BOUNDARY MULTIPLIERS OF A FAMILY OF MÖBIUS INVARIANT FUNCTION SPACES

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 4, 6, 99 BOUNARY MULPLERS OF A FAMLY OF MÖBUS NVARAN FUNCON SPACES Guanlong Bao and ordi Pau Shantou University, eartment of Mathematics Shantou, Guangdong 5563, P. R. China; glbaoah@63.com Universitat de Barcelona, eartament de Matemática Alicada i Analisi 87 Barcelona, Sain; jordi.au@ub.edu Abstract. For < < and < s <, we consider the function saces Q s() that aear naturally as the sace of boundary values of a certain family of analytic Möbius invariant function saces on the the unit disk. n this aer, we give a comlete descrition of the ointwise multiliers going from Q s () to Q r () for all ranges of <, < and < s,r <. he sectra of such multilication oerators is also obtained.. ntroduction An imortant roblem of studying function saces is to characterize the ointwise multiliers of such saces. For Banach function saces X and Y, denote by M(X,Y) the class of all ointwise multiliers from X to Y. Namely, M(X,Y) = {f: fg Y for all g X}. f X = Y, we just write M(X,Y) as M(X) for the collection of multiliers of X. For any g M(X,Y), denote by M g the multilication oerator induced by g, that is, M g (f) = gf. By the closed grah theorem, M g is a bounded oerator. n this aer we characterize the ointwise multiliers between a certain family of function saces on the unit circle. hese saces aear in a natural way as the boundary values of a certain family of analytic Möbius invariant saces on the disk [3] that has been attracted much attention recently. enote by the boundary of the unit disk in the comlex lane C. Let H() be the sace of all analytic functions on and let H be the class of bounded analytic functions on. For < < and s, consider the analytic Besov tye sace B (s) consisting of those functions f H() with ( / f B(s) = f (z) ( z ) da(z)) +s <, where da denotes the Lebesgue measure on. A norm in B (s) is given by f() + f B(s). For a, let σ a (z) = a z az, z, doi:.586/aasfm.6.43 Mathematics Subject Classification: Primary 3H5, 3, 46E5. Key words: Pointwise multiliers, Carleson measures, Blaschke roducts, Q s () saces. G. Bao is suorted in art by NSF of China (No. 3734).. Pau was suorted by SGR grant 4SGR89 (Generalitat de Catalunya) and GCY grants MM-793-C- and MM P (MCy/MEC).

2 Guanlong Bao and ordi Pau be a Möbius transformation of. he sace B = B () is Möbius invariant in the sense that f σ a B = f B, and B is the classical irichlet sace. For s > and < <, we denote by Q s () the Möbius invariant sace generated by B (s), that is, f Q s () if f H() and f = su f σ Q s() a B <. (s) a he saces Q s () are secial cases of a class of Möbius invariant function saces studied in [38] and coincide withf(,,s) wheref(,q,s) is the family of function saces studied in [3] and [3]. n articular, for s > the saces Q s() are the same and equal to the Bloch sace B (the maximal Möbius invariant sace) which consists of all functions f H() with f B = su( z ) f (z) <. z When =, one has Q s () = Q s() the holomorhic Q saces introduced in [4] and widely studied in the monograhs [3, 3]. n articular, Q () = BMOA, the sace of analytic functions with bounded mean oscillation []. Essén and Xiao [9] gave that if < s < and f is in the Hardy sace H, then f Q s () if and only if f Q s (), the sace of functions f L () with f Q = su s() s f(ζ) f(η) ζ η s dζ dη <, where is the length of an arc of the unit circle (a version of these saces for several real variables was studied in [8]). f s >, Xiao [9] ointed out that Q s () are equal to BMO(), the sace of bounded mean oscillation on. For >, via the ohn Nirenberg inequality (see [, 4]), one gets f BMO() su where f is the average of f over, that is f = f(ζ) dζ. f(ζ) f dζ, n view of that it is natural to consider, for < < and s >, the saces Q s () consisting of functions f L () such that (.) f = su f(ζ) f(η) dζ dη <. Q s() s ζ η s A true norm in Q s() is given by f,q s () = f L () + f Q s (). We are going to study these saces, and we will see that if f is in the Hardy sace H, then f Q s () if and only if f Q s (). Also, we give a comlete descrition of the ointwise multiliers M(Q s (),Q r ()) for <, < and < s,r <. t is worth mentioning that Stegenga [4] characterized the multiliers of bounded mean oscillation saces on the unit circle (see also [7]), and Brown and Shields [5] described the ointwise multiliers of the Bloch sace. A characterization of the ointwise multiliers M(Q s ()) was obtained in [8] roving a conjecture stated in [8]. See [, 7,, 5, 7, 35] for more results on ointwise multiliers of function saces. he following is the main result of the aer.

3 Boundary multiliers of a family of Möbius invariant function saces heorem.. Let <, < and < s,r <. hen the following are true. () f and s r, then f M(Q s (),Q r ()) if and only if f L () and ( (.) su log ) f(ζ) f(η) dζ dη <. r ζ η r () Let > and s r. f s > r, then f M(Q s (),Q r ()) if and only if f L () and f satisfies (.). f s r, then M(Q s (),Q r ()) = {}. (3) f s > r, then M(Q s (),Q r ()) = {}. Note that, when = = and s = r, art () of heorem. roves Conjecture.5 stated in [8]. However, as seen in the roof, this conjecture is an immediate consequence of the results and methods in [8]. Next we give an alication. Let be a bounded linear oerator on a Banach sace X. he sectrum of is defined as σ() = {λ C: λe is not invertible}, where E is the identity oerator on X. Allen and Colonna [] gave sectra of multilication oerators on the Bloch sace. n this aer, we also consider the sectra of multilication oerators on Q s() saces. For f Q s(), let R(f) be the essential range of f. Namely, R(f) is the set of all λ in C for which {ζ : f(ζ) λ < ε} has ositive measure for every ε >. By [6,. 57], if f L (), then R(f) is a comact subset of C. heorem.. Suose < < and < s <. Let f be the symbol of a bounded multilication oerator M f on Q s () sace. hen σ(m f) = R(f). he aer is organized as follows. n Section, we give some reliminaries as well as basic roerties of Q s () saces, such as inclusion relations or a characterization in terms of Carleson tye measures. he roof of heorem. is given in Section 3. Particularly interesting is the roof of art (3), where we are in need to use the results of Nagel, Rudin and Shairo [5], on tangential boundary behavior of functions in weighted analytic Besov saces. n Section 4, we rove heorem., and in the last section, we give the analytic versions of heorems. and.. hroughout this aer, for a ositive number λ and an arc, denote by λ the arc with the same center as and with the length λ. he symbol A B means that A B A. We say that A B if there exists a constant C such that A CB.. Preliminaries and basic roerties An imortant tool to study function saces is Carleson tye measures. Given an arc on, the Carleson sector S() is given by S() = {rζ : < r <, ζ }. π For s >, a ositive Borel measure µ on is said to be an s-carleson measure if µ(s()) su <. s

4 Guanlong Bao and ordi Pau When s =, we get the classical Carleson measures, characterizing when H L (,µ), where for < <, H denotes the classical Hardy sace [7] of functions f H() for which su M (r,f) <. <r< Here ( π / M (r,f) = f(re iθ ) dθ). π By [4], µ is an s-carleson measure if and only if ( a Because su a f σ a B (s) = āz ) s dµ(z) <. f (z) ( z ) ( σ a (z) ) s da(z), we can immediately see that f Q s() if and only if f (z) ( z ) +s da(z) is an s-carleson measure. A functionf H() is called an inner function if it is anh -function with radial limits of modulus one almost everywhere on the unit circle. A sequence {a k } k= is said to be a Blaschke sequence if ( a k ) <. k= he above condition imlies the convergence of the corresonding Blaschke roduct B defined as a k a k z B(z) = a k a k z. k= We also need the characterizations of inner functions in Q s() saces. Essén and Xiao [9] characterized inner functions in Q s () saces. Later, Pérez-González and Rättyä [] described inner functions in Q s () saces as follows. heorem A. Let < s < and > max{s, s}. hen an inner function belongs to Q s () if and only if it is a Blaschke roduct associated with a sequence {z k } k= which satisfies that k ( z k ) s δ zk is an s-carleson measure, that is ( σa (z k ) )s <. su a k= he roof of heorem. will also use the Rademacher functions {r j (t)} j= defined by, < t <, r (t) =, < t <,, t =,,. r n (t) = r ( n t), n =,,. See [39, Chater V, Vol. ] or [7, Aendix A] for roerties of these functions. n articular, we will use Khinchine s inequality.

5 Boundary multiliers of a family of Möbius invariant function saces 3 heorem B. (Khinchine s inequality) f{c k } k= l, then the series k= c kr k (t) converges almost everywhere. Furthermore, for < < there exist ositive constants A, B such that for every sequence {c k } k= l we have Q s A ( k= c k ) c k r k (t) k= dt B ( k= c k ). For < < and < s <, the following result shows that we can regard () as a Banach sace of functions modulo constants. Lemma.. Let < < and < s <. hen Q s() BMO(). Furthermore, Q s () is comlete with resect to (.). Proof. Let f Q s(). For any arc, it follows from Hölder s inequality that f(e it ) f dt ( s f(e it ) f(e iθ ) dθdt ) f(e it ) f(e iθ ) / dθdt, e it e iθ s because e it e iθ when e it and e iθ are in. hus Q s () BMO() with f BMO() f Q s (). Now let{f m } be a Cauchy sequence inq s (). hen it is also a Cauchy sequence in BMO(). Hence f m f in BMO() for some f in BMO() and there exists a subsequence {f mk } {f m } such that lim k f mk (e it ) = f(e it ), for a.e. e it. By Fatou s lemma, it follows easily that f mk f Q s () liminf l f m l f mk Q s (), which imlies that f mk f in Q s (). Since this finishes the roof. f k f Q s () f mk f Q s () + f mk f k Q s (),.. Characterizations of Q s () saces. o rove our main results in this aer, some characterizations of Q s () saces are necessary. Given f L (), let f be the Poisson extension of f, that is, f(z) = π π f(e iθ ) z dθ, z. e iθ z We will characterize Q s() saces in terms of Carleson tye measures. Before doing that, we state and rove some auxiliary results. Lemma.. Suose > and < s <. Let f L () and let F C () with lim r F(re it ) = f(e it ) for a.e. e it. For any arc, we have f(e is ) f(e iθ ) dθds F(z) ( z ) +s da(z). e is e iθ s S(3) n articular, if F(z) ( z ) +s da(z) is an s-carleson measure, then f Q s (). Proof. We use an argument used in [6] and [3,. 94]. After a change of variables, it is easy to see (it is done in the same way as in [6] or [3, Chater 7]

6 4 Guanlong Bao and ordi Pau where the case = was obtained) that f(e is ) f(e iθ ) dθds e is e iθ s For any arc with < π 3 f(e i(θ+t) ) f(e iθ ) ( ) f(e i(θ+t) ) f(e iθ ) dθ dt. t s and t (, ), set r = t. hen f(e i(θ+t) ) F(re i(θ+t) ) + F(re i(θ+t) ) F(re iθ ) + F(re iθ ) f(e iθ ) r F(xe i(θ+t) ) dx+ t F(re i(θ+u) ) du+ Since >, we can use Minkowski s inequality to obtain ( ( / f(e i(θ+t) ) f(e iθ ) dθ F(xe i(θ+t) ) dθ) dx ( t ( + ( ( r 3 = ( )+( ). r / ( ( F(re i(θ+u) ) dθ) du) + / ( t ( F(xe iθ ) dθ) dx) + r r F(xe iθ ) dx. ) F(xe iθ ) dθ) / dx F(re i(θ+u) ) dθ) / du For > and < s <, alying the Hardy inequality (see [6,. 7]) gives ( t ( t ( / )dt = F(( y)e iθ ) dθ) dy) dt s t s 3 ( ) ( ) F(( y)e iθ ) dθ y +s dy s 3 ( ) ( ) = F(xe iθ ) dθ ( x) +s dx s 3 F(z) ( z ) +s da(z). We also have S(3) t ( )dt s = S(3) t s ( t ( t +s ( 3 3 F(re iθ ) dθ) / du) dt ) F(( t)e iθ ) dθ dt F(z) ( z ) +s da(z). Combining the above estimates, we obtain the desired result. We also need the following result which is a generalization of Stegenga s estimate in [5]. Lemma.3. For > and < s <, let f L () and let, be two arcs on centered at e iθ with 3. hen there exists a constant C deending only ) )

7 Boundary multiliers of a family of Möbius invariant function saces 5 on and s such that f(z) ( z ) +s da(z) S() C [ ( ) f(e iθ ) f(e it ) dθdt+ +s f(e i(t+θ) ) f e iθ e it s dt ]. t 3 t Proof. Without loss of generality, one may assume that θ =. Following Stegenga, we let φ be a function with φ such that φ = on 3, su φ 3, and (.) φ(e iθ ) φ(e it ) eiθ e it for all θ, t [, π). Now we write f = (f f )φ+(f f )( φ)+f = f +f +f 3. Since f 3 is constant, we have f 3 =. For z = re iθ in the Carleson box S(), π f (e it ) f (z) dt f(e it ) f ( r) +(θ t) dt t 3 t and hence S() ( ) f (z) ( z ) +s da(z) +s f(e it ) f dt. t 3 t For the integral over S() of f, relacing S() with the unit disc and using Proosition 4. in [3], we obtain f (z) ( z ) +s f (e iθ ) f (e it ) da(z) dθdt S() e iθ e it s f (e iθ ) f (e it ) eit 3 f (e iθ ) f (e it ) dθdt+ dθdt e iθ e it e iθ e it s e iθ e iθ e it s eiθ 3 f (e iθ ) f (e it ) + dθdt e iθ e it s e it For estimating, we see first that, due to the condition (.), for e iθ, e it, f (e iθ ) f (e it ) f(e iθ ) f(e it ) + e iθ e it f(e it ) f. By Hölder s inequality, we deduce that f(e it ) f ( ) dθdt = f(e it ) f e iθ e it s e iθ e it +s dθ dt f(e it ) f s dt f(e iθ ) f(e it ) dθdt s f(e iθ ) f(e it ) e iθ e it s dθdt.

8 6 Guanlong Bao and ordi Pau hus, f(e iθ ) f(e it ) dθdt. e iθ e it s For, using that f (e iθ ) = for e iθ, one gets eit 3 f (e iθ ) f (e it ) eit 3 f(e it ) f = dθdt dθdt e iθ e iθ e it s e iθ e iθ e it s f(e it ) f s f(e iθ ) f(e it ) dt dθdt. e iθ e it s he estimate of 3 is similar to. he above inequalities imlies the lemma. he following theorem characterizes Q s() saces in terms of Carleson tye measures. t generalizes the corresonding result of Q s () saces in [6]. heorem.4. Let > and < s <. Suose f L (). he following conditions are equivalent. (a) f Q s(). (b) f(z) ( z ) +s da(z) is an s-carleson measure. (c) su a f(ζ) f(η) ζ η s ( a ζ a η a ) s dζ dη <. Proof. We first show that (b) is equivalent to (c). By Proosition 4. in [3], one gets π π (.) f(z) ( z ) +s f(e iθ ) f(e it ) da(z) dθdt e iθ e it s for all f L (). Note that f σ a = f σ a for any a. Relacing f by f σ a in the above formula and making a change of variables, we get ( ) a f(z) ( z ) +s s da(z) āz π π ( ) f(e iθ ) f(e it ) a s dθdt. e iθ e it s ae iθ ae it his gives that (b) is equivalent to (c). By Lemma., we see that (b) imlies (a). Next we verify that (a) imlies (b). Let f Q s (). hen f is also in BMO() by Lemma.. For an arc centered at e iθ, let = 3. hen Lemma.3 gives f(z) ( z ) +s da(z) S() ( ) f(e iθ ) f(e it ) dθdt+ +s f(e i(t+θ) ) f e iθ e it s dt. t 3 t Since, by [3,. 7], f(e i(t+θ) ) f dt t 3 t f BMO(), we have su f(z) ( z ) +s da(z) f s Q s() + f BMO(). S()

9 he roof is comlete. Boundary multiliers of a family of Möbius invariant function saces 7 f > and < s <, then Q s () BMOA H. hus functions in Q s () have boundary values. As noticed before, an analytic function f belongs to Q s () if and only if f (z) ( z ) +s da(z) is an s-carleson measure. Combining this with heorem.4, one gets the following result immediately. Corollary.5. Let > and < s <. Suose f H. hen f Q s() if and only if f Q s (). Remark. Let > and < s <. We say that f L s if f L () and f L s = π π f(e it ) f(e iθ ) e it e iθ s dθdt <. he condition (c) of heorem.4 gives that f Q s () if and only if su f σ a L s <. a hus if we set f Q s () = su f σ a L s, a then Q s() is a Möbius invariant sace in the sense of that f Q s () = f σ a Q s () for any f Q s () and a... nclusion relations. Alying heorem.4 and Corollary.5, we can obtain a comlete icture on the inclusion relations between different Q s () saces. heorem.6. Let <, < and < s,r <. () f, then Q s () Q r () if and only if s r. () f >, then Q s () Q s r () if and only if > r. Proof. We first consider the inclusion relation between the analytic sacesq s () and Q r (). Note that Q s () is a subset of the Bloch sace B. Let f Q s (). f and s r, then su f (z) ( z ) ( σ a (z) ) r da(z) a f B f (z) ( z ) ( σ a (z) )s da(z), su a which gives Q s () Q r (). By [34, heorem 7], if >, B (s) B (r) if and only if s > r that f Q s () if and only if su f σ a f(a) B (s) <. a. Note hus Q s () Q r () for > and s > r. For s > r, it is easy to construct a Blaschke sequence {z k } satisfying that k ( z k ) s δ zk is an s-carleson measure and k ( z k ) r δ zk is not an r-carleson measure (see Lemma 3.3 where such a sequence is constructed with even more roerties). Alying heorem A, we get that the corresonding Blaschke roduct B satisfies B Q s ()\Q r ().

10 8 Guanlong Bao and ordi Pau Let s < r. By [3, heorem 5.5], the lacunary series g(z) = k= k ( s + r ) z k Q s ()\Q r (). f >, s < r and s = r, the lacunary series h(z) = k= k s k z k Q s ()\Q r (). herefore, if, then Q s () Q r () if and only if s r. f >, then Q s () Q r () if and only if s > r. Hence it is enough to rove that the inclusion relations between Q s () and Q r () are the same as the inclusion relations between Q s () and Q r (). Suose Q s () Q r (). Let g Q s (). Without loss of generality we may assume that g is real valued. enote by g the harmonic conjugate function of ĝ. Set h = ĝ + i g. he Cauchy Riemann equations give ĝ(z) h (z). hen heorem.4 shows that h (z) ( z ) +s da(z) is ans-carleson measure. hus h Q s (). So h is also in Q r (). hen ĝ(z) ( z ) +r da(z) is an r- r (). On the other r (). he roof Carleson measure, that is g Q r (). Hence Q s () Q hand, if Q s () Q r (), then Corollary.5 shows Q s () Q is comlete. Remark. f < < <, < r < s < and s r, it is easy to check that any lacunary series in Q s () must be in Q r (), but Q s () Q r (). hus we are in need to use inner functions to determine the inclusion relation in the roof of heorem Logarithmic Carleson tye measures. Let α and s >. A ositive Borel measure µ on is called an α-logarithmic s-carleson measure if ( log ) α µ(s()) <. s su By [33], µ is an α-logarithmic s-carleson measure if and only if ( ) α ( ) a s log dµ(z) <. a āz su a Condition (.) in heorem. can be described in terms of α-logarithmic s-carleson measure as follows. Lemma.7. Let < < and < r <. hen the following conditions are equivalent. ( () su log ) f(ζ) f(η) dζ dη <. r ζ η r () f(z) ( z ) +r da(z) is a -logarithmic r-carleson measure. Proof. For an arc, by Lemma., we have f(ζ) f(η) dζ dη f(z) ( z ) +r da(z). ζ η r hus () imlies (). S(3)

11 Boundary multiliers of a family of Möbius invariant function saces 9 Let () hold. Without loss of generality, let be any arc centered at. hen f(e iθ ) f(e it ) dθdt ( ). log Combining this with Hölder s inequality, we deduce that ( ) / f(e iθ ) f dθ f(e iθ ) f dθ ( ) / f(e iθ ) f(e it ) dθdt log. Let = 3. Using the above estimate and a same argument in [8,. 499], we get f(e it ) f dt t 3 t log 3. Combining this with Lemma.3, we get that () is true. We also need the following result from []. Lemma C. Let > and s >. Let µ be a nonnegative Borel measure on. f µ is a -logarithmic s-carleson measure, then f(z) dµ(z) f B (s) for all f B (s). Now we are ready for the roofs of the main results of the aer. 3. Proof of heorem. 3.. Proof of art (). Assume first that f M(Q s (),Q r ()). Set h w (z) = log, w. From [, Lemma.6], wz su h w Q s () <, w for all < < and < s <. his together with Corollary.5 shows that h w (e iθ ) = log belongs to Q we iθ s () uniformly for w, and hence the same is true for g w = Reh w. For any arc centered at e it with < /3, take a = ( )e it. hen g a (e iθ ) log for all e iθ. Since f is a ointwise multilier, then fg a Q r () and it follows by Lemma. that fg a BMO(). By [4, Lemma.6], f(ζ)g a (ζ) dζ log fg a BMO() log fg a Q r (). hus which shows f L (). Since f(ζ) dζ su g a Q s () <, a g a (e iθ )(f(e it ) f(e iθ )) = g a (e it )f(e it ) g a (e iθ )f(e iθ )+f(e it ) ( g a (e iθ ) g a (e it ) ),

12 Guanlong Bao and ordi Pau we get r g a (e iθ )(f(e it ) f(e iθ )) dθdt g e iθ e it r a f Q r () + f L () g a Note that g a (e iθ ) log for all e iθ. hus f(e it ) f(e iθ ) dθdt e iθ e it r su a r ), ( log. Q s () which gives (.). Next, suose that f L () and (.) holds. We need to show f M(Q s (), Q r ()). he roof of this imlication is based on a technique develoed in [8] (see also []). By Lemma.7, f(z) ( z ) +r da(z) is a -logarithmic r-carleson measure. hus ( ) log a f(z) ( z ) ( σ a (z) ) r da(z) <. For all g Q s (), we need to rove gf Q r (). Since f ĝ is an extension of gf, by Lemma., it is enough to rove that (a) := ( f ĝ)(z) ( z ) ( σ a (z) ) r da(z) C for some ositive constant C not deending on the oint a. Using heorem.4, we have (a) f(z) ĝ(z) ( z ) ( σ a (z) ) r da(z) + f(z) ĝ(z) ( z ) ( σ a (z) ) r da(z) f L () g Q r () + f(z) ĝ(z) ( z ) ( σ a (z) ) r da(z). f and s r, heorem.6 gives Q s () Q r (), and by the closed-grah theorem, g Q r () g Q s () for all g Q s (). Hence, we have (a) f L () g Q s () + ĝ(z) f(z) ( z ) ( σ a (z) ) r da(z). Without loss of generality, we may assume that g is real valued. Let g be the harmonic conjugate function of ĝ. Set h = ĝ + i g. he Cauchy Riemann equations give ĝ(z) h (z). hen h Q s () Q r () and ĝ(z) h(z). Hence ĝ(z) f(z) ( z ) ( σ a (z) ) r da(z) h(z) f(z) ( z ) ( σ a (z) ) r da(z) h(a) f(z) ( z ) ( σ a (z) ) r da(z) + h(z) h(a) f(z) ( z ) ( σ a (z) ) r da(z) +.

13 Boundary multiliers of a family of Möbius invariant function saces Since Q s () is a subsace of the Bloch sace B, then any function h Q s () has the following growth: h(z) h B log z h Q s () log z for all z. hus ( log su a a ) f(z) ( z ) ( σ a (z) ) r da(z) <. Alying Lemma C, we see that r = ( a h(z) h(a) ) f(z) ( z ) +r da(z) ( āz) r ( ) ( a ) ( h() h(a) r h(z) h(a) + ( z ) da(z)) +r ( āz) r ( ) ( a ) r h B log + h (z) ( z ) ( σ a (z) ) r da(z) a h(z) h(a) + ( z ) ( σ āz a (z) ) r da(z) h h(z) h(a) Q s () + ( z ) ( σ āz a (z) ) r da(z). By [, Proosition.8], one gets h(z) h(a) āz Combining the above estimates, we obtain ( z ) ( σ a (z) ) r da(z) h Q r (). su(a) <. a hus f M(Q s (),Q r ()). 3.. Proof of art (). Let > and s r. f s gives the inclusion Q s () Q r (). Hence M(Q r (),Q r ()) M(Q s (),Q r ()). > r, heorem.6 Checking the roof in (), one gets that f M(Q s (),Q r ()) if and only if f L () and f satisfies (.). n case that s r, as has been observed in the roof of heorem.6, there is a lacunary Fourier series in Q s ()\Q r (). hen we get M(Q s (),Q r ()) = {} as a consequence of the following result. Lemma 3.. Let <, < and < s, r <. f there exists a lacunary Fourier series g(e iθ ) = k= a ke ikθ Q s ()\Q r (), then M(Q s (), Q r ()) = {}. Proof. We adat an argument from []. By Corollary.5 and the lacunary series characterization of Q s () saces in [3, heorem 5.5], we see that g Q a s () k k( s) < k=

14 Guanlong Bao and ordi Pau and a k k( r) =. k= Let {r k (t)} k= be the sequence of Rademacher functions and consider the function g t (z) = r k (t)a k z k, t. k= hen g t Q s () with g t Q s () g Q s (). Suose f M(Q s (),Q r ()). For any a, we have ( ) f(ζ)(g t (ζ) g t (η)) a r gt (f,a)dt := dζ dη dt ζ η r ζ a η a ( ) f(ζ)g t (ζ) f(η)g t (η) a r dζ dη dt ζ η r ζ a η a ( ) (f(ζ) f(η))g t (η) a r + dζ dη dt. ζ η r ζ a η a Hence, using Fubini s heorem, we obtain gt (f,a)dt fg t dt Q r () ( f(ζ) f(η) + ζ η r Since f M(Q s (),Q r ()), then fg t Q r () g Q inequality g t (η) dt ( k= a k ) )( ) g t (η) a r dt dζ dη. ζ a η a s () g H g Q s (),. Also, by Khinchine s because Q s () H. Combining these estimates with heorem.4 we have (3.) gt (f,a)dt g Q s () + g Q s () f <, Q r () because, as Q s (), then f = f Q r (). Now, if f, then there exists a ositive constant C such that (3.) π f(e iθ ) dθ C. Fubini s theorem and a change of variables give gt (f,)dt π dh h r π f(e iθ ) dθ g t (e iθ ) g t (e i(θ+h) ) dt. Alying Khinchine s inequality again, we see that ( ) g t (e iθ ) g t (e i(θ+h) ) dt a k e ikh. k=

15 Hence, by (3.), Boundary multiliers of a family of Möbius invariant function saces 3 gt (f,)dt π π ( dh h r k= a k e ikh ) h r g t (e iθ ) g t (e i(θ+h) ) dt, π dh f(e iθ ) dθ for any θ [, π). hus, using (.) and the Khinchine inequality, we have gt (f,)dt π dh h r π g t (e iθ ) g t (e i(θ+h) ) dtdθ g t (ζ) g t (η) ζ η r dζ dη dt g t (z) ( z ) +r da(z)dt ( ) g t (z) dt ( z ) +r da(z) (M ( z,g )) ( z ) +r da(z). Since g (z) is also a lacunary series, it is well known that M ( z,g ) M ( z,g ) (see [39] for examle). Hence gt (f,)dt (M ( z,g )) ( z ) +r da(z) g (z) ( z ) +r da(z) a k k( r) =. k= his contradicts (3.). hus M(Q s (),Q r ()) = {} Preliminaries for the roof of art (3). Lemma 3. shows that M(Q s (),Q r ()) is trivial for a wide range of values of the arameters, s,, r. However Lemma 3. miss the case that <, s > r and s r. n this case, Q s () Q r (), but any lacunary Fourier series in Q s () must be in Q r (). hus, we are in need to look for another method to determine that M(Q s (),Q r ()) is trivial in this case. We are going to use the tangential boundary aroximation results of Nagel, Rudin and Shairo [5] in order to handle this case. Given ζ and α >, let Γ α (ζ) = {z : ζz < α( z )} be a Stolz angle with vertex at ζ. f f H, then its non-tangential limit exists almost everywhere. Namely, for almost every ζ, the limit f(ζ) := lim z Γα(ζ) z ζ f(z) exists. n order to handle art (3), for every ζ, we need to construct a Blaschke sequence {a k } converging to ζ in a way that the associated Blaschke roduct is in

16 4 Guanlong Bao and ordi Pau Q s () \ Q r (). n view of heorem A, k ( a k ) s δ ak must be an s-carleson measure, but k ( a k ) r δ ak can not be an r-carleson measure. According to [3] this is not ossible if the sequence {a k } converges to ζ non-tangentially. For c > and δ >, consider the region { Ω δ,c (θ) = re iϕ : r > c ϕ θ δ} sin. hen Ω δ,c (θ) touches at e iθ tangentially. We say that a function h, defined in, has Ω δ -limit L at e iθ if h(z) L as z e iθ within Ω δ,c (θ) for every c. Nagel, Rudin and Shairo [5] obtained the following result. heorem. Suose <, f L (), < α <, and h(z) = π π π f(e iθ )dθ ( e iθ z) α, z. f α < and δ = /( α), then the Ω δ -limit of h exists almost everywhere on. Forβ R and < <, the Hardy Sobolev saceh β consists of analytic functions f in such that β f H, where f(z) = k= a kz k is the aylor exansion of f and β f(z) = (+k) β a k z k. k= Proosition 3.. Let < < and < s < t <. Suose h B (s). hen the Ω /t -limit of h exists almost everywhere on. Proof. By heorem, it is enough to show that there exists f L () with h(z) = π π π f(e iθ )dθ ( e iθ z) t, z. hus we only need to rove that h H t. Note that [36, heorem.9] h(z) ( z ) +s t da(z) h (z) ( z ) +s da(z) <. + t hus t h B (+s t). Since s < t, then +s t < and therefore B (+ s t) H (see [3, Lemma.4] for examle). Hence we get h H t. he roof is comlete. n case that, it is known [] that one can take t = s in Proosition 3.. he following construction will be a key for the roof of art (3). Lemma 3.3. Let < r < s < and < t <. For every e iθ, there exists a Blaschke sequence {a k } k= satisfying the following conditions. (a) e iθ is the unique accumulation oint of {a k }. (b) {a k } Ω /t,c (θ) for some c >. (c) k ( a k ) s δ ak is an s-carleson measure. (d) k ( a k ) r δ ak is not an r-carleson measure. Proof. Set a k = ( k ε ) e iθ k, k =,,,

17 where ε > is taken so that Boundary multiliers of a family of Möbius invariant function saces 5 r( t) < ε < min{r,s( t)} and θ k = k t ε +θ. Clearly, {a k } is a Blaschke sequence and e iθ is its unique accumulation oint with {a k } Ω /t,c (θ) for some c >. o rove that k ( a k ) s δ ak is an s-carleson measure, it is enough to consider sufficiently small arcs centered at e iθ. Since ε < s( t) < s, we deduce that ( a k ) s k s ε k s ε x s s ε ε dx t s, a k S() a k S() θ k θ k ( ) ε t ( ) ε t which gives that k ( a k ) s δ ak is an s-carleson measure. On the other hand, for r( t) < ε < r, one gets ( a k ) r x r r ε r r ε dx t r, as. ( ) ε t hus k ( a k ) r δ ak is not an r-carleson measure. he roof is comlete. We also need the following estimate. Lemma 3.4. Let < q < and < r <. Let f L q () and S an inner function. For a, (a) := f q (z)( S(z) ) q ( z ) ( σ a (z) ) r da(z) ( ) f(ζ) q S(ζ) S(η)) q a r dζ dη. ζ η r ζ a η a Proof. We first consider the case a =. Using Fubini s heorem, we see that ( ) () = f(ζ) q ( S(z) ) q ( z ) r da(z) dζ ζ z ( ) f(ζ) q S(ζ) S(z) q ( z ) r da(z) dζ. ζ z Note that S(ζ) S(z) q S(ζ) S(η) q z η z dη. Consequently, by the estimate [7], ( z ) r ζ z η z da(z) ζ η r, we have ( ) ( z ) r () ζ z η z da(z) f(ζ) q S(ζ) S(η) q dη dζ f(ζ) q S(ζ) S(η) q dη dζ. ζ η r hat is, f q (z)( S(z) ) q ( z ) r f(ζ) q S(ζ) S(η) q da(z) dη dζ. ζ η r

18 6 Guanlong Bao and ordi Pau Relacing f and S by f σ a and S σ a in the above inequality resectively and changing the variables, the desired result follows Proof of art (3). f s > r, for any e iθ, we take the sequence {a k } k= constructed in Lemma 3.3, and let B be the corresonding Blaschke roduct. hen heorem A shows that B Q s (). f f M(Q s (),Q r ()), then fb Q r () and ( ) f(ζ)(b(ζ) B(η)) a r dζ dη ζ η r ζ a η a su a fb Q r () + f <. Q r () hen Lemma 3.4 gives su a f (z)( B(z) ) ( z ) ( σ a (z) ) r da(z) <. Since the sequence {a k } is Carleson Newman, that is, ( σa (a k ) ) <, su a k= we have (see [3,. 9], for examle) that ( ( B(z) ) ( σak (z) )) his gives (3.3) su a k= E(a k ) k= ( σak (z) ), z. k= f ( (z) σak (z) ) ( z ) ( σ a (z) ) r da(z) <, where E(a k ) = {w : σ ak (w) < /} is a seudo-hyerbolic disk centered at a k. t is well known that ( z ) a k z ( a k ) for all z E(a k ). Furthermore, by [37, Lemma 4.3], σ a (z) σ a (a k ) for all a and z E(a k ). his, (3.3) and subharmonicity yield ( f (a k )) ( σ a (a k ) ) r <. Since su a k= su a ( σ a (a k ) ) r =, k= this forces f (a k ). Note that f Q r () and lim k a k = e iθ. hen g = f + i f Q r () B (r) because of heorem.4, since the Cauchy Riemann equations give g (z) ( f )(z). ake t > r in Lemma 3.3, and aly Proosition 3., to get ( f +i f )(e iθ ) = lim k ( f +i f )(a k )

19 Boundary multiliers of a family of Möbius invariant function saces 7 for almost every e iθ. his imlies f (e iθ ) = lim k f (ak ) = a.e. e iθ. f we take the normalized f with f () =, then the analytic function f + i f vanishes on almost everywhere. Hence f +i f vanishes on the whole disk. hen f (z), z. his imlies f(ζ) = for almost every ζ. he roof is comlete. 4. Proof of heorem. Let f be the symbol of a bounded multilication oerator M f on Q s() sace. f λ σ(m f ), then M f λe is invertible. Clearly, the inverse oerator of M f λe is M. By the oen maing theorem, M is also bounded on Q s (), and f λ f λ heorem. gives f λ L (). hen f(ζ) λ < a.e. ζ. f λ L (), Namely, the set ( ζ : f(ζ) λ < f λ L () ) has measure zero. hus λ R(f). Conversely, let λ R(f). hen there exists some ositive constant δ such that the set {ζ : f(ζ) λ < δ} has measure zero. Hence f(ζ) λ, a.e. ζ. δ hus M f λe is injective. Using f M(Q s ()) and heorem., we obtain ( su log ) f(ζ) λ f(η) λ dζ dη <. s ζ η s Alying heorem. again, one gets f λ M(Q s ()). hen for any g Q s (), we obtain g f λ Q s() and g (M f λe) f λ = g. hen M f λe is surjective. hus M f λe is invertible and hence λ σ(m f ). 5. he analytic version of heorems. and. heorem 5.. Let <, < and < s,r <. hen the following are true. () f and s r, then f M(Q s (),Q r ()) if and only if f H and ( (5.) su log ) f(ζ) f(η) dζ dη <. r ζ η r () Let > ands r. f s > r, then f M(Q s (),Q r ()) if and only iff H andf satisfies (5.). f s r, thenm(q s (),Q r ()) = {}.

20 8 Guanlong Bao and ordi Pau (3) f s > r, then M(Q s (),Q r ()) = {}. Proof. ust follow the roof of heorem.. Now we give a different roof of (3) using zero sets. Set ( z n = ) e iθn, n =,3,, n /t where r < t < s and θ n = n k= k + n, n =,3,. By [9, heorem 8], n ( z n ) s δ zn is an s-carleson measure. Also, since n= ( z n ) t =, it follows from the roof of [5, heorem 5.] that {z n } is not a zero set of B (r) (look also at the roof of Proosition 3.). Let B be the Blaschke roduct with zero sequence {z n }. hen B Q s ()\Q r (). f f M(Q s (),Q r ()), then fb Q r () B (r). f f, then there exists an inner function S and an outer function O such that f = SO. By [, Proosition 4.], OB B (r). Hence {z n } is a zero set of B (r). his is a contradiction. hus f. Next we rove the analytic version of heorem., without using heorem 5.. heorem 5.. Suose < < and < s <. Let f be the symbol of a bounded multilication oerator M f on Q s () sace. hen σ(m f) = f(). Proof. Let λ f(). Note that M f λe = M f λ. Clearly, M f λ is not invertible. hus λ σ(m f ). Since σ(m f ) is comact, we get f() σ(m f ). Let λ f(). hen there exists a ositive constant C such that inf f(z) λ > C, z which shows g(z) = f(z) λ H. hus M f λe is injective. Clearly, f H. For any h Q s(), fh Q s(). Consequently, for any a, g (z)h(z) ( z ) ( σ a (z) ) s da(z) f (z) h(z) f(z) λ ( z ) ( σ a (z) ) s da(z) f (z)h(z) ( z ) ( σ a (z) ) s da(z) + fh Q s() fh Q s() + f H h Q s(). f(z)h (z) ( z ) ( σ a (z) ) s da(z) hus gh Q s(). hen we get g M(Q s()). t follows thatm f λe is surjective. Hence λ σ(m f ). he roof is comlete. Acknowledgements. he work was done while G. Bao visited the eartment of Alied Mathematics and Analysis, University of Barcelona in 5. He thanks the suort given by the MUB during his visit.

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