∆pz,rqωpzqdApzq 1q »
∆pz,rqωpzqqq1 dApzq q11
¤C|∆pz,rq|, zPD, we writeωPCq. Additionally, we setC8 Yq¡1Cq.
Theorem 2.10. [43, Theorem 3.9] Let0 p 8,ωPC8and GDbe measurable. If
|GX∆pz,rq| ¥δ|∆pz,rq|, zPD, (2.6) for someδ¡0and rP p0, 1q, then G is a dominating set for Aωp.
The condition (2.6) is equivalent to the existence of a constantδ0 δ0 ¡0 such that|GXS| ¡δ0|S|for all Carleson squaresS. Unsurprisingly, the pseudohyperbolic disc∆pa,rqcan also be replaced by a suitable Euclidean disc such asDpa,ηp1 |a|qq with a fixed 0 η 1. The proofs of these equivalences can be found in [40], where Luecking showed that ifωis a standard weight, then (2.6) is also a necessary condition for Gto be a dominating set for Apω.
2.6 CARLESON AND SAMPLING MEASURES
In his famous proof of the Corona theorem [10], Lennart Carleson needed a contin-uous embedding of the Hardy spaces to Lp spaces of the disc and proved a result on measures of the disc. These measures have since been defined and studied for many spaces of analytic functions and are called Carleson measures.
Let 0 q 8andXbe a space of analytic functions in the unit disc. A positive Borel measureµonDis aq-Carleson measure for the spaceXif the identity operator is bounded fromXtoLqµ, that is,
}f}Lqµ À }f}X, f PX.
In the following, the space X in the general definition above will be the weighted Bermgan space Apω induced byωP pD. These measures have been studied in [50, 51, 54] and can be characterized via the properties of the following weighted maximal function. For a weightω, a positive Borel measureµonDandα¡0, set
Mω,αpµqpzq sup
zPSpaq
µpSpaqq
pωpSpaqqqα, zPD.
Ifα1, we simply writeMωpµq. We also use the notationΓpzqfor the non-tangential approach region defined by
Γpzq
"
ζPD:|θargpζq| 1 2
1|ζ|
r
*
, zreiθ PDzt0u,
and denote the related tent by Tpζq tz P D : ζ P Γpzqu. The following theo-rem by Peláez and Rättyä gives the above mentioned characterization ofq-Carleson measures for weighted Bergman spaces induced by a weight in the classDp.
Theorem 2.11. [51, Theorem 1] Let 0 p,q 8, ω P pD and µ be a positive Borel measure onD.
(a) If p¡q, the following statements are equivalent:
(i) µis a q-Carleson measure for Apω; (ii) The function
Bµpzq
»
Γpzq
dµpζq
ωpTpζqq, zPDzt0u, belongs to L
pqp
ω ; (iii) Mωpµq PL
pqp
ω .
(b) µis a p-Carleson measure for Aωp if and only if Mωpµq PL8. (c) If q¡p, the following statements are equivalent:
(i) µis a q-Carleson measure for Apω; (ii) Mω,q{ppµq PL8;
(iii) The function zÞÑ pωµpp∆pz,rqqSpzqqqq{p belongs to L8 for any fixed rP p0, 1q.
Carleson measures for different spaces of analytic functions have many applica-tions in function and operator theory. In particular, they are often useful in proving the boundedness of linear operators acting on spaces of analytic functions. The q-Carleson measures for Aωp and their characterization above are used frequently throughout the proofs in PapersII–IV.
For 0 p 8 and a weight ω in D, a positive Borel measure µ on D is a sampling measure forAωp if
»
D|fpzq|pdµpzq }f}pAp
ω, f PAωp.
Note that the measuresµsatisfying the inequality ”À” are thep-Carleson measures for Apω defined and characterized above.
Luecking characterized the sampling measures for Bergman spaces induced by a standard weight in [45]. To state the result in question, we need some additional notation and the concept of weak convergence of measures. Beginning with the lat-ter, we say that a sequence of measurespµnqonDconverges weakly to a measureµ, denoted byµnáµ, if
»
D
hpzqdµnpzq Ñ
»
D
hpzqdµpzq, hPCcpDq,
whereCcpDqis the class of nonnegative, continuous and compactly supported func-tions inD. The following theorem is one of the steps in Luecking’s characterization of sampling measures.
Theorem 2.12. [45, Theorem 1] Let0 p 8and α¡ 1, and letpµnqbe a sequence of p-Carleson measures for Aαpsuch thatsupaPD µp1|a|qnp∆pa,1{2qq2 α 8. Thenpµnqhas a weakly convergent subsequence. Moreover, ifµn áµ, then
nlimÑ8
»
D|fpzq|pdµnpzq
»
D|fpzq|pdµpzq for all f PAαp.
Continuing towards the characterization of sampling measures, letMbe the set of Möbius transformations of Dandµbe a positive Borel measure onD. For each ϕPM, define the measureµϕsuch thatµϕpEq µpϕ1pEqqfor all measurable sets ED, and denote the collection of these measuresµϕbyµpMq. Finally, letWµbe the collection of all weak limits of sequences of measures inµpMq.
Theorem 2.13. [45, Theorem 5] Let0 p 8,α¡ 1andµbe a p-Carleson measure for Aαp. Then the following statements are equivalent:
(i) µis a sampling measure for Aαp;
(ii) There exists a q P p0,pq such that the support of every measure in Wµ is a set of uniqueness for Aqα.
(iii) There exists aβ¡αsuch that the support of every measure in Wµis a set of unique-ness for Apβ.
Unfortunately, while Theorems 2.9 and 2.12 are generalized for Apω withω PD in PaperII, the whole proof of Theorem 2.13 cannot be generalized for weights in Dp. The problem is that the weight ω can be such that compositions of functions
f PAωp with Möbius transformations are not controllable in norm.
The inequality ”Á” in the definition of sampling measures is closely related to dominating sets for the space, as can be seen from the following result. For r P p0, 1q, a positive Borel measure µ and a weight ω on D, define kω,rpzq µp∆pz,rqq{ωp∆pz,rqqfor allzPD.
Theorem 2.14. [42, Theorem 4.2] Let0 p 8,ε¡0andµbe a positive Borel measure onD. Then there exists rP p0, 1qsuch that
»
D|fpζq|pdApζq À
»
D|fpζq|pdµpζq, f PAp, wheneversupaPD Apµp∆pa,1{2q∆pa,1{2qq 8and the set G!
zPD:kA,r ¡εsupaPD Apµp∆pa,1{2q∆pa,1{2qq) is a dominating set for Ap.
3 Linear differential equations and growth of solutions
In this section, we present some facts related to linear differential equations. In particular, we discuss methods of measuring the growth of solutions and coefficients of the equation
fpnq An1fpn1q A1f1 A0f An, n¥2. (3.1) In our considerations, the coefficients Ak are analytic in the unit disc or some other starlike domain in the complex plane. We also consider the homogeneous equation
fpnq An1fpn1q A1f1 A0f 0, n¥2, (3.2) and especially the important and much studied second order case
f2 A f 0, (3.3)
particularly when the domain considered is the unit disc.
It is a well-known fact that the growth of solutions of (3.1) is connected to the growth of the coefficientsAk. For example, one can force all solutions to be bounded by choosing coefficients that grow slowly enough. On the other hand, by taking coefficients of sufficiently fast growth, one can find solutions that grow faster than any pre-given function.
The growth of fast growing solutions of (3.1) is typically measured in terms of Nevanlinna theory [21,23]. For solutions that grow slowly, other methods often yield better results. Some techniques used in the literature are Gronwall’s lemma [19], Herold’s comparison theorem [24], Picard’s successive approximations [11, 17] and estimates based on Carleson measures for different spaces of analytic functions [22, 39, 50, 59]. Additionally, Wiman-Valiron theory can be used effectively in the case of the complex plane [38].
3.1 INTEGRATION METHOD AND SOLUTIONS IN THE GROWTH SPACE
In PaperI, we present conditions that force all solutions of (3.1) or their derivatives to the growth space Hω8. We achieve this goal via a method based on successive integration and estimates on the weightω. The main tools in the integration method are the Fundamental Theorem of Calculus, stating that
fpzq
»z
0 f1pζqdζ fp0q (3.4)
for all functions f analytic on the integration path from 0 toz, and Cauchy’s integral formula
fpnqpzq n! 2πi
»
C
fpζq
pζzqn 1dζ, nPNY t0u, (3.5)
where C is a closed curve around the point z, and f is analytic on C and in the domain enclosed by it. These formulas allow us to express an analytic function by means of its derivative and vice versa and thus make it possible for us to construct estimates on the norm of the solution after applying the equation (3.1).
In certain special cases of when the derivatives of the solutions belong to Hω8, one can force the solutions to the Bloch orQK spaces. For example, in PaperI, we improve on the following results where sufficient conditions are given such that the solutions lie in theQKspace.
Theorem 3.1. [39, Theorem 2.1] Let0 c 3{2, and let K satisfy
»8
0 s12c sup
0¤t¤1
Kpstq
Kptq ds 8. (3.6)
Then there exists a constantααpn,c,Kq ¡0such that if the coefficients Ajof(3.2)satisfy }Aj}H8nj ¤α, for all j1, . . . ,n1, and}A0}H8nc ¤α, then all solutions of(3.2)belong to QK.
Theorem 3.2. [39, Theorem 2.6] Let the non-decreasing function K satisfy (3.6) with c 1. Then there exists a constant β βpn,Kq ¡0 such that if}Aj}H8nj ¤ β, for all j1, . . . ,n1, and}A0}H8
n1 ¤β, then all solutions of (3.2)belong to QK.
The conditions given for the coefficientsAkin Theorems 3.1 and 3.2 seem slightly illogical in that the norm used to measure the growth of A0does not quite conform to the norms of the other coefficients. Indeed, it seems natural that Theorem 3.1 should hold if the condition }A0}H8nc ¤ αis replaced by }A0}Hn8 ¤ α, and analo-gously Theorem 3.2 should hold if}A0}Hn18 ¤βis replaced by}A0}Hn8 ¤β. These predictions are based on the relation
}f}H8p sup
zPD
|f1pzq|p1 |z|qp 1 |fp0q|, f PHpDq, (3.7) which is easily obtained from (3.4) and (3.5). The formula (3.7) implies that|fpkqpzq|
grows roughly as|fpjqpzq|p1 |z|qjk, and thus, if we want all of the terms fpnqand Akfpkq,k0, . . . ,n1, in equation (3.2) to have similar growth, then|Akpzq|should grow roughly asAjpzqp1 |z|qkj forj,k0, . . . ,n1. Our results in PaperIshow that this is in fact the case.
Another way to restrict the growth of a function is to set a bound on its Taylor coefficients.
Theorem 3.3. [39, Theorem 2.4] Let Apzq °8
n0anzn, an P C. If |an| ¤ 1 for all nPNY t0u, then all solutions of (3.3)belong to the Dirichlet space.
Theorem 3.3 is not sharp, which we show in PaperI. More specifically, we find a condition for the Maclaurin coefficients ak which allows|ak| kαÑ 8, 0 α 1{2, while simultaneously implying the assertion of Theorem 3.3.