# Integral Criteria for Weighted Bloch Functions

Authors: A. El-Sayed Ahmed; M. A. Bakhit
Journal: Journal of Mathematics
 (2021)

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## Abstract

The present manuscript gives analytic characterizations and interesting technique that involves the study of generalϖ-Besov classes of analytic functions by the help of analytic ϖ-Bloch functions. Certain special functions significant in both ϖ-Besov-norms and ϖ-Bloch norms framework and to introduce new important families of analytic classes. Interesting motivation of this concerned paper is to construct some new analytic function classes of general ϖ-Besov-type spaces via integrals on concerned functions view points. The introduced analytic ϖ-Bloch and ϖ-Besov type of functions with some interesting properties for these classes of function spaces are established within the constructions of their norms. Using the defined analytic function spaces, various important relations are also derived.


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## Body

## 1. Introduction

Operator theory and special classes of holomorphic function spaces have interesting significant roles in different branches of pure and applied mathematics as well as in recent studies of theoretical physics. Some concerned problems arising in operator and measure theories are framed in terms of certain classes of function spaces. Most of these classes can be studied and treated by using numerous kinds of measures and some types of operators which provide and evolve new means of recent studies in mathematical analysis. Besov and Bloch-type classes are the focus of such studies. These types of spaces are extremely applied in computational mathematical analysis and theoretical physics problems. In addition, these special function classes allow the derivation of different useful identities in a fairly straightforward way and help in introducing new families of function classes. Throughout this concerned paper, the following notations and definitions will be used.LetHD denote the class of all concerned holomorphic functions on the concerned unit disk D. For a∈D, the known concerned Möbius transformation φaw is symbolized by(1)φaw=a−w1−a¯w,withw∈D.For a specific pointa∈D and 0<R<1, the supposed pseudo-hyperbolic disk Da,R with the supposed center a and supposed radius R is symbolized also by Da,R=φaRD.The supposed concerned pseudo-hyperbolic diskDa,R can be considered an Euclidean disk: with a specific center and radius being 1−R2a/1−R2a2and1−a2R/1−R2a2, respectively ( [1]). Let ζ denote the concerned normalized specific area of Lebesgue-type measure on D. We will need the following interesting known identity:(2)1−φaw2=1−a21−w21−a¯w2=1−w2φa′w.Fora∈D, by concerned usual substitution w=φaw, we must consider the known Jacobian change in the concerned estimates given by dζw=φa′w2dζw. One should remark that φaφaw=w, and thus φa−1w=φaw. For a,w∈D and 0<r<1, the supposed pseudo-hyperbolic disc Da,R is denoted by Da,R=w∈D:φaw<R.Let the concerned function of Green’s type with concerned logarithmic singularity at the specific pointa∈D(3)gw,a=log1−a¯ww−a=log1φaw.Assume thatCh and Ch∗ are any two concerned quantities which depend on a holomorphic-type function h on D. These concerned quantities are said to be equivalent and symbolized by Ch≈Ch∗; when we find a concerned finite positive constant Υ not depending on the holomorphic function h, for every holomorphic-type function h on D, the following inequalities hold:(4)1ΥCh∗≤Ch≤ΥCh∗.When the concerned quantitiesCh and Ch∗ are equivalent, we deduce that(5)Ch<∞⇔Ch∗<∞.Note that we sayΘ1≲Θ2 (for two concerned functions Θ1 and Θ2) when we find a specific concerned constant γ>0 for which Θ1≤γΘ2.Next the following new general concerned function classes will be defined.Definition 1.
Letϖ be a given concerned bounded nondecreasing and continuous function ϖ:D⟶ℝ+. The concerned function h∈HD is said to belong to the weighted ϖ-Bloch-type space ℬϖ, when(6)hℬϖ=supw∈Dϖ1−wh′w<∞.Definition 2.
Letϖ be a given concerned bounded nondecreasing and continuous function ϖ:D⟶ℝ+. The concerned function h∈HD is said to belong to the little weighted ϖ-Bloch-type space ℬϖ,0, when(7)hℬϖ,0=limw⟶1−ϖ1−wh′w=0.Remark 1.
One should obviously note that these new analytic-type classes are more general than the well-known Bloch and little Bloch analytic-type classes. Ifϖt=tαwithα∈0,∞, then the analytic α-Bloch classes which were introduced and studied in [2] will be followed. In addition, when α=1, we obtain the known analytic Bloch-type class that is given in [3].The following new analytic weighted function classes will be introduced.Definition 3.
For a bounded nondecreasing continuous functionϖ:D⟶ℝ+, let 0<p<∞, and a holomorphic function h∈HD is said to belong to the ϖ-Besov space Bp,ϖφ, when(8)hϖ,p;φp=supa∈D∫Dh′wp1−w2n−2ϖp1−w21−|φaw|pdζw<∞.Definition 4.
For a bounded nondecreasing continuous functionϖ:D⟶0,∞, let 0<p<∞, and an analytic function h in D is said to belong to the ϖ-Besov space Bp,ϖg, when(9)hϖ,p;gp=supa∈D∫Dh′wp1−w2n−2ϖp1−w2gw,apdζw<∞.Definition 5.
(see [1]). Let h∈HD and let 1<p<∞. If(10)hBpp=supw∈D∫Dh′wp1−w2p−2dζw<∞,then h belongs to the Besov space Bp.The aim of the current paper is to establish with concerned proofs various results on analytic-type function spaces with the help of holomorphicℬϖ-function type classes in some holomorphic concerned general Besov functions satisfying more extended general integral norm conditions portrayed by some general weights in the known concerned complex disk. The new results generalize and evolve some results in [1, 2, 4]. To substantiate the authenticity of the obtained results and to clear the importance of the defined function classes, some related concerned corollaries are furnished obviously too.Remark 2.
It is an interesting considerable remark to mention the generalizations of some holomorphic-type function spaces inℂn (see [1,5–10]) and others. In the same way, there are some extensions by the use of hypercomplex functions (see [11–15]) and others.


## 2. Some Integral Criteria forℬϖ Functions

Some concerned interesting integral-type criteria for the analytic classesℬϖ and the analytic general Besov classes are established in this concerned present section. The obtained analytic results are generalizing and evolving the corresponding results in [1, 2, 16] and others.Now, suppose thatφaW=a−w/1−aw is a Möbius transformation of D, let Da,r=w∈D:φaw<r, and let gw,a=log1/φaw be the known Green’s function on D with a specific concerned logarithmic singularity at the point a∈D. Next, an interesting general result will be established.Theorem 1.
Letr∈0,1,p∈0,∞,q∈1,∞. In addition, we let h∈HD. Therefore, the next concerned general quantities are equivalent:(a)hℬϖp.(b)
For0<p<∞, we have(11)supa∈D1Da,r1−p/2∬Dh′wpϖp1−wdζw<∞.(c) For0<p<∞, we have(12)supa∈D∬Dh′wp1−|w|p−2ϖp1−wdζw<∞.(d) For0<p<∞ and 1<q<∞, we have(13)supa∈D∬Dh′wp1−|w|p−2ϖp1−w1−φaw2qdζw<∞.(e) For0<p<∞, we have(14)supa∈D∬Dh′wplog1wϖp1−wφa′w2dζw<∞.(f)(15)supa∈D∬Dh′wpgw,ap1−wp−2ϖp1−wdζw<∞.Proof.
For any concerned analytic-type functiong on D,gp is a concerned subharmonic function, and thus we deduce that(16)g0p≤1πr2∬D0,rgwpdζw.
Employingg by h′∘φa, the next inequality can be obtained:(17)h′ap≤1πr2∬D0,rh′∘φawpdζw=1πr2∬D0,rh′wp1−φaw221−w22dζw.
In view of the following concerned relations,(18)1−φaw21−w2=φa′w, also1−φaw21−w2≤41−a2a,w∈Dsee14.
Thus, the next inequality can be deduced:(19)h′ap≤16πr21−a22∬Da,rh′wpdζw.
Hence, using the known specific estimations1−a22∼1−w22∼Da,r, with w∈Da,r, the following concerned inequalities can be deduced:(20)h′ap1−apϖp1−a≤161−apϖp1−aπr21−a22∬Da,rh′wpdζw.
Because1−a2∼1−a22,(21)h′ap1−apϖp1−a≤161−apαϖp1−aπr21−a22∬Da,rh′wpdζw≤16ηπr2Da,r1−pα/2∬Dh′wpϖp1−wdζw=mrDa,r1−p/2∬Dh′wpϖp1−wdζw,where η is a specific positive constant which is greater than zero and mr=16η/πr2 is a concerned constant which depends on r. Hence, for the two quantities (a) and (b), we have(22)a≤k1b,where k1 stands for a positive concerned constant.
Now, sinceDa,r∼1−w22,∀w∈Da,r, so it is clear that b∼c. From, the concerned inequality(23)1−φaw2>1−r2,the following estimates can be simply obtained:(24)∬Da,rh′wp1−wp−2ϖp1−wdζw=∬Da,rh′wp1−wp−2ϖp1−w1−φaw2q1−φaw2qdζw≤11−r2q∬Da,rh′wp1−wp−2ϖp1−w1−φaw2qdζw.
Therefore, for the two quantities (c) and (d), we have(25)c≤k2d,where k2 stands for a positive concerned constant.
It is known that1−φaw2≤2gw,a for all w,a∈D, and thus we can infer that the concerned quantity (d) is less than or equal to a concerned positive constant times the concerned quantity (f).
Using the next fundamental inequalities(26)∬Da,rh′wp1−wp−2ϖp1−wgw,apdζw=∬Da,rh′φawp1−|φaw|pϖp1−φawlog1wqdζw1−w22≤hℬϖαp∬Da,rlog1wqdζw1−w22,where(27)T=∬Da,rlog1wq1−w2−2dζw<∞,we can easily infer that the concerned quantity (e) is less than or equal to a specific positive concerned constant times the concerned quantity (a).
With the help of the known inequality1−w2≤2log1/w, where w∈D, setting q=2 in the concerned quantity (d), we simply deduce that the concerned quantity (d) is also less than or equal to the concerned quantity (e). For the final step in the concerned proof, we consider the following estimates.(28)J=∬Da,rh′wplog1wqϖp1−wφa′w2dζw=∬D1/4+∬D/D1/4h′wplog1wqϖp1−wφa′w2dζw=J1+J2,with w∈D1/4=w:w<1/4,φa′w2=1−a2/1−a¯w4≤1/1−w4≤4/34, and thus(29)J1=∬D1/4∈J1h′wplog1wpϖp1−wφa′w2dζw≤hℬϖαp∬D1/4log1/w1−wpφa′w2dζw≤hℬϖαp43p+4∬D1/4log1wpdζw=43p+4Tphℬϖαp,where the integral(30)Tp=∬D1/4log1wpdζw<∞.
For each specific pointw∈D/1/4, the following useful estimates hold:(31)log1w≤41−w2≤81−w,which implies that(32)J2≤8∬D/D1/4h′wplog1wqϖp1−wφa′w2dζw≤8phℬϖp∬D/D1/4φa′w2dζw≤k3hℬϖp,where k3 is a concerned positive constant. Thus, the concerned general quantity (e) is less than or equal to a concerned positive constant times the concerned general quantity (a). Thus, the interesting proof of Theorem 1 is completely established clearly. In Theorem 1, assuming that ϖ1−w≡1, the following concerned corollary can be obtained clearly.Corollary 1.
Letr∈0,1p∈0,∞,q∈1,∞ and let h∈HD. Therefore, the next concerned general quantities are equivalent:a∗hℬϖp.b∗ For 0<p<∞, we have(33)supa∈D1Da,r1−p/2∬Dh′wpdζw<∞.c∗ For 0<p<∞, we have(34)supa∈D∬Dh′wp1−wp−2dζw<∞.d∗ For 0<p<∞ and 1<q<∞, we have(35)supa∈D∬Dh′wp1−wp−21−φaw2qdζw<∞.e∗ For 0<p<∞, we have(36)supa∈D∬Dh′wplog1wpφaw2dζw<∞.f∗(37)supa∈D∬Dh′wpgw,ap1−wp−2dζw<∞.
For the concerned holomorphic function classesℬϖ,0, we give the following corresponding result induced from Theorem 1 on the specific boundary of the concerned unit disc D.Theorem 2.
Letr∈0,1p∈0,∞,q∈1,∞ and let h∈HD. Therefore, the next concerned general quantities are equivalent:a1hℬϖ,0p.a2 For 0<p<∞, we have(38)lima⟶1−supa∈D1Da,r1−p/2∬Dh′wpϖp1−wdζw<∞.a3 For 0<p<∞, we have(39)lima⟶1−supa∈D∬Dh′wpϖp1−wp−2ϖp1−wdζw<∞.a4 For 0<p<∞ and 1<q<∞, we have(40)lima⟶1−supa∈D∬Dh′wpϖp1−wp−2ϖp1−w1−φaw2dζw<∞.a5 For 0<p<∞, we have(41)lima⟶1−supa∈D∬Dh′wplog1wpϖp1−wφa′w2dζw<∞.a6(42)lima⟶1−supa∈D∬Dh′wpgw,ap1−wp−2ϖp1−wdζw<∞.In Theorem2, assuming that ϖ1−w≡1, the next interesting corollary can be established directly.Corollary 2.
Letr∈0,1p∈0,∞,q∈1,∞ and let h∈HD. Therefore, the next concerned general statements are equivalent:b1hℬϖ,0p.b2 For 0<p<∞, we have(43)lima⟶1−supa∈D1Da,r1−p/2∬Dh′wpdζw<∞.b3 For 0<p<∞, we have(44)lima⟶1−supa∈D∬Dh′wp1−wp−2dζw<∞.b4 For 0<p<∞ and 1<q<∞, we have(45)lima⟶1−supa∈D∬Dh′wp1−wp−21−φaw2qdζw<∞.b5 For 0<p<∞, we have(46)lima⟶1−supa∈D∬Dh′wplog1wpφa′w2dζw<∞.b6(47)lima⟶1−supa∈D∬Dh′wpgw,ap1−wp−2dζw<∞.


## 3. More Analytic Characterizations

Extension of Kwon et al.’s result (see [4]) is also established in this current section with the help of holomorphic ℬϖ-type functions. Holomorphic function significants in both weighted ϖ-Besov norms and ℬϖ-Bloch norms framework and to introduce some new concerned families of holomorphic-type function classes.Theorem 3.
Let0<p<∞,0≤s<∞, with 0<p<s+1 and let h∈HD. Thus,(48)hℬϖp≈supa∈D∫Dh′wp1−w2p−2ϖp1−w1−φaw2pgw,a−sdζw.Proof.
In view of condition (b) in Theorem1, we deduce that(49)hℬϖp≈wsupa∈D∫Dh′wp1−w2p−2ϖp1−wdζw.
Set a concerned finite specific positive constantδR by(50)δR:1−R2plog11−R2−s.
Thus, we deduce that(51)hℬϖp≈supa∈D∫Da,Rh′wp1−w2p−2ϖp1−wlog11−R2−s1−R2pdζw.
Using the inequality(52)φaw2<R2⇔1−φaw2>1−R2,we infer that(53)hℬϖp≈supa∈D∫Dh′wp1−w2p−2ϖp1−w1−φaw2pgw,a−sdζw.
Thus,(54)hℬϖp≤k∗supa∈D∫Dh′wp1−w2p−2ϖp1−w1−φaw2pgw,a−sdζw.where k∗ is a specific positive constant that does not depend on a.
Now for the other direction, in view of changing variable base, that is replacingw by φaw, the following estimates can be deduced:(55)supa∈D∫Dh′wp1−w2p−2ϖp1−w1−φaw2pgw,a−sdζw≤hℬϖpsupa∈D∫D1−w2p−21−φaw2pgw,a−sdζw=hℬϖpsupa∈D∫D1−φaw2p1−w2plog1w−s1−a21−a¯w22dζw.
With the help of the known equality (1), the following estimates can be obtained:(56)supa∈D∫Dh′wp1−w2p−2ϖp1−w1−φaw2pgw,a−sdζw≤hℬϖpsupa∈D∫D1−w2p−2log1w−sdζw≤23p−2hℬϖp∫011+R−s1−Rp−s−2dr≤4π23p−s−2hℬϖp∫011−Rp−s−2dR<23p−sπhℬϖpIR,p,s,where the specific integral IR,p,s=∫011−Rp−s−2dR<+∞ when p<s+1. Therefore,(57)hℬϖp≥k1∗supa∈D∫Dh′wp1−w2p−2ϖp1−w1−φaw2pgw,a−sdζw.where the specific constant(58)k1∗=12s−3pπIp,s.
Thus, the concerned proof of our desired result is completely obtained.Remark 3.
Various research studies on advanced operator theory using numerous weighted classes of holomorphic functions have been actively appearing in some joyful important areas of mathematical sciences such as the theory of dynamical systems, the known probability theory, recent research in mathematical physics, and some branches of quantum mechanics. Such holomorphic weighted classes of concerned function spaces are still under interest of considerations in the aforementioned applications. The defined holomorphicϖ-Bloch spaces as well as the holomorphic ϖ-Besov spaces can be also applied in such interesting applications.


## 4. Conclusion

General analytic characterizations for some concerned extended classes of holomorphic Banach of function spaces of Bloch and Besov type are established and discussed in this article. The concerned proofs are obtained using two types of concerned holomorphic functions inD. Both the considered functions and the proofs methods have concerned parameters which make deep help to the obtained results. As specific results, various new theorems and lemmas for the considered function classes are well derived. Further, the obtained concerned generalized results proved that the used function classes give better performance compared with existing results in the literature. The n-th partial derivative concerned quantities with general formulas including the extended type of functions are also well derived. The well-known analytic Bloch, the analytic Besov, and the Zygmund spaces are introduced as specific concerned special cases of the new defined classes. The concerned computations of the extended Bloch-type norm and the extended Besov-type norms related to these special cases can be established too.The analytic characterizations of integral norms, the extended Besov space, and the specific analytic Bloch norms will give interesting general results and recent developments within this fascinating area of function spaces. Such concerned advanced aspects may be considered in some areas of further studies.



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*Source: 9994532-2021-04-23.xml*