Soe Reaks on the Bounday Behavios of the Hady Spaces Tao Qian and Jinxun Wang In eoy of Jaie Kelle Abstact. Soe estiates and bounday popeties fo functions in the Hady spaces ae given. Matheatics Subject Classification 1). Piay 3G35, 31B5; Seconday 4B3, 31B5. Keywods. onogenic, onogenic Hady space, haonic Hady space, Cauchy s estiate. 1. Intoduction Let D = {z = x + iy C : z < 1} be the open unit disc. The holoophic Hady space H p D) 1 p < ) consists of all functions f that ae holoophic in D and satisfy 1 π 1/p f p = fe iθ ) dθ) p <. <<1 π Getting close to the bounday of D, singulaities ay happen fo functions in H p D), whee we have the well known estiate cf. [3]) 1 z fz) C p f p fo 1 p <. By using the density of the holoophic polynoials cf. [9]), o that of the Poisson integals cf. [5]), one can pove that z 1 1 z )1/p fz) = fo 1 p <, which is oe pecise than the pevious inequality nea the bounday. This wok was poted by Macao FDCT 56/1/A3 and eseach gant of the Univesity of Macau No. UL17/8-Y4/MAT/QT1/FST.
Tao Qian and Jinxun Wang In the case p =, H p D) is of paticula ipotance. It is a Hilbet space with the inne poduct f, g = π fe iθ )ge iθ )dθ, f, g H D). In a nube of pactical applications as the undelying space H D) plays an ipotant ole e.g., in signal pocessing, iage pocessing and coding theoy). Obseving that fo any function f H D), we have 1 a f, φ a = 1 a fa), whee φ a z) = 1 az is a unit vecto of H D) with the paaete a D. By the afoeentioned popety, we get f, φ a =, a 1 which iplies that thee exists a D such that f, φ a attains the axiu value. This is cucial fo the signal adaptive decoposition ethods as a vaiation and ealization of geedy algoith) intoduced in [5, 8]. In this note, we give a genealization of the above esult to highe diensions, of which the special cases have been applied to the adaptive decoposition of functions of seveal vaiables [6, 7]). Ou ethod is a odification of the classic ethod see [3, page 18]), which depends on soe oe delicate estiates. Befoe we state ou ain esults, let us fist have a quick eview of soe basic knowledge on Cliffod algeba and Cliffod analysis. Let e 1,..., e be basic eleents satisfying e i e j + e j e i = δ ij, i, j = 1,..., n, whee δ ij equals 1 if i = j and othewise. Let R +1 = {x = + x 1 e 1 + + x e : x i R, i } be identified with the usual + 1)-diensional Euclidean space. The eal Cliffod algeba geneated by e 1,..., e, denoted by A, is an associative algeba in which each eleent is of the fo x = T x T e T, whee x T R, e T = e i1 e i e il and T = {1 i 1 < i < < i l } uns ove all odeed subsets of {1,..., } and x =, e = e = 1. The no and the conjugate of x ae defined by x = T x T )1/ and x = T x T e T espectively, whee e T = e il e i e i1 and e i = e i fo i, e = e. We have fo any x, y, z A, xy = y x, xy)z = xyz) and xy / x y. A function fx) = T f T x)e T C 1 Ω, A ) is said to be left onogenic in the open set Ω R +1 if and only if it satisfies the genealized Cauchy-Rieann equation Df = i= e i f x i =, whee the Diac opeato D is defined by D = + = e i x i. If f is left onogenic, then each coponent of f is a eal-valued haonic function. Fo oe infoation about the onogenic function theoy, see []. Let B x, ρ) = {y R +1 : y x < ρ} be the open ball in R +1, which is centeed at x and of adius ρ. Fo siplicity, we denote B = B, 1).
Soe Reaks on the Bounday Behavios 3 The onogenic Hady space H p B ) 1 p < ), consists of all functions f that ae left onogenic in B and satisfy 1/p f p = fη) ds) p <, 1.1) <<1 η =1 whee ds is the aea eleent of B. We pove that Theoe 1.1. If f H p B ) 1 p < ), then 1 x ) α + p α fx) C,p, α f p, 1.) whee α = l, l 1,..., l ), α = i= l i and α = l x = x ξ = ξ, then we have unifoly in ξ = 1. l1 x 1 x l. Wite 1 1 x ) α + p α fx) = 1.3) Coesponding to this, we also pove soe popositions fo the onogenic Hady space H p R +1 + ), which consists of all functions f that ae left onogenic on the half space R +1 + = {x = + x R +1 : >, x = x 1 e 1 + + x e R } and satisfy f p = f + x) p dx <, 1.4) > R whee dx = dx 1 dx. We note that fo f H p R +1 + ) 1 p < ), the bounday values fx) = x + f + x) exist alost eveywhee and copise a function in L p R ), of which the Poisson integal coincides with f [4]). Theoe 1.. Suppose f H p R +1 + ) 1 p < ), then oeove, x x α + p + α f + x) = x α + p α fx) C,p, α f p ; 1.5) + x α + p holds unifoly with espect to x R, and holds unifoly in >. α f + x) = 1.6) x + x α + p α f + x) = 1.7) Reak 1.3. Siila discussions as in Section will show that Theoe 1.1 esp. Theoe 1.) holds fo the haonic Hady space H p B ) esp. H p R +1 + )) fo 1 < p <, whee by definition, a function f lies in H p B ) esp. H p R +1 + )) eans that f is haonic in B esp. R +1 + ) and 1.1) esp. 1.4)) holds. But fo the case p = 1, 1.3) esp. 1.6) and 1.7)) ay not hold fo H 1 B ) esp. H 1 R +1 + )). Fo exaple, f, x 1 ) = x x +x 1 H 1 R +), but f, x 1 ) does not unifoly tend to zeo as +.
4 Tao Qian and Jinxun Wang. Poof of the Theoes Poof of Theoe 1.1. Fo Cauchy s estiate cf. [1]) we know that hence α fx) C, α 1 x ) α ax fy), y B x, 1 x ) 1 x ) α + p α fx) C, α ax 1 y ) p fy). y B x, 1 x ) So, to pove 1.) and 1.3), it is enough to show that and 1 x ) p fx) C,p f p.1) 1 1 x ) p fx) =.) fo 1 p <. Denote by V = C 1 ) +1 the volue of the ball B x, 1 ), wite y = y η = ρη, note that and y x y x = ρ, y x = ρη ξ = η ξ) ρ)η η ξ ρ η ξ y x, so y B x, 1 ) iplies { 1 < ρ < 1, η ξ < 1 )/. Hence, fo 1 p <, we have 1 ) /p fx) = 1 ) /p V 1 1 ) /p V 1 1 ) /p V 1 B x,1 ) B x,1 ) 1 1 ρ 1 ) /p V 1 1 ) fy)dy fy) p dy η ξ <1 )/ <ρ<1 1 ) /p V 1 1 ) <ρ<1 = C,p f p. η ξ < 1 ) η =1 fρη) p dsdρ fρη) p ds fρη) p ds.3)
Soe Reaks on the Bounday Behavios 5.1) is now poved. On the othe hand,.3) C,p η ξ < 1 ) 1/p. fρη) ds) p <ρ<1 Note that as a function of η, <ρ<1 fρη) L p B ), and the easue of the set {η : η ξ < 1 )/} tends to zeo as 1,.) follows by the absolute continuity of the Lebesgue integal. Poof of Theoe 1.. By Cauchy s estiate we have hence α fx) C, α x α ax y B x,/) fy), x α + p α fx) C, α ax y B x,/) y/p fy). So, the poof of 1.5) and 1.6) is now educed to the poof of the following and x /p fx) C,p f p.4) x /p + fx) = x x/p fx) =.5) + fo 1 p <. Once these have been poved, the poof of 1.7) will be educed to the poof of f + x) =.6) x + unifoly with espect to [a, b], + ). Denote by V x = C x +1 the volue of the ball B x, x ), then fo 1 p <, x /p fx) = x /p Vx 1 fy + y)dy B x, ) 1/p x /p fy + y) dy) p.7) so.4) is veified. x /p x /p V 1 V 1 V 1 = C,p f p, B x, ) 3x fy + y) p dydy R fy + y) p dy y > R.8)
6 Tao Qian and Jinxun Wang On the othe hand, when is sall,.7) x /p x /p V 1 V 1 3x C,p y x C,p y x y x y, 3 ) fy + y) p dydy y x fy + y) p dy y, 3 ) fy + y) p dy 1/p. fy + y) dy) p y > Note that as a function of y, y> fy + y) L p R ) and the easue of the set {y : y x x } tends to zeo as +, by the absolute continuity of the Lebesgue integal we have When is lage,.8) x /p + x/p fx) =. V 1 C,p R y, 3 ) R fy + y) p dy y, 3 ) fy + y) p dy holds unifoly with espect to x R, and fy + y) fy + y) L p R ) fo 1 p <. y, 3 ) y > Also, fo.4) we know that which iplies fy + y) x /p C,p f p, y, 3 ) + y, 3 ) fy + y) = holds unifoly with espect to y R. By the Lebesgue s doinated convegence theoe we have + x/p fx) =.
Soe Reaks on the Bounday Behavios 7 Now we poceed to pove.6). Since f + x) = Γ +1 ) π +1 +1 Γ b ) π +1 = C y >N x y + x ) fy)dy +1 R R = C I 1 + I ), fy) dy x y + a ) +1 fy) dy x y + a ) +1 + y N by Hölde s inequality, ) I 1 x y + a ) +1)p 1/p dy y >N R ) y + a ) +1)p 1/p dy 1/p, C,p fy) dy) p y >N y >N fy) dy x y + a ) +1 y >N fy) p dy fy) p dy whee 1 p + 1 p = 1. Because fy) L p R ), I 1 is sall povided N is lage enough. With N fixed, I C x +1 fy) dy x + ), y N due to fy) is integable on {y : y N}, that poves.6). The poof of Theoe 1. is now coplete. ) Refeences [1] S. Axle, P. Boudon, W. Raey, Haonic Function Theoy. nd ed., GTM, Spinge-Velag, 1. [] F. Backx, R. Delanghe, F. Soen, Cliffod Analysis. London: Pitan Advanced Publishing Poga, 198. [3] John B. Ganett, Bounded Analytic Functions. Revised 1st ed., GTM, Spinge- Velag, 7. [4] J. E. Gilbet, Magaet A. M. Muay, Cliffod Algebas and Diac Opeatos in Haonic Analysis. Cabidge: Cabidge Univesity Pess, 1991. [5] T. Qian, Intinsic ono-coponent decoposition of functions: An advance of Fouie theoy. Math. Meth. Appl. Sci., 33 1), 88 891. [6] T. Qian, W. Spößig, J. X. Wang, Adaptive Fouie decoposition of functions in quatenionic Hady spaces. Math. Meth. Appl. Sci., to appea. [7] T. Qian, J. X. Wang, Y. Yang, Matching pusuits aong shifted Cauchy kenels in highe-diensional spaces. Pepint.
8 Tao Qian and Jinxun Wang [8] T. Qian, Y. B. Wang, Adaptive Fouie seies a vaiation of geedy algoith. Advances in Coputational Matheatics, 34 11), 79 93. [9] K. Zhu, Spaces of Holoophic Functions in the Unit Ball. GTM, Spinge- Velag, 5. Tao Qian Depatent of Matheatics Faculty of Science and Technology Univesity of Macau Taipa, Macao, China e-ail: fsttq@uac.o Jinxun Wang Depatent of Matheatics Faculty of Science and Technology Univesity of Macau Taipa, Macao, China e-ail: wjxpyh@gail.co