 s 1 a 2 ) 3αq Dfx) db 1 + R ω x. 1 a ) Ea,R) Now, using Lemma 3.1, we obtain Dfx) 1 x 2 ) 3αq ) sdbx Gx, a) B 1 0) ω 1 x ) ) 1 R 2 s 1 a 2 ) 3αq πr 3 1 R 2 ) 2 1 a 2 ) 3 Dfa) 1 + R ω 1 a ) 34) = πr3 1 R 2 ) 2+s 1 a 2 ) 3αq 34) R) s ω 1 a ) which imlies that, Dfa), 1 a 2 ) 3αq ω 1 a ) Dfa) 482) R) s πr 3 1 R 2 ) 2+s 1 a 2 ) 3 This comletes the roof. B 1 0) Dfx) 1 x 2 ) 3αq ω 1 x ) Gx, a) ) sdbx. Corollary 3.1. From roosition 3.1, we get for α, 1, 2 < q <, s > 0, and ω : 0, 1] 0, ) that Fω,G, α q, s). Proosition 3.2. Let α, 1, 2 < q <, s > 2, and ω : 0, 1] 0, ). Let f be a hyerholomorhic function in B 1 0), a B 1 0);

8 94 A. Kamal and T. I. Yassen a < 1 and f. Then, we have that Dfx) 1 x 2 ) 3αq ) sdbx Gx, a) kb γ ω 1 x ) ωf)), B 1 0) where γ =. Proof. From see [8, 10]), we have Then, 1 x 2 ) 3α 2 ω 1 x ) Dfa) B α ωf). Dfx) 1 x 2 ) 3αq B 1 0) ω 1 x ) ) f)) γ sdbx Gx, a). B 1 0) Gx, a) ) sdbx Using the equality where Gx, a) = 1 ϕ ax) 2 1 ax = 1 a 2 )1 x 2 ) 1 ax 3, 1) 1 x 1 ax 1 + x, 1 a 1 ax 1 + a 2. 2) Then, we get Dfx) 1 x 2 ) 3αq ) sdbx Gx, a) B 1 0) ω 1 x ) 1 a 2 ) s 1 x 2 ) s f)) γ db B 1 0) 1 ax 3s x. 2 2s 1 a ) s 1 x ) s f)) γ B 1 0) 1 x s )1 a ) db x. 2s 2 2s 1 f)) γ 1 a ) db x. s kb γ ωf)). B 1 0) Therefore, the roof of roosition is comlete.

9 On Hyerholomorhic Fω,G α, q, s) Saces 95 Corollary 3.2. From roosition 3.2, we get for α, 1, 2 < q <, s > 2, and ω : 0, 1] 0, ) that F α ω,g, q, s). The results in Corollary 3.1 and Corollary 3.2 rove the following theorem, which give to us the characterization for the hyerholomorhic weighted Bloch sace by the integral norms of Fω,G α, q, s) saces of hyerholomorhic functions. Theorem 3.1. Let f be a hyerholomorhic function in B 1 0). Then for α, 1, 2 < q <, s > 2, and ω : 0, 1] 0, ), we have = F α ω,g, q, s). For characterization the little hyerholomorhic weighted Bloch sace, used the the same arguments in the revious theorem to rove the following theorem. Theorem 3.2. Let f be a hyerholomorhic function in B 1 0). Then for α, 1, 2 < q <, s > 2, and ω : 0, 1] 0, ), we have,0 = Fω,G,0, α q, s). Theorem 3.3. Let 0 < R < 1 and ω : 0, 1] 0, ). Then for the hyerholomorhic function f in B 1 0), the following are equivalent a) f B ω. b) For each 2 < q <, 1 α <, and 0 < < Dfx) 1 x 2 ) 3αq ) sdbx Gx, a) < +. ω 1 x ) su a B 1 0) B 1 0) c) For each 2 < q <, 1 α <, and 0 < < Dfx) 1 x 2 ) 3αq db ω x < +. 1 x ) su a B 1 0) su a B 1 0) Ma,R) d) For each 2 < q <, 1 α <, and 0 < < Ma, R) αq 2 + 2s 3 D fx) db ω x < +. 1 a ) Ma,R)

10 96 A. Kamal and T. I. Yassen Proof. To rove a) imlies b). Using 1) and 2), we have su a B 1 0) B 1 0) su a B 1 0) Dfx) 1 x 2 ) 3αq ω 1 x ) Dfx) 1 x 2 ) 3αq ω 1 x ) B 1 0) Gx, a) ) sdbx Gx, a) ) sdbx. Using the same stes as in Proosition 3.2, we obtain su a B 1 0) B 1 0) 2 2s f K 1 f <. B 1 0) Dfx) 1 x 2 ) 3αq ω 1 x ) 1 1 a ) db s x Gx, a) ) sdbx b) imlies c), using the same stes as in Proosition 3.1, we deduce that su Dfx) 1 x 2 ) 3αq ) sdbx Gx, a) a B 1 0) B 1 0) ω 1 x ) ) 1 R 2 s su Dfx) 1 x 2 ) 3αq db 2 a B 1 0) ω x. 1 x ) Ma,R) For c) imlies d), we use the fact 1 x 2 ) 3 Ma, R) see [12]). Then su a B 1 0) Ma,R) αq Ma, R) 2 + 2s 3 ω 1 a ) Ma, R), x Dfx) 1 x 2 ) 3αq db ω x 1 x ) su Dfx) db x. a B 1 0) Ma,R)

11 On Hyerholomorhic Fω,G α, q, s) Saces 97 For d) imlies a). Form Lemma 2.1, we have = Now, since Dfa) 1 a 2 ) 3αq ω 1 a ) a 2 ) 3αq πr 3 1 R 2 ) 2 1 a 2 ) 3 ω 1 a ) a 2 ) 3αq πr 3 1 R 2 ) 2 1 a 2 ) 3 ω 1 a ) 1 R2 a 2 ) 3αq R 3αq 1 R 2 a 2 ) 3αq R 3αq Ma,R) Ma,R) Ma, R) = 1 a 2 ) 3 1 R 2 a 2 ) 3 R3. Also, we used the following inequalities Dfx) dbx. Dfx) dbx 1 R 2 1 R 2 a R 2 and 1 a 2 1 R 2 a a 2. Then, we have Dfa) 1 a 2 ) 3αq ω 1 a ) Ma, R) αq 2 + 2s 3 πr 3 1 R 2 ) 2 1 a 2 ) 3 ω 1 a ) 1 R2 a 2 ) 3αq R 3αq Ma,R) Ma, R) αq 2 + 2s 3 πr 3 1 R 2 ) 2 1 R 2 ) 3 ω 1 a ) 1 + R2 ) 3αq R 3αq = R 2 ) 3 αq 2 + 2s 3 ) αq 31+ πr 2 + 2s 3 ) 1 R 2 ) Therefore, our theorem is roved. Ma,R) Dfx) db x Dfx) dbx αq 2 + 2s 3 Ma, R) 2+3 ω 1 a ) Ma,R) Dfx) db x. From Theorem 3.3, using the same arguments, we directly obtain the following theorem.

12 98 A. Kamal and T. I. Yassen Theorem 3.4. Let 0 < R < 1 and ω : 0, 1] 0, ). Then for the hyerholomorhic function f in B 1 0), the following are equivalent a) f B ω,0. b) For each 2 < q <, 1 α <, and 0 < < lim a 1 B 1 0) Dfx) 1 x 2 ) 3αq ω 1 x ) Gx, a) ) sdbx = 0. c) For each 2 < q <, 1 α <, and 0 < < lim a 1 Ma,R) Dfx) 1 x 2 ) 3αq ω 1 x ) db x = 0. d) For each 2 < q <, 1 α <, and 0 < < αq Ma, R) 2 + 2s 3 lim D fx) db a 1 ω x = 0. 1 a ) Ma,R) The following theorem give another relation between the quaternion α sace and the quaternion valued-functions sace Fω,G α, q, s). Theorem 3.5. Let f be a hyerholomorhic function in B 1 0). Then for α, 1, 2 < q <, β, s > 0, and ω : 0, 1] 0, ), we have f B ω B 1 0) Dfx) 1 x 2 ) 3αq ω 1 x ) Proof. From Theorem 3.3, we have f Dfx) 1 x 2 ) 3αq Ma,R) ω 1 x ) Let the constant ) 1 R CR) = 1 R 2 ) β 2 s, 1 + R since CR) deending on R is finite, then 1 ϕ a x) 2 ) β Gx, a) ) s dbx. db x.

13 On Hyerholomorhic Fω,G α, q, s) Saces 99 f su a B 1 0) Ma,R) Dfx) 1 x 2 ) 3αq ω 1 x ) 3) ) 1 R 1 R 2 ) β 2 s db x. 1 + R Since x Ma, R), then ϕ a x) < R, ϕ a x) 2 < R 2, and 1 ϕ a x) 2 > 1 R 2. Then, we have f su a B 1 0) λ su a B 1 0) Ma,R) B 1 0) Dfx) 1 x 2 ) 3αq ω 1 x ) Dfx) 1 x 2 ) 3αq ω 1 x ) Conversely, we have Dfx) 1 x 2 ) 3αq su a B 1 0) f B ω B 1 0) su a B 1 0) B 1 0) ω 1 x ) Using 1) and 2), we obtain Dfx) 1 x 2 ) 3αq su a B 1 0) f B ω B 1 0) su a B 1 0) 2 2β+s) f B ω Then, we obtain f λ su a B 1 0) B 1 0) B 1 0) su a B 1 0) ) 1 R 1 R 2 ) β 2 s db x 1 + R 4) 1 ϕ a x) 2 ) β Gx, a) ) s dbx. 1 ϕ a x) 2 ) β Gx, a) ) s dbx 1 ϕ a x) 2 ) β Gx, a) ) s dbx. 5) ω 1 x ) 1 ϕ a x) 2 ) β Gx, a) ) s dbx 1 x 2 ) β 1 a 2 ) β 1 x 2 ) s 1 a 2 ) s 1 x ) β 1 a ) 2s 1 x ) db x. s 1 a ) β s db x. 6) B 1 0) D fx) 1 x 2 ) 3αq ω 1 x ) 7) 1 ϕ a x) 2 ) β Gx, a) ) s dbx.

14 100 A. Kamal and T. I. Yassen From 4) and 7) the roof is comlete. 4. Acknowledgements The authors would like to thank the referees for their useful comments, which imroved the original manuscrit. References [1] R. Ablamowicz and B. Fauser, Mathematics of Clifford - a Male ackage for Clifford and Graßmann algebras, Adv.Al. Clifford Algebr. 15 2) 2005), [2] R. Ablamowicz, Comutations with Clifford and Graßmann algebras, Adv. Al. Clifford Algebr ) 2009), [3] S. Bernstein, Harmonic Q saces, Comut. Meth. Fun. Theory. 9 1) 2009), [4] K.M. Dyakonov, Weighted Bloch saces, H and BMOA, J. Lond. Math. Soc. II. Ser. 65 2) 2002), [5] A. El-Sayed Ahmed, On weighted α-besov saces and α-bloch saces of quaternion-valued functions, N. Fun. Anal. Ot., )2008), [6] A. El-Sayed Ahmed and S. Omran, Weighted classes of quaternion-valued functions, Banach J. Math. Anal. 6 2) 2012), [7] A. El-Sayed Ahmed, A. Kamal and T. I. Yassen, Characterizations for Q K,ω, q) tye functions by series exansions with Hadamard gas, CUBO. A Math. J ), [8] A. El-Sayed Ahmed and Fatima Asiri, Characterizations of weighted Bloch Sace by Q,ω -tye Saces of quaternion-valued functions, J. Comut. Theor. Nanosc ), [9] A. Kamal, A. El-Sayed Ahmed and T. Yassen, Carleson measures and Hadamard roducts in some general analytic function saces, J. Comut. Theor. Nsc ), [10] K. Gürlebeck, U. Kähler, M. Shairo, and L. M. Tovar, On Q saces of quaternion-valued functions, Comlex Variables, ), [11] K. Gürlebeck and A. El-Sayed Ahmed, Integral norms for hyerholomorhic Bloch-functions in the unit ball of IR 3, Progress in Analysis Begehr et al., eds.) Kluwer Academic, 2003), [12] K. Gürlebeck and A. El-Sayed Ahmed, On B q Saces of Hyerholomorhic Functions and the Bloch Sace in R 3, Finite or Infinite Dim. Com. Anal. Al. 2004), [13] K. Gürlebeck, K. Habetha, and W. Srößig, Holomorhic functions in the lane and n-dimensional sace, Birkhaüser Verlag, Basel 2008).

15 On Hyerholomorhic Fω,G α, q, s) Saces 101 [14] H.R. Malonek, Quaternions in alied sciences, a historical ersective of a mathematical concet, 17th Inter.Conf. on the Al. of Comuter Science and Mathematics on Architecture and Civil Engineering, Weimar 2003). [15] R. A. Rashwan, A. El-Sayed Ahmed and Alaa Kamal, Integral characterizations of weighted Bloch saces and Q K,ω, q) saces, Math. tome ) 2) 2009), [16] L. F. Reséndis and L. M. Tovar, Besov-tye characterizations for Quaternionic Bloch functions,in: Le Hung Son et al Eds) finite or infinite comlex Analysis and its alications, Adv. Comlex Analysis and alications, Boston MA: Kluwer Academic Publishers) 2004), [17] R. Zhao, On a general family of function saces, Annal. Acad. Scien. Fennicae. Series A I. Math. Diss Helsinki: Suomalainen Tiedeakatemia, 1996), Alaa Kamal Deartment of Mathematics, Qassim University College of Sciences and Arts in Muthnib Qassim, KSA. alaa mohamed1@yahoo.com Taha Ibrahim Yassen Deartment of Mathematics faculty of Sciences Port Said University Port Said, Egyt taha hmour@yahoo.com