Bad Boundary Behavior in Star Invariant Subspaces II

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1 University of Richmond UR Scholarship Repository Math and Computer Science Faculty Publications Math and Computer Science 202 Bad Boundary Behavior in Star Invariant Subspaces II William T Ross University of Richmond, wross@richmondedu Andreas Hartmann Follow this and additional works at: Part of the Other Mathematics Commons Recommended Citation Ross, William T and Hartmann, Andreas, "Bad Boundary Behavior in Star Invariant Subspaces II" (202) Math and Computer Science Faculty Publications 5 This Article is brought to you for free and open access by the Math and Computer Science at UR Scholarship Repository It has been accepted for inclusion in Math and Computer Science Faculty Publications by an authorized administrator of UR Scholarship Repository For more information, please contact scholarshiprepository@richmondedu

2 BAD BOUNDARY BEHAVIOR IN STAR INVARIANT SUBSPACES II ANDREAS HARTMANN & WILLIAM T ROSS ABSTRACT We continue our study begun in [HR] concerning the radial growth of functions in the model spaces(ih 2 ) INTRODUCTION Suppose I = BS µ is an inner function with Blaschke factor B, with zeros{λ n } n in the open unit diskdrepeated according to multiplicity, and singular inner factors µ with associated positive singular measure µ on the unit circle T The following result was shown by Frostman in 942 for Blaschke products (see [Fro42] or [CL66]) and by Ahern-Clark for general inner functions [AC7, Lemma 3] Theorem (Frostman, 942; Ahern-Clark, 97) Let ζ T and I be inner with µ({ζ})=0 Then the following assertions are equivalent (2) () Every divisor ofi has a radial limit of modulus one atζ (2) Every divisor ofi has a radial limit atζ (3) The following condition holds λ n n ζ λ n + T ζ e it dµ(eit )< Based on a stronger condition than the above, Ahern and Clark [AC70] were able to characterize good non-tangential boundary behavior of functions in the model spaces(ih 2 ) of the classical Hardy spaceh 2 (see [Nik86] for a very complete treatment of model spaces) Theorem 3 ([AC70]) Let I= BS µ be an inner function with zeros{λ n } n and associated singular measureµ Forζ T, the following are equivalent: () Everyf (IH 2 ) has a radial limit atζ (2) The following condition holds λ n (4) n ζ λ n 2+ )< T ζ e it 2dµ(eit In this paper, we will study what happens when we are somewhere in between the Frostman condition (2) and the Ahern-Clark condition (4) In order to do so we will introduce an auxiliary function Let ϕ (0,+ ) R + be a positive increasing function such that Date: November 25, Mathematics Subject Classification 30J0, 30A2, 30A08 Key words and phrases Hardy spaces, star invariant subspaces, non-tangential limits, inner functions, unconditional sequences, generalized Carleson condition

3 2 ANDREAS HARTMANN & WILLIAM T ROSS () x ϕ(x) x is bounded, (2) x ϕ(x) x 2 is decreasing, (3) ϕ(x) ϕ(x+o(x)), x 0 Such a function ϕ will be called admissible One can check that functions like ϕ(x)=x p, p<2, and ϕ(x)=x p log(/x), <p<2, are admissible Our main result is the following Theorem 5 Let I= BS µ be an inner function with zeros{λ n } n and associated singular measure µ, ϕ an admissible function, and ζ T If (6) n λ n ϕ( ζ λ n ) + T ϕ( ζ e it ) dµ(eit )<, then every f (IH 2 ) satisfies ϕ( r) (7) f(rζ) r When ϕ(x)=xthen we are in the Frostman situation (2) and no restriction is given for the growth off since generic functions inh 2 satisfy the growth condition f(rζ) =o( ) r On the other hand, when ϕ(x)=x 2 we reach the Ahern-Clark situation (4) For other ϕ such as ϕ(x)=x 3/2 or perhaps ϕ(x)=x 2 log(/x) we get that even though functions in(ih 2 ) can be poorly behaved (as in the title of this paper), the growth is controlled There is some history behind these types of problems When ϕ(x)=x 2N+2, where N = 0,,2,, Ahern and Clark [AC70] showed that (6) is equivalent to the condition thatf (j), 0 j N, have radial limits at ζ for everyf (IH 2 ) When ϕ(x)=x p, p (, ), Cohn [Coh86] showed that (6) is equivalent to the condition that every f H q IH q 0, where q=p(p ), has a finite radial limit atζ Why did we write this second paper? In [HR] we discussed controlled growth of functions from(bh 2 ), where B is a Blaschke product not satisfying the condition (4) of the Ahern-Clark theorem We have a general result but stated in very different terms, and using very different techniques, than the paper here In particular, in [HR] we obtain two-sided estimates for the reproducing kernels which yields more precise results The results presented here are one-sided estimates but are for general inner functions and not just Blaschke products 2 PROOF OF THE MAIN RESULT It is well known that(ih 2 ) is a reproducing kernel Hilbert space with kernel function kλ I I(λ)I(z) (z) = λz It suffices to prove Theorem 5 for ζ= If denotes the norm in H 2, the estimate in (5) follows from the following result along with the obvious estimate f(r) f k I r, f (IH 2 ), r (0,)

4 BAD BOUNDARY BEHAVIOR IN STAR INVARIANT SUBSPACES II 3 Theorem 2 Let I= BS µ be an inner function with zeros{λ n } n and associated singular measureµand ϕ be an admissible function If (22) then n λ n ϕ( λ n ) + T ϕ( e it ) dµ(eit )<, (23) k I r 2 ϕ( r) ( r) 2 Proof Our first observation is that sincex ϕ(x)/x is bounded, (22) implies condition (2) By Theorem this implies that lim r B(r) =lim r S µ (r) = Hence kr I 2 = I(r) 2 = exp(log( I(r) 2 )) = exp(log( B(r) 2 +log S µ (r) 2 )), r 2 r 2 r 2 and sincelog B(r) 0and log S µ (r) 0when r, we get k I r 2 = exp(log B(r) 2 +log S µ (r) 2 ) r 2 (+(log B(r) 2 +log S µ (r) 2 )+o(log B(r) 2 +log S µ (r) 2 )) = r 2 log B(r) 2 +log S µ (r) 2 r 2 Thus to prove the estimate in (23) we need to prove (24) and (25) (26) Case : the Blaschke product B log B(r) 2 r 2 log S µ (r) 2 r 2 ϕ( r) ( r) 2 ϕ( r) ( r) 2 First note that from the Frostman condition (2) we get λ n λ n 0 In particular, from a certain index n 0 on the points λ n, n n 0, will be pseudohyperbolically far from the radius[0,), ie, there is aδ such that for everyn n 0 and r [0,), This implies log A well known calculation shows that b λn (r) δ b λn (r) 2 b λ n (r) 2 b λn (r) 2 = ( r2 )( λ n 2 ) rλ n 2

5 4 ANDREAS HARTMANN & WILLIAM T ROSS Thus (27) log B(r) 2 r 2 = λ n r 2 log 2 n b λn (z) 2 n λ n r 2 Now letλ n =r n e iθn We need the following two easy estimates: (28) ρe iθ 2 ( ρ) 2 +θ 2, ρ,θ 0 (29) ( z 2 + w 2 ) /2 z + w, z,w C In particular, λ n 2 ( r n ) 2 +θ 2 n We now remember condition (26) which implies that r n = λ n =o( λ n )=o(( r n )+θ n ) so that necessarily r n =o(θ n ) Hence λ n r 2 ( r n r) 2 +θ 2 n=( r n +r n ( r)) 2 +θ 2 n ( r) 2 +θ 2 n The estimate in (27) yields (20) log B(r) 2 r 2 n λ n 2 r n λ n r 2 n ( r) 2 +θn 2 ( r) 2 r n = + {n r<θ n} θn 2 r n r n + {n r<θ n} θn 2 {n r θ n} ( r) 2 ( r n ) {n r θ n} Let us discuss each summand in (20) individually For the first, we use the fact that ϕ is admissible and soϕ(θ) ϕ( e iθ ) to get r n {n r<θ n} θn 2 r n = {n r<θ n} ϕ(θn )θn/ 2 ϕ(θ n ) /2 r n {n r<θ n} ϕ(θ n ) bounded by assumption {n r<θ n} {n r<θ n} /2 r n ϕ(θ n )(θn/ϕ(θ 2 n )) 2 r n θn/ϕ(θ 4 n ) /2 By assumption, x ϕ(x)/x 2 is decreasing Hence we can bound θ 2 n/ϕ(θ n ) below in this last sum by( r) 2 /ϕ( r) Hence r n {n r<θ n} θn 2 ϕ( r) ( r) 2 {n r<θ n} r n ϕ(θ n ) /2 ϕ( r) ( r) 2

6 BAD BOUNDARY BEHAVIOR IN STAR INVARIANT SUBSPACES II 5 For the second sum in (20) we have ϕ(θn ) ( r n ) = ( r n ) {n r θ n} {n r θ n} ϕ(θn ) /2 ( r n ) n )ϕ(θ n ) {n r θ n} ϕ(θ n ) {n r θ n}( r bounded by assumption ϕ( r) n ) {n r θ n}( r where we have used the fact that ϕ is increasing Dividing through the square root of the sum, this last inequality (and then squaring) implies This verifies (24) Case 2: the singular inner factors µ ( r n ) ϕ( r) {n r θ n} This case is very similar to the first case Indeed, log S µ (r) 2 r 2 =2 T re iθ 2dµ(eiθ ) T ( r) 2 +θ 2dµ(eiθ ) where we have again used (28) As in the Blaschke situation we split the integral into two parts depending on which term in the denominator dominates: (2) log S µ (r) 2 r 2 {θ r θ} ( r) 2 +θ )+ ) 2dµ(eiθ {θ r θ} ( r) 2 +θ 2dµ(eiθ )+ {θ r θ} θ 2dµ(eiθ ( r) 2 dµ(e iθ ) {θ r θ} Let us consider the first integral {θ r θ} θ 2dµ(eiθ ) = {θ r θ} ϕ(θ)θ2 / ϕ(θ) dµ(eiθ ) /2 /2 ( {θ r θ} ϕ(θ) dµ(eiθ )) ( {θ r θ} θ 4 /ϕ(θ) dµ(eiθ )) Note that e iθ θ Then using the hypothesis of admissibility we have ϕ(θ) ϕ( e iθ ) and so ϕ(θ) dµ(eiθ ) ϕ( e iθ ) dµ(eiθ ) which is bounded by assumption Hence, by the Cauchy-Schwarz inequality, /2 ϕ {θ r θ} θ ) ( 2dµ(eiθ {θ r θ} θ 4 /ϕ(θ) dµ(eiθ )) =( 2 /2 (θ) )) {θ r θ} ϕ(θ)θ 4dµ(eiθ /2, /2

7 6 ANDREAS HARTMANN & WILLIAM T ROSS Now using the fact thatx ϕ(x)/x 2 is decreasing we obtainϕ 2 (θ)/θ 4 (ϕ( r)) 2 /( r) 4 Hence ) ϕ( r) /2 {θ r θ} θ 2dµ(eiθ ( r) 2( {θ r θ} ϕ(θ) dµ(eiθ )) ϕ( r) ( r) 2 We turn to the second integral in (2) to get {θ r θ} dµ(e iθ ) = {θ r θ} ϕ(θ) ϕ(θ) dµ(e iθ ) /2 ( ϕ(θ)dµ(e iθ )) /2( {θ r θ} {θ r θ} ϕ(θ) dµ(eiθ )) We have already seen above that the second factor above is bounded by assumption Using the fact that ϕ is increasing we get /2 dµ(e iθ ) ( ϕ(θ)dµ(e iθ )) /2 ϕ( r)( dµ(e iθ )) {θ r θ} {θ r θ} {θ r θ} Dividing through by the integral (and then squaring), we obtain which verifies (25) {θ r θ} dµ(e iθ ) ϕ( r), 3 AN EXAMPLE The Blaschke situation was discussed in [HR] where we obtained two-sided estimates for the reproducing kernels It can be shown with concrete examples that the estimates from Theorem 2 are in general weaker than those obtained in [HR] for Blaschke products Let us discuss the simplest case, in fact close enough to a Blaschke product, that a singular inner functions µ with a discrete measureµ Let µ=α n δ ζn, n where δ ζn T and α n are positive numbers with n α n < guaranteeing that µ is a finite measure ont Let us fix ζ n =e iθn =e i/n, α n = n+ε, n=,2, Also let ϕ(t)=t γ which defines an admissible function for<γ<2 In order to have condition (22) it is necessary and sufficient to have α n n ϕ( e iθn ) n n +ε ϕ(/n) n which is equivalent toγ<ε We suppose that (3) By Theorem 2 we deduce that <ε<2 n γ n +ε= n k I r 2 ϕ( r) ( r) 2 =( r ) 2 γ n +ε γ<

8 BAD BOUNDARY BEHAVIOR IN STAR INVARIANT SUBSPACES II 7 In this situation we have f(r) ( r) γ/2, f (S µh 2 ), which is slower growth than the standard estimate f(r) ( r) /2, f H2 In this situation, it is actually possible to get a double-sided estimate for the reproducing kernel: since ϕ is admissible, Theorem implies that I(r) η T when r In particular forr (0,), this implies that r I(r) =exp( α 2 n n ζ n r 2) n α n r 2 ζ n r 2 Let us consider the reproducing kernel of(s µ H 2 ) at r=ρ N = 2 N Indeed, Now using (28) and so or, equivalently, k I ρ N 2 = I(ρ N) 2 ρ 2 N 2 N ( ( n n k I ρ N 2 n α n ζ n ρ N 2 2 N ρ 2 N ( exp( α n n ζ n ρ N 2)) α n /2 N ζ n ρ N 2)) ζ n ρ N 2 n N, α n /n 2 +/2 2N= n 2 N α n /n 2+ n>2 N α n /2 2N n 2 n 2 n +ε+22n N n>2 n +ε 2(2 ε)n N 2 ε = ( ) ρ N (32) kρ I N ( ) ρ N ε/2 (the estimate extends to the whole radius) As a consequence, the estimate from Theorem 2 is not optimal, though it is possible to come closer to it by choosing eg, ϕ(t)=t ε /log +γ (/t), γ>0

9 8 ANDREAS HARTMANN & WILLIAM T ROSS 4 A LOWER ESTIMATE We finish the paper with a construction of an f (S µ H 2 ), with µ the discrete measure discussed in the previous section, getting close to the growth given by the norm of the reproducing kernels thoughout the whole radius(0,) As in [HR] our construction will be based on unconditional sequences We need to recall some material on generalized interpolation in Hardy spaces for which we refer the reader to [Nik02, Section C3] Let I= n I n be a factorization of an inner function I into inner functions I n, n N The sequence{i n } n satisfies the generalized Carleson condition, sometimes called the Carleson-Vasyunin condition, which we will write{i n } n (CV), if there is aδ>0such that (4) I(z) δinf n I n(z), z D In the special case of a Blaschke product B= B Λ with simple zeros Λ={λ n } n and I n = b λn, this is equivalent to the well-known Carleson conditioninf n B Λ {λn}(λ n ) δ>0 If{I n } n (CV) then{(i n H 2 ) } n is an unconditional basis for(ih 2 ) meaning that everyf (IH 2 ) can be written uniquely as with In our situation we havei=s µ and f=f n, f n (I n H 2 ), n f 2 f n 2 n I n =e α z+ζ n n z ζ n The corresponding spaces(i n H 2 ) are known to be isometrically isomorphic to the Paley- Wiener space of analytic functions of exponential type α n /2 and square integrable on the real axis In this situation a sufficient condition for (4) is known: µ({ζ n })µ({ζ k }) sup < n k n ζ n ζ k 2 (see [Nik86, Corollary 6, p 247]) So, sinceε>by (3), we have /n sup +ε /k +ε n k n /n /k =sup 2 n k n /n ε /k ε n k 2 π2 3 < Hence(IH 2 ) is an l 2 -sum of Paley-Wiener spaces (each of which possesses for instance the harmonic unconditional basis) In particular, picking the sequence{k n } n, where λ n =r n ζ n =r n e i/n, r n = n, K n = kin λ n k In λ n (I nh 2 ),

10 BAD BOUNDARY BEHAVIOR IN STAR INVARIANT SUBSPACES II 9 is an unconditional sequence in(ih 2 ) Observe that Λ={λ n } n is not a Blaschke sequence We can introduce the family of functions f β =β n K n n where f β 2 n β n 2 < Let us estimate the norms k In λ n First observe that Hence so that λ n +ζ n r n + α n =α n λ n ζ n r n = 2 /n n +ε /n = 2 /n 0, n n ε k In λ n 2 = I n(λ n ) 2 r 2 n = exp(α λ n+ζ n nλ n ζ n ) r n 2α n ( r n ) 2, αn n (+ε) k In λ n ( r n ) 2= /n I n(λ n ) exp(log I n (λ n ) ) = r n r n (+α r n+ nr ) n r n =n /2 ε/2 =n ( ε)/2 Observe now that theλ n s belong to a Stolz domain with vertex at Indeed, λ n = r n =/n ζ n λ n (this follows from (28)) For fixedβ={β n } n withβ n 0we compute Ref β (λ N ) β n n (ε )/2 Re I n(λ n )I n (λ N ) n λ n λ N We have already seen thatr I n (λ n ),n, and We have to consider I n (λ n ) α n +r n r n 2 n ε α n λ N +ζ n λ N ζ n For n or N big enough, Re(λ N +ζ n ) Im(λ N +ζ n ) λ N +ζ n We thus have to consider the denominator We observe that by Lemma 28 λ N ζ n = ζ n λ N ( r N )+ n N = N + n N n ifn N (42) N ifn>n As a consequence, Again: α n λ N +ζ n λ N ζ n 0, n I n (λ N ) +α n λ N +ζ n λ N ζ n

11 0 ANDREAS HARTMANN & WILLIAM T ROSS Hence r n + I n (λ n )I n (λ N ) (+α n r n )(+α λ N +ζ n +r n ζ n +λ N n ) α n +α n λ N ζ n r n ζ n λ N = α n ( +r n + ζ n+λ N (+r n )(ζ n λ N )+( r n )(ζ n +λ N ) )=α n r n ζ n λ N ( r n )(ζ n λ N ) From here we have (43) ζ n r n λ N = 2α n ( r n )(ζ n λ N ) =2α ζ n r n λ N nζ n ( r n )(ζ n λ N ) = 2α n ζ n λ n λ N ( r n )(ζ n λ N ) I n (λ n )I n (λ N ) λ n λ N We claim that at least forn 2N, 2α n ζ n ( r n )(ζ n λ N ) = 2 n ε ζ n ζ n λ N Indeed, ζ n ζ n Re ζ n λ N ζ n λ N ζ n ζ n λ N = ζ nλ N ζ n λ N 2, so that for the claim to hold it is sufficient to check that forn 2N We have already seen in (42) that Now ζ n λ N Re( ζ n λ N ) ζ n λ N N, n 2N Re( ζ n λ N )= r N Re(e i(/n /N) )= ( N )(cos( n N )) N, n 2N, which proves the claim We thus can pass in (43) to real parts so that forn 2N Re( I n(λ n )I n (λ N ) λ n λ N ) Re( 2 n ε ζ n ζ n λ N ) 2 n ε Re( ζ nλ N ζ n λ N 2) 2 n ε /N /n 2 +(/n /N) 2 2 n ε /N (/N) 2 N nε, when n 2N Hence Re( ζ n λ N ) β n Ref β (λ N ) β n N n n ( ε)/2 ζ n λ N 2 n 2N n (+ε)/2

12 BAD BOUNDARY BEHAVIOR IN STAR INVARIANT SUBSPACES II Pick for instance β n = n (+η)/2, where η> 0 is arbitrary, so that obvioulsy β n 0 and β l 2 Then ε/2 η/2 Ref β (λ N ) N n 2N n +(ε+η)/2 N ( N (ε+η)/2=n ε/2 η/2 λ N ) where η > 0 is arbitrarily small Compare this with the estimate of the reproducing kernel (32) With better choices of β it is of course clear that we can come closer to the maximal growth given by the reproducing kernel Finally, we point out that wheni(z) whenz in a fixed Stolz domain, it is, in general, particularly difficult to decide whether or not a sequence of reproducing kernels for(ih 2 ), with the parameter in a Stolz domain with vertex at, is an unconditional basis or not Even when sup n I(λ n ) <, there is a characterization known for unconditional basis which is, in general, difficult to check REFERENCES [AC70] P R Ahern and D N Clark, Radial limits and invariant subspaces, Amer J Math 92 (970), [AC7], Radial nth derivatives of Blaschke products, Math Scand 28 (97), [CL66] E F Collingwood and A J Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No 56, Cambridge University Press, Cambridge, 966 [Coh86] W S Cohn, Radial limits and star invariant subspaces of bounded mean oscillation, Amer J Math 08 (986), no 3, [Fro42] O Frostman, Sur les produits de Blaschke, Kungl Fysiografiska Sällskapets i Lund Förhandlingar [Proc Roy Physiog Soc Lund] 2 (942), no 5, [HR] A Hartmann and WT Ross, Bad boundary behavior in backward shift invariant subspaces I, preprint (20) [Nik86] N K Nikol skiĭ, Treatise on the shift operator, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 273, Springer-Verlag, Berlin, 986, Spectral function theory, With an appendix by S V Hruščev [S V Khrushchëv] and V V Peller, Translated from the Russian by Jaak Peetre [Nik02] N K Nikolski, Operators, functions, and systems: an easy reading Vol 2, Mathematical Surveys and Monographs, vol 93, American Mathematical Society, Providence, RI, 2002, Model operators and systems, Translated from the French by Andreas Hartmann and revised by the author INSTITUT DE MATHÉMATIQUES DE BORDEAUX, UNIVERSITÉ BORDEAUX I, 35 COURS DE LA LIBÉRATION, TALENCE, FRANCE DEPARTMENT OF MATHEMATICS, UNIVERSITY OF RICHMOND, VA 2373, USA address: hartmann@mathu-bordeauxfr, wross@richmondedu