Interpolation in the Nevanlinna and Smirnov classes and harmonic majorants
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(2) INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS ANDREAS HARTMANN, XAVIER MASSANEDA, ARTUR NICOLAU, & PASCAL THOMAS A BSTRACT. We consider a free interpolation problem in Nevanlinna and Smirnov classes and find a characterization of the corresponding interpolating sequences in terms of the existence of harmonic majorants of certain functions. We also consider the related problem of characterizing positive functions in the disk having a harmonic majorant. An answer is given in terms of a dual relation which involves positive measures in the disk with bounded Poisson balayage. We deduce necessary and sufficient geometric conditions, both expressed in terms of certain maximal functions.. C ONTENTS 1. Introduction and statement of results. 2. 1.1. Interpolating sequences for the Nevanlinna Class. 2. 1.2. Positive harmonic majorants. 4. 1.3. Geometric criteria for interpolation. 7. 2. Preliminaries. 9. 3. From harmonic majorants to interpolation. 10. 4. From interpolation to harmonic majorants. 11. 5. The trace spaces. 13. 6. Harmonic majorants and measures with bounded balayage. 14. 7. Weaker conditions for the existence of harmonic majorants. 20. 8. Proofs of the geometric conditions. 23. 9. Hardy-Orlicz classes. 28. References. 30. Date: April 30, 2004. 1991 Mathematics Subject Classification. 30E05, 32A35. Key words and phrases. free interpolation, Smirnov and Nevanlinna class, harmonic majorants, Poisson balayage. All authors supported by the PICS program no 1019 of Generalitat de Catalunya and CNRS. First and third author also supported by European Commission Research Training Network HPRN-CT-2000-00116. Second author also supported by the DGICYT grant BFM2002-04072-C02-01 and the CIRIT grant 2001-SGR00172. Third author also supported by DGICYT grant BFM2002-00571 and the CIRIT grant 2001-SGR00431. 1.
(3) 2. A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. 1. I NTRODUCTION. . AND STATEMENT OF RESULTS. . 1.1. Interpolating sequences for the Nevanlinna Class. Let be a discrete sequence of points in the unit disk . For a space of holomorphic functions , the interpolation problem consists in describing the trace of on , i.e. the set of restrictions , regarded as a sequence space. One approach is to fix a target space and look for conditions so that . An alternative approach, known as free interpolation, is to require that be ideal, i.e. stable under multiplication by . See [Nik02, Section C.3.1 (Volume 2)], in particular, Theorem C.3.1.4, for functional analytic motivations. This approach is natural for those spaces that are stable under multiplication by , the space of bounded holomorphic functions on . For Hardy and Bergman spaces both definitions turn out to be equivalent, with the usual choice of as an space with an appropriate weight (see [ShHSh], [Se93]).. . . . . . . . . .
(4). The situation changes for the non-Banach classes we have in mind, namely the Nevanlinna class. $&!#"%('*+-) ,/.13 02 !546 798;:=<?> @;ACBEDGFIH. and the related Smirnov class. 6 J $&!!"%(' +-) , . 3 02 !54 6 798;:=<?> K-ACB +-) , . 3 02 !54 6 7L:=<?> @-ACBMHON. WT VOXa` )bY dc\d !54 6 7 c @eDfF . PQ9RS S UTWVOX S )(Y [Z7[ #54 6 RS \D]F_^ * g f = h. We briefly discuss the known results. Naftaleviˇc [Na56] described the sequences for which the trace coincides with the sequence space Na (we state the precise result after Proposition 1.12). The choice of Na is motivated by the fact that for , and this growth is attained. Unfortunately, the growth condition imposed in Na forces the sequences to be confined in a finite union of Stolz angles. Consequently a big class of Carleson sequences (i.e. sequences such that ), namely those containing a subsequence tending tangentially to the boundary, cannot be interpolating in the sense of Naftaleviˇc. This does not seem natural, for is in the multiplier space of . In a sense, the target space Na is “too big”. Further comments on Naftaleviˇc’s result can be found in [HaMa01] and below, after Proposition 1.12.. . . QP 9RCS S ji S k) Y dZld !54 6 R S j DmF_^ 6 . 6. For the Smirnov class, Yanagihara [Ya74] proved that in order that contains the space , it is sufficient that is a Carleson sequence. Ya However there are Carleson sequences such that does not embed into Ya [Ya74, Theorem 3] : the target space Ya is “too small”.. . We now turn to the definition of free interpolation.. . @ j n] , i.e. whenever 9Ro op and rqso ot. rqso5R5o ok Definition. Let be a space of holomorphic functions in . A sequence nu is called free f J y w x z . interpolating for if v is ideal. We denote Remark 1.1. For any function algebra containing the constants, v is ideal if and only if nv N =. Definition. A sequence space is called ideal if , then also ..
(5) INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. rq S S1 = r q7S-7 Z\ S wyxCz 6 n wyxz . It is then clear that. Z\ uq7S . 3. L7 Z W S1 v. The inclusion is obviously necessary. In order to see that it is sufficient notice that, by assumption, for any there exists such that . Thus, if , the sequence of values can be interpolated by .. i S ) Y dZld DfF. Z3. Free interpolation for these classes entails the existence of nonzero functions vanishing on all except a given . Hence the Blaschke condition is necessary and will be assumed throughout this paper.. S. v S S with zero-sequence S dZl Z Z c Z c ' . Define Y ) Y !54 Se Z @ ' thenif c Z J c / if c. Given the Blaschke product . . Here . , denote . SJ. . S .
(6) . Definition. We say that a Borel measurable function defined on the unit disk admits a positive harmonic majorant if and only if there exists a positive harmonic function on the unit disk such for any . that . c c c = denote the space of harmonic functions in and ! 6 the subspace of its Let positive functions. Consider also the Poisson kernel in : " c$#&%e " ` %e Re ' %%)( +cc * ),% Y dc\c\ 0 N 0 Y Y Our characterization of interpolating sequences for the Nevanlinna class is as follows. Note that the existence of a harmonic majorant occurs at two junctures: first, to decide which sequences of points are free interpolating, second, to identify the trace space that arises for those sequences which are indeed free interpolating. Theorem 1.2. Let. . . be a sequence in . The following statements are equivalent:. (a) is a free interpolating sequence for the Nevanlinna class (b) The trace space is given by:. .. Z 2 !54 6 R S , Z J C^ N (c) admits a harmonic majorant. P 6 S ^ , 7 (d) There exists 354 such that for any sequence of nonnegative numbers S98 6 S Z S8 6 S !54 Se Z @ '2: 3I; TWVO<>X = S 8 6 S " Se %e N .-v ]PQ9RS S 0/1 6 . such that . We recall that any positive harmonic function on the unit disk is the Poisson integral of a positive measure on the unit circle, . c . "@?BADC. c . <>= " ` %Q MA A e% N. We will say that a harmonic function is quasi-bounded if and only if it admits an absolutely continuous boundary measure (for the reasons for this terminology, see [He69, pp. 6–7]). The analogous result for the Smirnov class will, as can be expected, involve quasi-bounded harmonic functions..
(7) 4. A. A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. . Let denotes the normalized Lebesgue measure in . Also, for a nonnegative function on the unit disk, let denote the associated non-tangential maximal function (see (1.1) below). Theorem 1.3. Let. . 6. 0 Z\ 2 #54C6 R S #7Z J C^ N. be a sequence in . The following statements are equivalent:. (a) is a free interpolating sequence for the Smirnov class (b) The trace space is given by. 6 - uPQ9RCS S 0/1 J 6 quasi-bounded (c) !#" admits a quasi-bounded harmonic majorant. 8 6 S 9 denotes the set of nonegative sequences TW VO .
(8) (d) o % , where X Z
(9) SCo S P 6 S ^ such that T VM<>X = S i 6 S " Se %e : . 7 ; yP ) % ] %e ^; DGF , and (e) (i) TWVO X 3
(10) o !#" S i 6 S Z for any sequence of sequences of nonnegative numbers (ii) o % P 6 S
(11) o ^ 9 such that o !% #" Si 6 S
(12) o " SQ %e almost everywhere on . 7. TWVMX. D F. The classical Carleson condition characterizing interpolating sequences for bounded analytic = functions in the unit disk is , hence statements (c) in both results above can be viewed as Carleson-type conditions. A. In: view of Theorems 1.2 and 1.3, it seems natural to ask whether the measure such that "@?,A+C can be obtained from in a canonical way. We do not have an answer to this question, but with Propositions 1.12 and 1.13 it is easy to construct examples that!#discard natural " $" A candidates, such as the (weighted) sum of Dirac masses , or Poisson " % % (see definition below). balayage measures % . A i S ) Y [Z7[ eS e MA e . i S [Z7[ S S k) Y. 1.2. Positive harmonic majorants. The conditions in Theorems 1.2 and 1.3 (d) arise in the solution of a problem of independent interest: &(' Problem. Which functions admit a (quasi-bounded) harmonic majorant?. Y 6 Answers to this problem lead to rather precise theorems about the permissible decrease of the. modulus of bounded holomorphic functions, e.g. Corollary 1.5 below. See [Hay], [LySe97]; [EiEs] also provides a survey of such results. The existence of harmonic majorants is relevant as well to the study of zero-sequences for Bergman and related spaces of holomorphic functions [Lu96]. An answer to the problem of positive harmonic majorants can be given in dual terms (see [BNT] for another characterization). The Poisson balayage (or swept-out function) of a finite A positive measure in the closed unit disk is defined as A " A . %e . = ` %e A c5 . %. (N. We will be interested in the class of measures having bounded balayage. Recall that Carleson measures are those finite positive measures whose balayage has bounded mean oscillation (see [Gar81, Theorem VI.1.6, p. 229]); this is also an easy consequence of the -BMO duality (see [Gar81, Theorem VI.4.4, p. 245]). Hence positive measures with bounded balayage form a. '.
(13) INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. 5. subclass of the usual Carleson measures. It is easy to see (cf. Section 6) that positive measures with bounded balayageA are precisely those which operate againstA positive harmonic functions, 3 such that that is, those measures for which there exists a constant 3. . = c A A c . :. . . . 3 . for any positive harmonic function in the unit disk .. G uP A. TWVMX A %e : ) ^CN Theorem 1.4. Let be a nonnegative Borel function on the unit disk . The following statements are equivalent: : c5 for all c . (a) There exists a (positive) harmonic function such that c5 (b) There exists a constant 3 3 such that TWVM X . = c MA A c : 3 N ), while the sufficiency follows The necessity of condition (b) is obvious (e. g. 3 Define. . positive Borel measures on. . such that ; <>= . from a convenient version of a classical result in Convex Analysis, known as Minkowski-Farkas Lemma. The characterization of interpolating sequences in the Nevanlinna class in dual terms given by condition (d) in Theorem 1.2 follows from this result. This can be applied to study the decrease of a non-zero bounded analytic function in the disk along a given non-Blaschke sequence.. CS S 7 Z @ aD ;S Z *. . . a sequence of positive Corollary 1.5. Let be a separated non-Blaschke sequence and values. Then there exists a non-zero function with , , if and only if is the union of a Blaschke sequence and a sequence for which there exists a universal 3 constant 3 such that. . !54 ' : 3ITWVM<X = ; 6 . for any sequence of nonnegative numbers . 8. . 6. . . . . . 8. . . 6 " . . %Q . . In a similar way, Theorem 1.3 (d), (e) are obtained as an application of the following analogue of Theorem 1.4 for quasi-bounded harmonic functions (i.e. for the Smirnov class).. . Theorem 1.6. Let be a nonnegative Borel function on the unit disk . The following statements are equivalent:. c . c5 . (a) There exists a (positive) quasi-bounded harmonic function such that all . ? & ? # # (b) There is a convex increasing function with that admits a harmonic majorant on ; A 9 (c) .. o !% #" T VMX . Co A . f F . Ff . : . !% !" ( F. c . for such.
(14) TWVOX3 y P % ] %e ^; DvF , and !#" . = A A o1 for any sequence P A o ^gn (ii) o % . everywhere on ,. 6. A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. (d) (i) . such that. o !% #". . A o %e . almost. Condition (b) is inspired by a characterization of quasi-bounded harmonic functions given in Armitage and Gardiner’s book [ArGa, Theorem 1.3.9, p. 10]. For the problem of harmonic majorants it is desirable to obtain criteria which, although only necessary or sufficient, are more geometric and easier to check than the duality conditions of Theorems 1.4 and 1.6. and aperture is defined by Recall that the Stolz angle with vertex %. uP %e c u c Y % : ) Y c\ 0 ^CN In our considerations the angle is of no importance, so we will write %e for the generic Stolz ' angle with aperture . Given a function from to 6 , the non-tangential maximal function is defined as 7 %e TWVM
(15) X N (1.1) ; ' Recall that denotes the normalized Lebesgue measure on . Consider the weak- space IP; measurable TWVOX &P %1 7 %Q K 4 ^ DfF_^; ' 3 and let uP; measurable !% #" &P %1 7 %e @ 4 ^; ^CN ' 3 . It is well-known that the non-tangential maximal function of the Poisson transform of a pos ' itive finite measure belongs to (see [Gar81, Theorem 5.1, p. 28]). A more A ' 3 careful analysis shows that if is absolutely continuous, then its Poisson transform is in . This and some easy estimates imply the following result. ' . Proposition 1.7. (a) If admits a harmonic majorant, then ' 3 . (b) If admits a positive quasi-bounded harmonic majorant, then . @ " ? C @ " ? C ' (c) If , then the function admits A as a quasi-bounded . . . harmonic majorant.. . As far as necessary conditions are concerned, there is a way to improve the previous result by using the Hardy-Littlewood maximal function. Given , this is defined as. TWVMX )
(16) . # .
(17). where the supremum is taken over all arcs For. . . containing .. %e WT VM =X c %e T VM =X c b TW VM X
(18)
(19)
(20) ` # ` `
(21) ; is the characteristic function of a set and ` is the “Privalov shadow” interval
(22) ]P ` % u Cc %Q ^CN define. . where (1.2). .
(23) ' ' 3 . .. INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. Proposition 1.8. (a) If admits a harmonic majorant, then (b) If admits a quasi-bounded harmonic majorant, then . . 7. We will give some examples in Proposition 7.4 that show that this is indeed stronger than the necessary condition given in the first part of Proposition 1.7, but still falls short of giving a sufficient condition for the existence of a harmonic majorant. 1.3. Geometric criteria for interpolation. We would like to obtain some geometric implications of the analytic conditions given in Theorems 1.2 and 1.3. To begin with, we would like to state the maybe surprising result that separated Blaschke sequences (with respect to the hyperbolic distance) are interpolating for the Smirnov class (and hence the Nevanlinna class). Recall that a sequence is called separated if , where # 24 . 9 S !x S Z Z c$# ` j K
(24)
(25) c Y
(26)
(27) #
(28)
(29) c.
(30)
(31) )Y is the pseudo-hyperbolic distance. ! 4 Se Z @ ' can always be majorized by the values at Z For such sequences, the values. of the Poisson integral of an integrable function (see Proposition 4.1), thus the following corollary is immediate from Theorem 1.3. Corollary 1.9. Let. be a separated Blaschke sequence. Then. fJwyxz 6. (hence. GJwyxz . ).. More precise conditions can be deduced from Propositions 1.7, 1.8 and (c) in Theorems 1.2 and 1.3. be a sequence in . f J y w x z then ' j . If fJwyxz 6 then ' 3 . (a) If t ' (b) If then /wyxCz 6 (and hence /wyxCz ).. Corollary 1.10. Let. Notice that the necessary conditions obtained by replacing by in an immediate consequence of the estimate . . GJwyxz 6 , then !54 Se ' " S #!"" (1.3) %(' ) Y dZld , Z\ K N GJwyxz , then (b) If TWS9VMX ) Y dZld !54 , Se Z\ K ' DGF N (1.4) (c) If is Blaschke and (1.5) S98 ) Y dZld !54 , Se Z\ K ' DfF # GJwyxz 6 (and so f/wyxCz as well). then This result implies the following Carleson-type conditions.. Corollary 1.11. (a) If. . in (a) also hold. This.
(32) 8. A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. 6. . Condition (1.3) already appeared in [Ya74, Theorem 1] as a necessary condition for the sequence space Ya (as defined in the beginning of Section 1.1) to be included in the trace of . Condition (1.4) is discussed in Proposition 1.12 and the corollary thereafter.. '. In some situations the conditions above are indeed a characterization of interpolating sequences. For instance, the weak -condition characterizes interpolating sequences lying on a radius, while for sequences approaching the unit circle very tangentially the characterization is given by the strong -condition. This is collected in the next results.. '. nv lies in a finite union of Stolz angles. f y w x z 6 (a) fwyxz if and only if (1.3) holds. (b) if and only if (1.4) holds.. Proposition 1.12. Assume that. . It should be mentioned that (b) can also be derived from Naftaleviˇc’s result [Na56, Theorem 3]. On the other hand, his full characterization of the sequences such that Na can also be deduced from Theorem 1.2. Corollary (Naftaleviˇc, 1956). angles and (1.4) holds.. . Na. if and only if. is contained in a finite union of Stolz. Let us consider the other geometric extreme, sequences which in particular only approach the circle in a tangential fashion. Write. 8 S ) Y dZld S # S where stands for the Dirac measure at Z . I wyxCz A Proposition 1.13. If has bounded balayage, then A . (1.6). if and only if. I wyxz 6 , and. if and only if (1.5) holds. A . has bounded balayage implies in particular that approaches Note that the condition that the circle tangentially. In Section 8, we will see more concrete conditions of geometric separation A which are sufficient to imply that has bounded balayage (Proposition 8.2). A . When has bounded balayage, the trace space will embed into Yanagihara’s target space. More precisely, the following result holds.. G 6nG n . Proposition 1.14. The following are equivalent: (a) Ya , (b) A Ya , (c) has bounded balayage, i.e.. T VMX ; <>= i S ) Y dZld " S %e DGF . 6 . These are automatically in Yanagihara considered the sequences such that wyxz 6 , since for any Blachke sequence . Conversely, Lemma 8.1 (see Section 8) implies that n - , thus if wyxz 6 , then by Theorem 1.3(b) 6 . Therefore Ya. Ya. Ya. Ya. Theorem 1.3 characterizes in particular the sequences studied by Yanagihara.. . . 6 n . Altogether, free interpolation for the Nevanlinna and Smirnov classes can be described in terms of the intermediate target spaces - and - . Notice first that always - and.
(33) Gwyxz 6 if and only if 6 _ is proved at the beginning of Section 5). So, n .- f(this J y w x z - , and if and only if n .-vnG . - . Observe also that nG - G INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. Ya. 9. Na. The paper is organized as follows. The next section is devoted to collecting some basic results on functions in the Nevanlinna class. In Section 3 we prove the sufficiency for interpolation of the conditions (c) of Theorems 1.3 and 1.2. We essentially use a result by Garnett allowing interpolation by functions on sequences which are denser than Carleson sequences, under some decrease assumptions on the interpolated values. In Section 4 we study the necessity of these conditions. We first observe that in the product appearing in Theorem 1.2, only the fac tors with close to are relevant. Then we split the sequence into four pieces, thereby reducing the interpolation problem, in a way, to that on separated sequences. The trace space characterization will be discussed in Section 5. In Section 6 we consider measures with bounded balayage, show that they operate against positive harmonic functions and prove Theorems 1.4 and 1.6. In Section 7, we prove Proposition 1.8, and provide examples to show that the sufficient condition is not necessary, and the necessary condition not sufficient. Section 8 is devoted to the proofs of Corollary 1.11, Propositions 1.12, 1.14, and 1.13, as well as the deduction of Naftaleviˇc’s result from Theorem 1.2. Also, we give examples of measures with bounded balayage. In the final section, we exploit the reasoning of Section 3 to construct non-Carleson interpolating sequences for “big” Hardy-Orlicz classes.. . SQ Z . Z. SQ Z . Z. Acknowledgements. The authors wish to express special thanks to Jean-Baptiste HiriartUrruty for introducing them to Farkas’ Lemma, to Stephen Gardiner for pointing out an efficient characterization of quasi-bounded harmonic functions, and to Alexander Borichev for Lemma 6.6, and his discussions with us about harmonic majorants. 2. P RELIMINARIES We next recall some standard facts about the structure of the Nevanlinna and Smirnov classes (general references are e.g. [Gar81], [Nik02] or [RosRov]). A function. . is called outer if it can be written in the form. X . <>= %%!( cc #54\ %e A %e 0# , 4 a.e. on and !54 g ' Y j . Such a function is the quotient _ ' 5 where ,3 ) outer functions ' # 5 with < : !4
(34) , # + . In particular, the weight0 of two bounded ) , we ) have is given by the boundary values of 0 ' . Setting. #54 c K " ? 0 C c5 . <= " ` %e %e A %e N This formula allows us to freely switch between assertions about outer functions and the associated measures ( A .
(35) h such that
(36) ) almost Another important family in this context
(37) are inner functions: everywhere ] onP o . o Any
(38) inner function can be factorized into a Blaschke product carrying the zeros Z ^ of , and a singular inner function defined by X . <>= % ( c A A %e # c Y %Y c 7 c . 3. . .
(39) 10. A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. for some positive Borel measure. A. J 6. singular with respect to Lebesgue measure.. '. According to the Riesz-Smirnov factorization, any function. . #. ' # : , is0 singular inner, is a Blaschke product and J are0 represented ) as ) 0 p ' ' #. : . 0 0 with < outer, <. , < singular inner, is a Blaschke product and ) ) In view of the Riesz-Smirnov factorization described above, the essential difference between Nevanlinna and Smirnov functions is the extra singular factor appearing in the denominator in the Nevanlinna case. This is reflected in the corresponding result for free interpolation in by where. '#. . . is represented as. are outer with. . Similarly, functions. the fact that . . is bounded by a harmonic function, not necessarily quasi-bounded. 3. F ROM. HARMONIC MAJORANTS TO INTERPOLATION. Sp OS \Z K. n . , set . The key result to the proof of the For a given Blaschke sequence sufficient condition is the following theorem by Garnett [Gar77], that we cite for our purpose in a slightly weaker form (see also [Nik02] as a general source, in particular C.3.3.3(g) (Volume 2) for more results of this kind). ? & ? Theorem. Let be a decreasing function such that . If a # # satisfies sequence. F Y R 9u RCS S f. Ff . . 3 MA D]F. R S : S !54 : S # Z / # J then there exists a function such that R . #54UL: S ( Z . Observe that according to our former notation we have ) in the Nevanlinna As we have already noted in Remark 1.1, in order to have free interpolation and n 6 respectively. Our aim will and Smirnov classes, it is sufficient that K n 6. be to accommodate the decrease given in Garnett’s result by an appropriate function in or . This is the crucial step in the proof given hereafter of the sufficiency of conditions (c) in both Theorems 1.3 and 1.2.. I 6 c (3.1) . <>= %)% ( cc A A %e Y has positive real part in the disk. By Smirnov’s theorem, is an outer function in some
(40) , A.4.2.3 (Volume 1)). Also X is in the D ) , and therefore in 6 (see [Nik02], in!5particular a 4 S : J Nevanlinna class. By assumption we have Re Z\ , Z . ) . Proof of sufficiency of 1.3 (c) and 1.2 (c). The proof will be presented for the more difficult case ! . Then is the Poisson of the Nevanlinna class. So, assume that majorizes A integral of a positive measure on the circle and the function.
(41) +( d Z\ K . ( & , which obviously satisfies the hypothesis of Garnett’s theorem, , ) which is0 still outer in 6 . We have the estimate + 0 ( Z @ + ( Re Z\ ( #54 : S 9#54a9) : S # 0 0 ) 0 W OS S hence the sequence defined by eS !5) 4U9: S # Z J # Z\ is bounded by . q rq7S S by a function in , split ) In order to interpolate an arbitrary qsSj qsS eS X S Y Z W S 9!4 : S Z N X Y Z W q S eS X Z W S S is bounded, we can apply Garnett’s result to Since by hypothesis Y interpolate the sequence RSj q S eS X S Y Z W S 9#54 : S # Z J # _ by a function . Now X is a function in with p Gq . A The proof for the Smirnov case is obtained by observing that if the measure is absolutely continuous, then X is in the Smirnov class and so is the interpolating function . INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. Take now and set. 11. . . . . . . . . . . . 4. F ROM. INTERPOLATION TO HARMONIC MAJORANTS. S. Z . !54 eS Z @ ' Z. We first show that in order to construct the appropriate function estimating we which are close to . This is in only need to consider the factors of given by points accordance with the results for some related spaces of functions [HaMa01, Theorem 1], and it obviously implies Corollary 1.9.. , there exists a quasi-bounded ) c # X Y and is given by (3.1) with c N. ' !4 S
(42) S SQ c K : c # ` Y 6 , where and therefore an outer function A A A , such that S
(43) S ` Se c @ c K #
(44) Proof. We shall use the intervals ` introduced in (1.2). In [NPT, p. 124, lines 3 to 17], it is proved that the function given by 6 3 S8 %e #. %e 6 where 3 is an appropriate positive constant, is suitable. At this juncture, the separation hypoth# Proposition 4.1. Let be a Blaschke sequence. " ? C For any . ,. , such that positive harmonic function .
(45). . . .
(46). . . esis made in [NPT, Lemma 4] is no longer used..
(47) 12. A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. Proof of the necessity of 1.3 (c) and 1.2 (c). We will use a dyadic partition of the disk: for any in , let. o +-, + o : + o o I P = : ? < > ? +-,M+ # (4.1) CB ( ^
(48) # D N ) and the associated Whitney partition in “dyadic squares”: o :f8 o ' o u @ P ; 8 : :
(49) o ? < > ? < > + + # D ) Y ^CN (4.2) ) Y o Observe that the hyperbolic diameter of each Whitney square is bounded between two absolute constants. < ' < such that each piece < lies in a union of We split the sequence into four pieces: dyadic squares that are uniformly separated from each other. More precisely, set
(50) ' # '
(51) ' is given by P o ^ o (for the remaining three sequences we respectively where the family P o 6 ' ^ o , P o 6 ' ^ o and 0 0 P o 6 ' 6 ' ^ o ). In order to avoid technical difficulties choose 0 0 those containing 0 0 points of 0 (in0 case is empty there is nothing to prove). we count only In what follows we will argue on one sequence, say ' . The arguments are the same for the other sequences.
(52) ' Our first observation is that, by construction, for # , , #
(53) #x $ c. # . 2 4 # ` o for some fixed . In what follows, the letters , ... will stand for indices in of the form # 9 # : bD + . The closed rectangles are compact in so that ' can0 only contain a finite number of points (they contain at least one point, by assumption). Therefore D( S "g#x
(54) , SQ Z @ S P ' (note that we consider the entire Blaschke product associated with ). Take Z Z ^ , S Z ' @ . such that fJwyxz . Since ]n , there exists a function ' such that Assume now that ' Z ) if Z PP Z '' !^ if Z Z ^!@N
(55) . By the Riesz-Smirnov factorization we have. S " # ' # # ' is some function in 0 0 and is outer in J where is singular inner, : + 0 assume 0 <. ) , ) # . Hence ' Z ' @ : ,
(56) S " Z ' @ ' ) # ' # , Z Z @ ) 0 0 and
(57) S " Z ' @
(58) , Z ' # Z ' @ # N 0 0. (4.3). ' . .
(59) . . . Again, we can.
(60) !54 . INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. 13. is a negative 0 0 such that ) ' : 6 #54 ' # ' : !4 # # ' B Z Z K! B c5 c @# B Z Z K! # c # 0 0 0 0 0 0 hence ' ' # ' : # : # ' '! , Z Z @ , c c @ B Z Z K # c N 0 0 0 0 0 0 This yields (4.4) # Z K : # Z ' K :
(61) S " Z ' K 0 0 J ' . 0 0 for every Z P ' ' is separated. Let us now exploit Proposition 4.1. By construction, the sequence Z !^ 1n Therefore, there exists an outer function ' in the Smirnov class such that S " S Z ' @ ' Z ' @ # N Again, ' is a quotient of two bounded outer functions and we can suppose that ' is outer in : J with ' ) . Also, we can use Harnack’s inequality as above to get ' Z ' @
(62) ' Z K J ' . This together with (4.4) and our definition of Z ' give for every Z S , Z K
(63) S Z ' @ ,
(64) S " Z ' K M, S " S Z ' K # Z @ M ' Z @
(65) # ' 0 0 ' and ' . Set ' for every Z : and ; by construction, ' is outer with '. 0 + ) and ' is singular inner. < < 0 #$# , and define the Construct in a similar way functions , for the sequences < , < <. , and is singular inner. So, products J < , ' thereandexists /in , and < P ' + . Of course is outer whenever Z hence # #$#Q^ such that Z ) e S (4.5) , Z\ K Z Z @
(66) Z\ Z @ N !4 satisfies Z\2 #54 , Se Z\ K . The Therefore, the positive harmonic function 6 Y singular inner factors doY not occur in (4.3), proof for goes along the same lines, except that #. #. Since does not vanish and is bounded above by 1, the function B 6 harmonic function. By Harnack’s inequality, there exists an absolute constant . 0 #54 6) . 0. . . . . . . . . . . . and so will not appear in (4.5) either. 5. T HE. TRACE SPACES. In this short section we prove the trace space characterization of free interpolation given in Theorems 1.2 and 1.3.. = ]nG - G n .-. In order to see that (b) in each theorem implies free interpolation it suffices to observe that and use Remark 1.1.. For the proof of the converse, we will only consider the situation in the Nevanlinna class, since the case of the Smirnov class is again obtained by removing the singular part of the measure and the singular inner factors..
(67) 9RCS S Z RCS Z f ' M ' J ' ) 0 0 0 _ #54U J ) Z !4 6 RS 0 0 [ M A h A ) 0 Z J 9ReS S is such that there is a positive finite measure A with " ?,A+C Z that 6 RS . The suppose !4 Conversely, A A A Radon-Nikodym decomposition of is given by , where A . A (fA. ' is positive and AA is a positive finite singular measure. Let be the singular inner function associated with , and let be the function defined by 7 c X ' . <= %)( c %e MA %e # c N % c * Y !4 6 RS : !54 Z\ K , thus 6. By definition, is outer in and . Clearly, RS : Z @ . Since is ideal by assumption, there exists 3 interpolating LRCS S . 14. A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. Assume that and that is such: that , . Since can be written as. , where ,. ,. is singular inner with associated : A singular measure , and is an outer , we can define the A A function with. "@?BA+C positive finite measure which obviously satisfies , 0( .. . 6. H ARMONIC. MAJORANTS AND MEASURES WITH BOUNDED BALAYAGE. e ` Q A e uP A positive Borel measures on such that TWVM<>X = A %e : ^CN G ; ) A A = Proposition 6.1. Let be a positive Borel measure on the disk. Then 6 aA is finite for any A Let us start by proving that positive measures with bounded balayage " are precisely those which A A = % % % and operate against positive harmonic functions. Recall that . . such that positive harmonic function on6 the disk if and only if there exists some 4 balayage uniformly bounded by . Furthermore, the relevant constants are related:. has. A T VMX %e TWVOX . = aA A J 6 #& ) # and for any positive harmonic function , _" . = aA A N "@? % C , where % is a measure on . If A has balayage bounded by 6 , Proof. Let . = c A A c . <>= . = " ` %e A A c A % %Q : 6 % 6 N & " ` Conversely, since c %e o is a harmonic function for any fixed % , = " ` %e MA A c is pointwise defined. Pick a sequence % such that o #% !" . = " ` % o MA A c5 ; T VM<>X = . = " ` %e MA A c5 #. P where the supremum on the right hand side might a priori be infinite. Since the set 6 t ^ is uniformly bounded on compact sets in , a normal family argument P = aA ) A $ 1^ D_F . Observe that the mapping & c " ` % o is in for every shows that W T O V X o . Hence T VMX o = " ` % o A A c DfF . % , ; <>= . . . . This proves that announced.. A. has bounded balayage, and the equalities between constants that we had.
(68) INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. 15. The next result is a refined version of Theorem 1.4 stated in the introduction.. c . c . c TWVMX . A f D F N E_#x \P
(69) ! K # ^ N. Theorem 6.2. Let be a nonnegative Borel function on the unit disk. Then there exists a harmonic function such that 2 for any if and only if A = (6.1) Furthermore,. . That (6.1) is necessary is clear from the above considerations. In order to prove that it is sufficient, we will reduce ourselves to a discrete version of it. We will use the dyadic squares introduced in (4.2). As in the previous section, choose a point in each square, say. o c o o c o c o ) Y + X +@, M+ N Observe that by Harnack’s inequality, there exists a universal constant :" if c$ # c lie o (as defined %Q that : ' " ` %Q : " such ` ` %e , for in the same Whitney square in (4.2)), then . any % Lemma 6.3. The function satisfies condition (6.1) if and only if there exists a constant P 7 6=o ^ such that such that for any sequence of nonnegative coefficients 8 6 o " : ` %Q (6.2) ; TWVM<>X = o )# . then 8. (6.3). o 6 o
(70) WT VMX. . :. '. @:. Furthermore, 3 . . :. . 3 . . . . N . is an absolute constant. ) o o such that c o 2 TWVOX
(71) + and define the measure Proof of Lemma 6.3. Pick c A i o 6 o ` . Then, if P76 o ^ satisfies (6.2), A %e . = " ` %e MA A c 8 o 6 o " ` %e : 8 o 6 o " ` %e : JN , where 3. 4. . . . . So if satisfies (6.1), 8. o 6 o
(72) WT VMX. . . :. . . + 8o 6 o c o + . = A A : + . . N. The converse direction is easier, and left to the reader (it also follows from the proof of the theorem, below).. . n. We now need a classical result in convex analysis. Recall that the convex hull of a subset '. is defined as. 5x 8 - < R < R < < '
(73) . . #. < . #. 8. OH N < < ).
(74) P Z eZ # / ^ , then the conical convex hull of is defined as 6 u 5x 5x ' 6 8 - < R < R < # < H N < ' 5x 5x is a finite set, the conical convex hull is equal to its closure:. 16. A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. If we write. '. . . . . .
(75). . . . . When (for this and other facts, see [HULL]). The key fact for us will be the generalized form of the Minkowski-Farkas Lemma (see [HULL, Chapter III, Theorem 4.3.4]) that we cite here only for ' finite . Let # stand for the standard Euclidean scalar product in . : :. ' : . ' ' # . Theorem 6.4. Let # , , be such that : :. ' ' . Denote 7# # . Then the following properties are equivalent ' '. for # :. 9R PQLR ^k) n 8 ). J . : 8 f (a) For any 5x , # . 8 (b) # .. uP J. . R. K^ . . . . . ' , the coordinates are denoted by We: will: use the following special case. For a vector ' ' . , . Also, denotes the set of points of with nonnegative coordinates. : :. : ' ' Corollary 6.5. Given , , let # , and suppose ' . . Then the following properties are equivalent for that : . <. 6. A. ). R . P 6 R 6 ^ ) ) 6 6. J 6 : , # : : ) . (a) For any and for any #=N N N# A , # (b) There exist ) such that i ' ) ) ) < : 8 R < N ' P-: ' Proof. Let < ^ ' < be the canonical basis of and consider ]PQLR 9# ) # ) : : ^ PQ Y : < # # ) : : AM^CN Then 6 corresponds to the : in Theorem 6.4, from what we see that (a) implies that there exist : : : < # , ) , ) A ,8 - such that 8 # 9R L: < # N < 7# ) ' ) Y < ' i - ' and When applied to each coordinate, this yields ) < 8 R < < : 8 - R < N ' ' Y . . . . . . The converse implication is immediate.. Proof of Theorem 6.2. Suppose that satisfies (6.1). For each nonnegative integer , we define. R . . ". ` X " + -+ , W 33 o . .
(76). . . . . 0
(77). . ' . for. :. . :. +.
(78). Yf) #.
(79) o + o 3 Ag i and 6 O7P 6 o ^ 33 o. INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. 17.
(80). '6 0 o3 8 6 o " ` X " + +@, W : ) # for ) : : + Yf) H N 3 ' 6 3 3 6 o for gives a point in 6 . 0 Obviously, 6 is not empty: for instance and P76 o ^ 6 will satisfy (6.2) ) up to a constant. Indeed,) for any B ? # +-, , We claim that any + so that + +-, : B D ( + +-, , therefore by Harnack’s there is an index vD : + , ) inequality, for any c such that c\ " L: < > " ` `
(81) <
(82) ) > Y X + +-, W : " ` X + +-, N 0 02 and by Lemma 6.3 and the hypothesis, satisfies (6.3) with P ' 6 o ^ satisfies (6.2), Therefore ' constant . Corollary 6.5 then implies the existence of positive coefficients 3 with 0 sum equal to , such that ' : 0 8 3 " ` X " + +@, W . <>= " ` A % #
(83) T VMX . .
(84). . . ' . . . . . .
(85).
(86). .
(87). .
(88). .
(89).
(90).
(91). . .
(92).
(93).
(94). where %
(95) masses:. .
(96). . .
(97). .
(98). is the discrete measure on the circle given by the following combination of Dirac. ' 8 0 3
(99) <
(100) N % 0 02 Since the mass of % is uniformly bounded by , we can take a weak* limit % of this sequence of measures, so that for any # , :
(101) T VMX . <= " ` A % c o # "@? % C . Harnack’s inequality now implies that there is an absolute constant 3 ' such where . This proves the theorem, with the inequality that 3 ' c5 c for any c #x P # ^ : 3 ' : 303 ' N The constants 3 , and 3 ' only depend on the discretization we have chosen. Picking a discretization with smaller “squares”, we may make all three constants as close to as we wish. )
(102).
(103).
(104). . . Now we can prove Corollary 1.5.. . 7 Z\ K D ;S . Z . Proof of Corollary 1.5. Given a non-Blaschke sequence , arguing as in [NPT] one can show that there exists a function in the unit disk with for any if and only if is the union of a Blaschke sequence and a sequence for which there exists a positive for all harmonic function in the unit disk with . Then the result follows 24 from Theorem 1.4.. !4 Y. We finish this section with the proof of Theorem 1.6..
(105) . 18. . A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. Proof of Theorem 1.6. a d . Part (i) holds whenever admits a harmonic majorant, be it quasi-bounded or not (see Proposition 1.7), while (ii) follows from the dominated convergence theorem.. d c . We proceed by contradiction. Suppose that there exist 4 and a sequence of o A o measures such that (6.4) . C9o A A o KN Co A o A o Let . Then . <>= A o %Q A %e A oO A o yP ^; N A oJ Since . We apply the , their Carleson norms are uniformly bounded by some 3 3 4 direct part of [Gar81, Lemma I.5.5, p. 32] to ; the lemma is stated for harmonic functions, but harmonicity plays no role in the proof of the direct part. We obtain : 6 : 6
(106) A A 3 3 3 # A by (d) (i). Since the sequence tends to in , some subsequence must tend to almost everywhere, and applying (d) (ii) to that subsequence, we find a contradiction with (6.4). ' c b . We define a function on by ? C. ( # ( # for. . oM o yP o W o ^; . . ' 3 &P. ' 3. ' s^; . 6 Uo R5o o LRo is an increasing sequence of positive numbers tending ) to infinity, to be determined where o is given recursively by 3 and oU ( o 6 ' ( . Observe that each later, o is and ) defined on the whole real line_(they give supporting hyperplanes for )the polygonal convex " @ 6 ' . graph of ). We shall also use # for ok Co and We prove that admits a harmonic majorant using Theorem 1.4. Let @o TWVMX = o A A . If A , then . = c MA A c5 o 8 3 . Lo Co 6 ' o oM c A A c o 8 . = ? o6 oa c o6 o 6 ' c C A A c 3 Y 8 . = 3 c MA A c5 ( o . = ? o6 oM c o6 ' oa c C A A c ' Y 8 :GR 3 . = c5 MA A c ( o ' . = LR5o Y R5o ' oa c Y 6 A A c5 :GR 3 3 ( $ o 8 ' LR5o Y R5o ' -o N !!" o -o , we can choose an increasing sequence 9Reo such that #!"EolR5o F , but Since LR5o $o i ' Y R5o ' @o DGF , and we are done. b a . First notice that can be replaced by a function : with the same properties as and the additional explicit description: oa R5o ( o R o for ? Co # eo 6 ' C # . . . . . . . . . . . . . . . . . . . . . . . . .
(107) R5o R o. 3. Co o ' is an increasing sequence of positive numbers ) o i ' R o DfF . ) o o o i . ; thus. INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. . . 19. . where for , and 4 tending to infinity fast enough so that $ Define . o . . . . . . . . The following Lemma is due to Alexander Borichev. . 6 . 6 e A e . . N. such that whenever is bounded and Lemma 6.6. There exists an absolute constant 354 : : for some , then there exists quasi-bounded such that and : <= % % 3 . 3oM R o . 3. .. o. o. In order to prove (a) let be a harmonic majorant of . Each is then bounded and majorized by , hence by applying the previous lemma we find quasi-bounded such that : and : 3 . The series , since for converges in all and 8 8. o. o. 3 R o o Uo . :. 3. i o R o v 3 o ) D F. ' . o. #. and defines therefore a quasi-bounded harmonic majorant of .. "@"@?B AD C # + W . Let A denote the boundary measure of , . We use the standard dyadic decomposition of the circle . For any , let o be the union of the dyadic intervals
(108) o n Co Let 3 ) such that A
(109) o 4
(110) o N o
(111) o
(112) o 6 '
(113) o ' Note
(114) that cannot contain two contiguous intervals
(115) such thato , because o o C o ' n , a contradiction. Therefore, if n , then then A
(116) o : A
(117) o ' :
(118) o ' +
(119) o D + A
(120) o N
(121)
(122)
(123) For any interval , let be the interval of same center and triple length, and let , o o where the union is taken over all the dyadic intervals included in . We write A A 3 A %e+ ( <= A A A A ' ( A A # 0 0 where 3 4 is to be chosen. This measure is absolutely continuous with respect to arc length. "@? A C . Indeed, let c and suppose that there exist a 0 The function
(124) we are looking for is dyadic interval n , maximal among the dyadic intervals contained in , such that (6.5) . " ` %e A %e 3 ) N 0 Then clearly c5 c . We claim that if c is such that (6.5) does not hold for any
(125) maximal dyadic interval n , then c5 c5 , which will finish the proof. Proof of Lemma 6.6. Set i.e. the measure such that given in (4.1). . Under that assumption, since the level sets of the Poisson integral in (6.5) are arcs of circles
(126) connecting the extremities of , where they make a fixed angle with depending 6
(127)
(128)
(129) on 3 , we must have % D for any % and any maximal dyadic subinterval of , so that. c Y . . . . 0.
(130) ` %e for %
(131) are comparable, say to the value at its center % . Therefore for any . " ` %e A A %Q : 6 " ` % . A A %e : + 6 " ` % . A %e : + 6 0 . " ` %e A %Q + 6 0 . " ` %e A A %e N 3 0 6 6 Since is an increasing function of 3 , and therefore 6 4 decreasing function of 3 , we ) a therefore, 0 so that 3 + 0 , and 0 may choose a value of 3 4 large enough since is the union ) 0 0 of its maximal dyadic subintervals, . " ` %e A A %e . " ` %e A A %e . " ` %e A A %e N <>=9 " ` %e A A %e , and we are done. A <>=9 " By construction, ` %e A %e 20. A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. all the
(132) values such ,. ". . . 7. W EAKER. CONDITIONS FOR THE EXISTENCE OF HARMONIC MAJORANTS. In this section we state first a sufficient condition implied by a result of Borichev on a similar problem. On the other hand, we also prove the necessary condition of Proposition 1.8 and show that it is not sufficient. ' Theorem 7.1 (BNT). Given a collection of nonnegative data , there exists a finite positive measure % on such that. . if and only if. (7.1). %.
(133) o
(134) o . . P o ^1n 6. . o . TWVMX
(135) o 8 o
(136) o P
(137) o ^
(138) o . is a disjoint family. H DGF N. o ` i o ` 9: <?> L: < > )
(139) ` L : <?> N. This is an analogue of the discretized version of Theorem 1.2(d), (as in Lemma 6.3) obtained A
(140)
(141) , and by replacing the Poisson by considering only measures of type " by the “square” kernels kernel. `. Here of ..
(142). `. denote the intervals defined in (1.2) and. stands again for the characteristic function. The similarity of Theorem 1.2 with this result leads us to an:. Z J Sj. . P76KS ^. [Z7[ " ) Y ` provide a sufficient (but not necessary) conTheorem 7.1 together with the estimate ` dition for domination by true harmonic functions, which is clearly less restrictive than requiring ' that , but easier to check in concrete examples than the characterizing condition Open Question. 6 Is condition (d) in Theorem 1.2 still sufficient if we restrict it to that for any , or ?. of Theorem 1.4.. such.
(143) o WT VMX
(144). INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. 21. satisfies (7.1) admits a Corollary 7.2. Any positive function such that & " harmonic majorant. On the other hand, the positive harmonic function does not satisfy (7.1) for certain choices of .. c o o c . c. 6. ". `. ` ). 6. for Proof. It is well known and 6 easy to see that there exists a constant such that any (the constant depends on the aperture of the Stolz angle). Therefore, for any . "@? C. %. c . . .. 6. <=. %e A % %e . . . 6 %.
(145) o
(146) o . 6 . . o 6
(147) WT VOX. #. 6 % C is the harmonic majorant we are looking for. PQ # t ^ . Then the ) is not necessary, consider o To see
(148) that the condition any n o " intervals d' are all disjoint; +
(149) o d' ' , so that) condition (7.1) will fail however ` ) (the sum is comparable to ). In the same way Corollary 7.2 and Proposition 1.8 imply the following aso ,inwrite
(150) 1.11,
(151) o Corollary (the radial projection of the square to an arc of the result. For which proves that. "@?. . . . . circle).. Corollary 7.3. Assume that and that 8. is contained in a union. . of Whitney squares. TWVOX
(152) ) Y c @ d GST VM X
(153) !54 , Se Z @ ' H f D F P
(154) \ # ^ where the supremum is taken over all n such that . . . of center. c . #. is a disjoint family, then. is interpolating for the Nevanlinna class.. We move next to the proof of the necessary condition in terms of the Hardy-Littlewood maximal function.. 6 P. 6 < O Y " 6 <
(155) , )
(156) ( N 0 of a measurable set n ' . Y 0 measure For convenience we shall write here t for the Lebesgue : : Also, we only need to look at boundary points in a fixed bounded interval, say . ] P
(157) .
(158)
(159) .
(160) ? C Y ) ) , let # 4 ^ . For any , there exists c c and
(161) 4
(162) ` such For any Y ) ) that c (7.2) . 4 # i.e. c K
(163) `
(164) 4 N 6 f ' # and a disjoint family of By Vitali’s covering lemma, there exist an absolute constant : : 6
(165) ) ' intervals ( = , , such that i 5
(166) .
(167) ).
(168) Write c c ( . Note that since the point c is contained in the “tent” over ` (therefore in the tent over ) the points c are separated in the hyperbolic metric.. Proof of Proposition 1.8. (a) The problem.
(169) can be localized,. so we may work on the upper half plane, ( 4 , with # ( , restricting ourselves to positive harmonic functions which are Poisson integrals of positive measures with finite mass. Here the Poisson kernel is given by. * . ^.
(170) + 5 5
(171) ` ` + c " ,) . A A c A , ) . A c # ) ) ( (. 0 0 ) )
(172) and, by (7.2), c c 24 ` . ' ' , Therefore, since 6 + ' : +) 8 5 : 8
(173) ` : P 4 ^a : 3 N 22. A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. c. c . Now way : let following. define new points in
(174) the ( . Note that
(175) . We claim that 2 , where is a harmonic majorant of . Indeed, writing . . . ?,A+C. and ,. . (b) Similarly.. We now give two examples showing that the necessary condition of Proposition 1.8 is strictly stronger than that of Proposition 1.7 but still not sufficient. Proposition 7.4. (a) There are functions such that , but that do not admit a harmonic majorant. (b) There are functions such that , but .. ' ' . ' . Proof. The proof rests on the following family of examples. Note that it is easy to turn those examples into examples of sequences which are (or are not) interpolating for the Nevanlinna class.. 6 Z +. . . Again we will work on . Our functions will vanish everywhere on the upper half plane,. except on the sequence ( , where and . To ensure that :. we take ( . With this choice, it can be deduced from Proposition 8.2 (or the remark before Lemma 8.4), that a necessary and sufficient condition for the existence of a harmonic majorant is that , that is,. ] 6 '. Y 0. ) '. 8. (7.3) We note that. . . . Z DGF N +. "( # " " N ) P )P Z ^ which Henceforth we only study data ^ are increasing sequences positive Q P M. 6 ' of ' ' 6 6 numbers tending to infinity. We ( ^ forms an #x\P that 3 also"gassume Y D ^ increasing sequence. Let . The necessary condition arising from the ' ' reads fact that 8 3' : 3 (7.4) # 4 N
(176) . . . This condition will be assumed for both examples..
(177) # ( , and let P . 3 for , define 4 ^ ' Now, _ 1 " !x U P Y 6 ' ^ . Then,
(178)
(179)
(180)
(181) ' # ( # ' (
(182) INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. 23. . and. P 2 4 ^
(183)
(184)
(185)
(186). ' . . (7.5).
(187)
(188) . (. . ' . .
(189) . . + (.
(190) . 8. '. . . '
(191) . . . ' . . 3 '. (. +.
(192) . 8. . .
(193) . . . N. . Since
(194) , condition (7.4) becomes Inorder to prove (a), choose that remains bounded above, while the necessary and sufficient condition (see (7.3)) is. . . . 8. . . DvF N. 9 # 54 & ' #,5this condition fails, so that !
(195) 4 ' ' . Since ! 6 ' However, 3 !
(196) ' , and ' #54 Y ' . thus ' With. . . . Therefore equation (7.5) becomes. P 2 4 ^
(197)
(198)
(199)
(200). . !5) 4. (. . !5) 4. (.
(201)
(202) . :. + 3.
(203) 8.
(204) . :. !5) 4. ' 6 ' ). '. M ' W ,. admits no harmonic majorant.
(205) , then. . #5) 4. (. + !54 ! 4 ' ' ! 4 3 *. 3 . #. and this choice of does satisfy the necessary condition given in Proposition 1.8.. 3 ' ' ) ' ' ' + #54 ' P ^ ) ) 3 * ). !
(206) in the Lemma, choose ! To prove the second statement . With similar but easier calcula tions one sees that and . Therefore (7.5) becomes
(207) 8
(208)
(209) 4
(210)
(211) ' ( ( #
(212)
(213)
(214) . so the weak sition 1.7.. . '. condition fails for . !54. , even though satisfies the necessary condition in Propo. 8. P ROOFS. OF THE GEOMETRIC CONDITIONS. S n % ] . Proof of Corollary 1.11. Since.
(215). . %e # 54 , eS \Z K ' H # Z J #. to prove (a) and (b) it suffices to apply condition (a) of Corollary 1.10. Statement (c) follows from the next Lemma applied to . 9 = &(' Lemma 8.1. Let satisfy . Then admits a quasi-bounded harmonic majorant.. 5 Y. 6. i S M) Y [Z7[ Z DfF.
(216) 24. i S Z . ' j . A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. Proof. Let . By assumption result follows from Corollary 1.10(b) and Theorem 1.3.. and obviously . . :. . , hence the. (b) and (c) also follow directly from Theorem 1.2(d), by a simple argument based on the ' ,Parts = duality. o enough to consider Proof of Proposition 1.12. It is the case where is contained in only one ; < < ' < n , # N N N # , and % % , then Stolz angle. Indeed, if with ; !#" ; c K # ) ># ` % ` ) S ' ! 4 e S # 5 4 ' so that Z @ behaves asymptotically like , Z\ K in ; (here Z < ). Also,. . . . . . . . . we can assume that the sequence is radial (this means that we replace the initial sequence by one which is in a uniform pseudo-hyperbolic neighborhood of the initial one; by Harnack’s inequality such a perturbation does not change substantially the behavior of positive harmonic functions).. 6 ' j PZo^o n ;of o #54 S o ' ;o o @o @o ) Y dZ [ !#" o @ovZ @ ov S ) o o o 6 ' o @o @o 6 ' Y 8 o . %e o o %e # % EN ' , and Then. "@? C o " o. Z . Z #&%e 8 o %e MA %e 8 Co o ) [Z o [ . A %Q i Co o @o o @ o o #54 , S Z o @ ) ' Y N dZ ) Y [Z ) Y dZ ) Y This and Theorem 1.3 prove the assertion. A ' , the Dirac mass on The proof for the Nevanlinna class is even simpler. Set A ) From (1.4) we get !54 S Z o @ ' ) o " ?,A C Z o # dZ ) Y and we finish by applying Theorem 1.2.. According to Corollary 1.11 it is enough to prove the sufficiency of the conditions. Let us first show . In order to construct a function that condition (1.3) implies interpolation in ? meeting the requirement of Theorem 1.3(c) assume that is. # . Clearly there exists a arranged in increasing order and set: ,
(217)
(218) decreasing sequence with , , and . Now, if , set
(219)
(220) ,. , and set. . . . . . . . .. . . 6. . Proof of Proposition 1.13. By Corollary 1.11(c), we already know that (1.5) is a sufficient condi. Conversely, suppose that is interpolating for , that is, tion for to be interpolating for A = A admits a harmonic majorant. Since has bounded balayage, then , which is exactly (1.5).. . . A DGF A . Proof of Proposition 1.14. It is obvious that (a) implies (b). If we assume (c), will act against any positive harmonic function. Suppose . As seen in Section 5, there exists a positive.
(221) #54 6 p : . Thus, taking A S as in (1.6), 8 S ) Y d Zld !54 6 Z @ . = #54 6 Z\ K A A Z\ : . = Z MA A Z DGF N " S i S Finally, to prove that (b) implies (c), suppose that (c) doesn’t hold, i.e. is d l Z d is lower semi-continuous, this implies that ) Y unbounded. Since . Since is the. . ' such that <>= F . Taking an outer function 6 ' dual of , there exists ! 5 4 " ? C p with we see that S 8 ) Y dZld !54 Z\ K S 8 ) Y dZld . <= " S-t . <>= g F # INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. harmonic function so that. 25. . . so (b) doesn’t hold.. ] wyxz . . Proof of Naftaleviˇc’s theorem. Assume that is contained in a finite union of Stolz angles and (1.4) holds. By Proposition 1.12, , hence the trace is given by the majorization condition of Theorem 1.2(b). Taking as majorizing function the Poisson integral of the sum of the Dirac masses at the vertices, we see that Na .. Conversely, if . . . . is free interpolating and by Corollary Na then the trace is ideal, so 1.11(b) (1.4) holds. According to Theorem 1.2(b) and the definition of Na , the function. c ) Y [ Z7[ ' if c / Z J if c "@?BADC c and consider the intervals admits a harmonic majorant . Let c
(222) uP ` % 5c %Q ^CN A
(223) There exist constants and 3 such that ` 2 4 3 for any c such that c ) Y dc [ ' . in a finite union of Stolz angles, then there is an accumulation % If isofnot contained
(224) S point n such that n %e for any . Pick 4 ; then for Z P ,
(225) S S % and we can construct an infinite subsequence n ^ are such that the Privalov shadows . . . disjoint. This prevents from being the Poisson integral of a finite positive measure. We now give an example of a concrete separation condition implying that balayage.. A . has bounded. nv is contained in the union of a family of Whitney squares 4U W4 ' c @d c c @ 4 , where c Y is the center of and) Y is a positive function, with . for any # , . decreasing and . 3 DfF N GJwyxz if and only if fJwyxz 6 , and0 if and only if Then 8 dc K[ vS TWVM X
(226) !54 Se Z @ ' DfF
(227) + )Y . Proposition 8.2. Assume that such that . . .
(228) 26. A. HARTMANN, X. MASSANEDA, A. NICOLAU, P. THOMAS. A . Note that this covers some cases where does not have bounded balayage, even though another measure associated with the sequence will (see the proof).. L : < > # 8 centered at : <?> , of 8 L: <?> # 8 uP c I dc\ :G8 # 4U c5 B :f8 ^CN Y implies boundedness Y The next result is a Carleson-type condition) which of the balayage. : 98 A L: <?> 8 Lemma 8.3. Suppose that # W , where is a nondecreasing function on ? # + with . . 3 A DGF N A Then is a measure with bounded balayage. 0 In order to prove Proposition 8.2 consider the “Carleson window” side : . o A 9: <?> + o + o WT > VM X # DvF. A discrete version of this condition is 8. #. as can be checked by writing a Riemann sum.. B5 P c " ` L: <?> 1 ^ . This is a disk, tangent to the unit : <> O . Therefore B5 n L: < > #&3 for , say, with ) ) ) A A Using the distribution function B W and the fact that the measure is bounded, we get. Proof. For any 4 , let ( circle at the point , of radius an absolute constant. 354. A. the following estimate for the balay´ee of : " A A : =. . ` L: < > A c . 3 : 3 ' (. 5B A 3 ' ( . A B W A : 3 ' ( . A 9: <?> #&3 W A ' ' . ' . ' 3 A : 3 ' ( 3 . 3 A DfF N 0. We will now compare measures satisfying the condition in Lemma 8.3, measures with bounded balayage and Carleson measures. Each set is included in the next, and the examples will show that both inclusions are strict.. P o^ A o + 7 c5 A A c $o 8 o +-) , .13 02 W ) Y + o : <?> ACBeN =. ' )Y A One can check that is o a Carleson measure if and only if it has bounded balayage and this A o happens if and only if i DIF . Also satisfies the condition in Lemma 8.3 if and only if i o i Co DfF . ? A Example 2. Let be a nonnegative-valued function on the interval # . Let be the ) measure concentrated on the ray from the origin to given by ' ) 7 . A . = c A c . 3 A N Example 1. Let be a sequence of nonnegative reals. Let be the measure given in dual terms by concentrated on the circles centered at the origin of radius.
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(231) INTERPOLATION IN THE NEVANLINNA AND SMIRNOV CLASSES AND HARMONIC MAJORANTS. One can check that. A
(232). ' . ' A. is a Carleson measure if and only if there exists a constant. :. . A. . #. 4. 27. such that. . and is a measure with bounded balayage if and only if it satisfies the condition in Lemma 8.3,
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