Positive Schatten-Herz class Toeplitz operators on the ball

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1 Positive Schatte-Herz class Toelitz oerators o the ball Boo Rim Choe, Hyugwoo Koo ad Youg Joo Lee Abstract. O the harmoic Bergma sace of the ball, we give characterizatios for a arbitrary ositive Toelitz oerator to be a Schatte- Herz class oerator i terms of averagig fuctios ad Berezi trasforms. Cotets 1. Itroductio 1 2. Basic lemmas 3 3. Schatte-Herz class Toelitz oerators 6 Refereces Itroductio For a fixed iteger 2, let B = B deote the oe uit ball i R. For 0 <, let L = L (V ) be the Lebesgue saces o B where V deotes the Lebesgue volume measure o B. The harmoic Bergma sace b 2 is a closed subsace of L 2 cosistig of all comlex-valued harmoic fuctios o B. By the mea value roerty of harmoic fuctios, it is easily see that oit evaluatios are cotiuous o b 2. Thus, to each x B, there corresods a uiue R(x, ) b 2 which has the followig reroducig roerty: (1.1) f(x) = f(y)r(x, y) dy, x B B 2000 Mathematics Subject Classificatio. Primary 47B35, Secodary 31B05. Key words ad hrases. Toelitz oerator, Harmoic Bergma sace, Schatte class, Schatte-Herz class. The first two authors were suorted by Mid-career Researcher Program through NRF grat fuded by the MEST (R ). 1

2 2 B. R. CHOE, H. KOO AND Y. J. LEE for all f b 2. The exlicit formula of the kerel fuctio R(x, y) is well kow: { (1 R(x, y) = 1 B 1 x 2 y 2 ) 2 } [x, y] 4 x 2 y 2 [x, y] for x, y B where [x, y] = 1 2x y + x 2 y 2. Here, as i elsewhere, we write x y for the dot roduct of x, y R ad E = V (E) for the volume of Borel sets E B. Hece the kerel fuctio R(x, y) is real ad hece the comlex cojugatio i the itegral of (1.1) ca be removed. See [2] for more iformatio ad related facts. Let R be the Hilbert sace orthogoal rojectio from L 2 oto b 2. The reroducig roerty (1.1) yields the followig itegral reresetatio of R: (1.2) Rψ(x) = ψ(y)r(x, y) dy, x B B for fuctios ψ L 2. It is easily see that the rojectio R ca be exteded to a itegral oerator via (1.2) from L 1 ito the sace of all harmoic fuctios o B. It eve exteds to M, the sace of all comlex Borel measures o B. Namely, for each µ M, the itegral Rµ(x) = R(x, y) dµ(y), x B B defies a fuctio harmoic o B. For µ M, the Toelitz oerator T µ with symbol µ is defied by T µ f = R(fdµ) for f b 2 L. Note that T µ is defied o a dese subset of b 2, because bouded harmoic fuctios form a dese subset of b 2. A Toelitz oerator T µ is called ositive if µ is a ositive fiite Borel measure (hereafter we simly write µ 0). For ositive Toelitz oerators o harmoic Bergma saces, basic oerator theoretic roerties such as boudedess, comactess ad the membershi i the Schatte classes have bee studied o various settigs; see [5], [7], [11], [12] ad refereces therei. Aother asect of ositive Toelitz oerators has bee recetly studied. Namely, otio of the so-called Schatte-Herz classes S, (see Sectio 3) was itroduced ad studied i [10] i the holomorhic case o the uit disk. Harmoic aalogues were subseuetly studied i [6]. However, these earlier works are restricted to the case of 1,. I this aer we exted the characterizatio i [6] for Schatte-Herz class ositive Toelitz oerators to the full rage of arameters ad. To state our results we briefly itroduce some otatio. Give µ 0, µ r deotes the averagig fuctio over seudohyerbolic balls with radius r ad µ deotes the Berezi trasform. See Sectio 2 for relevat defiitios. Also, we let λ deote the measure o B defied by dλ(x) = (1 x 2 ) dx

3 SCHATTEN-HERZ CLASS TOEPLITZ OPERATORS 3 ad K(λ) deote the so-called Herz saces (see Sectio 3). The ext theorem is the mai results of this aer. I case 1 <, Theorem 1.1 (with slightly differet averagig fuctios) below has bee roved i [6]. I case 0 < < 1, the authors [4] have recetly obtaied the corresodig results o the harmoic Bergma sace of the uer halfsace. The cut-off oit 1 is shar i the theorem below. Theorem 1.1. Let 0 <, 0, 0 < r < 1 ad µ 0. The the followig two statemets are euivalet: (a) T µ S, ; (b) µ r K(λ). Moreover, if 1 <, the the above statemets are also euivalet to (c) µ K(λ). I Sectio 2 we ivestigate kow results o weighted L -behavior of averagig fuctios ad Berezi trasforms. I Sectio 3, we first rove Theorem 1.1 ad the rovide examles idicatig that the arameter rage reuired i Theorem 1.1 is best ossible. 2. Basic lemmas I this sectio we collect several kow results which will be used i our characterizatio. We first recall Möbius trasformatios o B. All relevat details ca be foud i [1, ]. Let a B. The caoical Möbius trasformatio φ a that exchages a ad 0 is give by φ a (x) = a + (1 a 2 )(a x ) for x B (ote φ a = T a i the otatio of [1]). Here x = x/ x 2 deotes the iversio of x with resect to the shere B. Avoidig x otatio, we have φ a (x) = (1 a 2 )(a x) + a x 2 a [x, a] 2. The ma φ a is a ivolutio of B, i.e., φ 1 a = φ a. The hyerbolic distace β(x, y) betwee two oits x, y B is give by β(x, y) = 1 2 log 1 + φ y(x) 1 φ y (x). As is well-kow, β is Möbius ivariat. Let ρ(x, y) = φ y (x). This ρ is also a Möbius ivariat distace o B. We shall work with this seudohyerbolic distace ρ. For a B ad r (0, 1), let E r (a) deote the seudohyerbolic ball with radius r ad ceter a. A straightforward calculatio shows that E r (a) is a Euclidea ball with (2.1) (ceter) = (1 r2 ) 1 a 2 r 2 a ad (radius) = (1 a 2 )r 1 a 2 r 2.

4 4 B. R. CHOE, H. KOO AND Y. J. LEE Give µ 0 ad r (0, 1), the averagig fuctio µ r ad the Berezi trasform µ are defied by ad µ r (x) = µ[e r(x)] E r (x) µ(x) = (1 x 2 ) B R(x, y) 2 dµ(y) for x B. While it is customary to ut R(x, x) 1 i lace of (1 x 2 ) i the defiitio of the Berezi trasform, we adoted the above defiitio for simlicity. For measurable fuctios f, we defie f r ad f similarly, wheever they are well defied. Give α real, we let L α = L (V α ) where V α deotes the weighted measure defied by dv α (x) = (1 x 2 ) α dx. For α = 0, we have L 0 = L. Note λ = V. Also, give a seuece a = {a m } i B, we let l,α (a) deote the -summable seuece sace weighted by {(1 a m 2 ) α }. For α = 0, we let l = l,0 (a). Give a iteger k 0, we let R k (x, y) be the reroducig kerel for the weighted harmoic Bergma sace with resect to the weight (1 x ) k. So, R 0 = R is the harmoic Bergma kerel metioed before. Exlicit formulas for these kerels are give i [8, (3.1)]. The followig lemma take from [5, Lemma 3.1] shows that averagig fuctios, whe radii are small eough, are domiated by Berezi trasform. Lemma 2.1. For a iteger k 0 ad µ 0, there exists some r k (0, 1) with the followig roerty: If 0 < r r k, the there exists a costat C = C(, k, r) such that µ r (a) C(1 a 2 ) +2k R k (x, a) 2 dµ(x), a B. I articular, µ r C µ for 0 < r r 0. B We also eed the fact that the L α-behavior of averagig fuctios of ositive measures is ideedet of radii. I what follows, L 0 deotes the sace of all fuctios f bouded o B ad f(x) 0 as x 1. Lemma 2.2. Let 0 <, r, δ (0, 1) ad α be real. Assume µ 0. The the followig statemets hold : (a) µ r L α if ad oly if µ δ L α; (b) µ r L 0 if ad oly if µ δ L 0. Proof. See [5, Proositio 3.6]. Let {a m } be a seuece i B ad r (0, 1). We say that {a m } is r- searated if the balls E r (a m ) are airwise disjoit or simly say that {a m } is searated if it is r-searated for some r. Also, we say that {a m } is a r- lattice if it is r 2 -searated ad B = me r (a m ). Oe ca exlicitly costruct

5 SCHATTEN-HERZ CLASS TOEPLITZ OPERATORS 5 a r-lattice by usig the same argumet as i [8]. Note that ay maximal r 2-searated seuece is a r-lattice. The followig lemma take from [5, Theorem 3.9] gives a iformatio o weighted L -behavior of averagig fuctios, as well as its discrete versio, ad Berezi trasforms. Lemma 2.3. Let 0 <, r, δ (0, 1) ad α be real. Let µ 0 ad a = {a m } be a r-lattice. The the followig two statemets are euivalet: (a) µ δ L α; (b) { µ r (a m )} l,+α (a). Moreover, if { max 1 + α, 1 + α, α + 1 } <, the the above statemets are also euivalet to (c) µ L α. For a ositive comact oerator T o a searable Hilbert sace H, there exist a orthoormal set {e m } i H ad a seuece {λ m } that decreases to 0 such that T x = m λ m x, e m e m for all x H where, deotes the ier roduct o H. For 0 < <, we say that a ositive oerator T belogs to the Schatte -class S (H) if { 1/ T := λm} <. m More geerally, give a comact oerator T o H, we say that T S (H) if the ositive oerator T = (T T ) 1/2 belogs to S (H) ad we defie T = T. Of course, we will take H = b 2 i our alicatios below ad, i that case, we ut S = S (b 2 ). Also, for 0 <, we use the otatio l for the summable seuece sace. We eed the followig characterizatio of Schatte class ositive Toelitz oerators which is take from [5, Theorem 4.5]. Lemma 2.4. Let 0 < <, µ 0 ad assume that {a m } is a r-lattice. The the followig three statemets are euivalet: (a) T µ S ; (b) { µ r (a m )} l ; (c) µ r L (λ). Moreover, if 1 <, the the above statemets are also euivalet to (d) µ L (λ).

6 6 B. R. CHOE, H. KOO AND Y. J. LEE 3. Schatte-Herz class Toelitz oerators I this sectio we rove Theorem 1.1. To itroduce the Herz sace, we first decomose B ito the disjoit uio of auli A k give by A k = {x B : 2 (k+1) 1 x < 2 k } for itegers k 0. For each k, we let χ k = χ Ak. Recall that χ E deotes the characteristic fuctio of E B. Also, give µ M, we let µχ k stad for the restrictio of µ to A k. For 0 <, ad α real, the Herz sace K,α cosists of fuctios ϕ L loc (V ) for which ϕ K,α := { 2 kα ϕχ k L } l <. Also, we let K,α 0 be the sace of all fuctios ϕ K,α such that { } 2 kα ϕχ k L l 0 ; recall that l 0 deotes the subsace of l cosistig of all comlex seueces vaishig at. Note that K,α K,α 0 for all <. For more iformatio o the Herz saces, see [9] ad refereces therei. Let 0 < < ad α be real. The, sice 1 x 2 2 k for x A k ad k 0, we have 2 kα ϕχ k L ϕ(x) (1 x 2 ) α dx A k ad thus (3.1) ϕ K,α for 0 <. I articular, we have ϕ K, { } l ϕχ k L α { } ϕχ k L (λ) l ad this estimate is valid eve for =, because / = 0 if =. For this reaso, we ut K (λ) = K, for the full rages 0 < ad 0. Note that K(λ) L (λ) for 0 <. That is, these two saces are the same as sets ad have euivalet orms. Next, we itroduce a discrete versio of Herz saces. Let a = {a m } be a arbitrary lattice. Give a comlex seuece ξ = {ξ m } ad k N, let ξχ k deote the seuece defied by (ξχ k ) m = ξ m χ k (a m ). Now, give 0 <, ad α real, we let l,α (a) be the mixed-orm sace of all comlex seueces ξ such that ξ l,α (a) := {2 kα ξχ k l } <. l

7 So, we have SCHATTEN-HERZ CLASS TOEPLITZ OPERATORS 7 ξ l,α (a) = k 2 kα ξ m a m A k, 0 <, <. Also, we say ξ l,α 0 (a) if { 2 kα ξχ k l } l 0. Fially, we let l,0 (a) = l (a). We ow itroduce the so-called Schatte-Herz class of Toelitz oerators. Let S deote the class of all bouded liear oerators o b 2 ad deote the oerator orm. Give 0 <,, the Schatte-Herz class S, is the class of all Toelitz oerators T µ such that T µχk S for each k ad the seuece { T µχk } belogs to l. The orm of T µ S, is defied by T µ, = { T µχk }. l Also, we say T µ S,0 if T µ S, ad { T µχk } l 0. Note that S, S,0 for all <. The followig observatio lays a key role i rovig that K,α -behavior or l,α -behavior of averagig fuctios are ideedet of radii. Theorem 3.1. Let 0 <,, r (0, 1) ad α be real. The the followig statemets hold for µ, τ 0: (a) { 2 kα µ r χ k L (τ)} l if ad oly if { 2 kα ( µχ k ) r L (τ)} l ; (b) { 2 kα µ r χ k L (τ)} l0 if ad oly if { 2 kα ( µχ k ) r L (τ)} l0. Before roceedig to the roof, we ote the followig coverig roerty which is a simle coseuece of (2.1): If r (0, 1) ad N = N(r) is a ositive iteger such that (3.2) 2 N r 1 r < 2N, the (3.3) E r (z) A j, z A k. Here ad i the roof below, we let A j = if j < 0. Proof. Let µ, τ 0 be give. We rove the roositio oly for < ; the case = is imlicit i the roof below. Choose N = N(r) as i (3.2) ad ut γ = γ,n = max{1, (2N + 1) 1 }. By (3.3) we ote for all k. Thus, we have µ r χ k 1 µ r χ k L (τ) γ ( µχ j ) r ( µχ j ) r L (τ)

8 8 B. R. CHOE, H. KOO AND Y. J. LEE for all k. Sice 2 kα 2 N α 2 jα for k N j k + N, it follows that 2 kα µ r χ k L (τ) 2 jα ( µχ j ) r L (τ) for all k. This imlies oe directio of (b). Also, it follows that 2 kα µ r χ k L (τ) γ 2 jα ( µχ j ) r L (τ) for all k. Thus, summig u both sides of the above over all k, we have 2 kα µ r χ k L (τ) (2N + 1) 2 jα ( µχ j ) r L (τ), k=0 which gives oe directio of (a). We ow rove the other directios. Note that E r (z) ca itersect A k with k 0 oly whe z A j by (3.3). Thus we have (µχ k ) r = j=0 (µχ k ) r χ j µ r χ j. Thus, a similar argumet yields the other directios of (a) ad (b). The roof is comlete. As a immediate coseuece of Lemma 2.2 ad Theorem 3.1 (with τ = V ), we have the followig Herz sace versio of Lemma 2.2. Corollary 3.2. Let 0 <, 0, δ, r (0, 1) ad α be real. Let µ 0. The µ r K,α if ad oly if µ δ K,α. Also, alyig Theorem 3.1 with discrete measures τ = m δ a m where δ x deotes the oit mass at x B, we have the followig. Corollary 3.3. Let 0 <,, r (0, 1) ad α be real. Let µ 0 ad assume that a = {a m } is a r-lattice. Put ξ k = 2 kα { (µχ k ) r (a m )} m l for k 0. The the followig statemets hold: (a) { µ r (a m )} l,α (a) if ad oly if {ξ k } l ; (b) { µ r (a m )} l,α 0 (a) if ad oly if {ξ k} l 0. We eed oe more fact take from [3]. Lemma 3.4. Let 1, 0 ad α be real. The the Berezi trasform is bouded o K,α if ad oly if < α + 1/ < 1. We ow rove the Herz sace versio of Lemma 2.3. The restricted rage i (3.4) below ca t be imroved; see Sectio 4. Theorem 3.5. Let 0 <, 0, r (0, 1) ad α be real. Let µ 0 ad assume that a = {a m } is a r-lattice. The the followig two statemets are euivalet:

9 (a) µ r K,α ; SCHATTEN-HERZ CLASS TOEPLITZ OPERATORS 9 (b) { µ r (a m )} l,α+ (a). Moreover, if α < 1 { < mi 1 α, 1 α } (3.4), the the above statemets are euivalet to (c) µ K,α. Proof. (a) (b): Let ξ k = 2 k( +α) { (µχ k ) r (a m )} l for each k. Note that if (µχ k ) r (a m ) > 0, the A k E r (a m ) ad thus 1 a m 2 k. Thus, for <, we have ξ k = { (µχ k ) r (a m )(1 a m ) +α } l (3.5) = { (µχ k ) r (a m )} l,+α (a) (µχ k ) r L α for all k by Lemma 2.3. It follows that µ r K,α { µ r χ k L α } l (3.1) { (µχ k ) r L α } l (Theorem 3.1) {ξ k } l (3.5) { µ r (a m )} l, +α (a) (Corollary 3.3), which comletes the roof for <. The roof for = is similar. (c) = (a): This follows from Lemma 2.1 ad Lemma 2.2. We ow assume (3.4) ad rove that (a) or (b) imlies (c). For 1, oe may use Lemma 3.4 ad [5, Lemma 3.8] to see that (a) imlies (c). So, we may further assume < 1 i the roof below. (b) = (c): Assume (b). First, cosider the case 0 < <. By the roof of Lemma 2.3, we have (3.6) µ(x) m (1 a m ) µ (1 x ) r (a m ) [x, a m ] 2 for x B. Let j ad k be give. Note that by a itegratio i olar coordiates ad Lemma 2.4 of [5] A j dx 2 [x, a] 2 j [1 (1 2 j 1 ) a ] j (2 j + 2 k ) 2 +1, a A k;

10 10 B. R. CHOE, H. KOO AND Y. J. LEE this estimate is uiform i j ad k. Hece, if a m A k, the we have (1 a m ) (1 x ) A j [x, a m ] 2 (1 x )α dx 2 k 2 j(+α) dx [x, a m ] 2 A j 2 k 2 j(+α+1) (2 j + 2 k ) 2 +1 = 2 k 2 j( α ) (1 + 2 j k ) Thus, settig ξ k = 2 k( +α) a m A k µ r (a m ) ad itegratig both sides of (3.6) over A j agaist the measure dv α (x), we obtai µ(x) (1 x ) α dx ξ k 2 (j k)( α ) A j (1 + 2 j k ) k Note that 2 (j k)( α) (1 + 2 j k ) (j k)( α) = 1 2 ( α) k j for k j ad 2 (j k)( α) (1 + 2 j k ) (j k)(+α+1) = 1 2 (+α+1) k j for k < j. Therefore, combiig these estimates, we have µ(x) (1 x ) α dx ξ k (3.7) A j 2 γ k j k for all j where γ = mi{ α, + α + 1}. Note γ > 0 by (3.4). Now, for <, we have by (3.7) ad Youg s ieuality µ K,α {ξ k } l = {ξ k} l. O the other had, for 0 <, we have agai by (3.7) µ K,α { } ξ k 2 γ k j ξ 1 k 2 γ k j / {ξ k} l j k k j as desired. The case = 0 also easily follows from (3.7). The roof is comlete. The followig versio of Theorem 1.1 is ow a simle coseuece of what we ve roved so far. Theorem 3.6. Let 0 <, 0 ad µ 0. Assume that a = {a m } is a r-lattice. The the followig three statemets are euivalet: (a) T µ S, ;

11 (b) µ r K (λ); (c) { µ r (a m )} l. SCHATTEN-HERZ CLASS TOEPLITZ OPERATORS 11 Moreover, if 1 <, the the above statemets are also euivalet to (d) µ K(λ). Proof. The roof of Theorem 3.1 show that all the associated orms are euivalet. We have (a) (b) by Lemma 2.4 ad Corollary 3.3. Hece the theorem follows from Theorem 3.5 (with α = ). I the rest of the aer, we show that the arameter rage (3.4) is shar. Throughout the sectio we cosider arbitrary 0 < ad α real, uless otherwise secified. We first recall the followig fact which is a coseuece of various examles i [3, Sectio 4]; If α + 1/ or α + 1/ 1, ad 0, the there exists some f 0 such that f K,α but f / K,α. Moreover, roofs i [3] show that examles of fuctios f above also satisfy f r K,α for each r (0, 1). Note that the above take care of examles we eed for the case α 0. For α < 0, ote that the arameter rages for which we eed examles i (3.4) reduce as follows: (3.8) 1 α 1. Give δ > 1, let Γ δ be the otagetial aroach regio with vertex e := (1, 0,..., 0) cosistig of all oits x B such that x e < δ(1 x ). Also, give γ 0, let f γ be the fuctio o B defied by ( ) 1 2 γ f γ (x) = (1 x ) log. 1 x The source for our examles for the arameter rage (3.8) will be fuctios of the form f γ χ Γδ with γ suitably chose. We first ote the followig oitwise estimates take from [5, Lemma 5.2]. Lemma 3.7. Give δ > 1 ad γ 0, the fuctio g = f γ χ Γδ has the followig roerties: (a) Give r (0, 1), f γ χ Γδ1 ĝ r f γ χ Γδ2 for some δ 1 ad δ 2 ; (b) If γ > 1, the g f γ 1 χ Γδ3 for some δ 3 ; (c) If γ 1, the g = o some oe set. Next, we ote the recise arameters for f γ χ Γδ to belog to K,α. Lemma 3.8. Let 0 <, 0 ad α be real. Give δ > 1 ad γ 0, ut g = f γ χ Γδ. The g K,α if ad oly if oe of the followig coditios holds:

12 12 B. R. CHOE, H. KOO AND Y. J. LEE (i) 1/ > 1 α/; (ii) 1/ = 1 α/ ad = ; (iii) 1/ = 1 α/ ad = 0 < γ; (iv) 1/ = 1 α/ ad γ > 1 > 0. Proof. Note A k Γ δ 2 k for all k. Thus we have 2 kα gχ k L 2 kα 2 k A k Γ δ 1 1 (1 + k) γ 2 k( (1 + k) γ for all k, which gives the desired result. 1+ α ) Now, usig Lemmas 3.7 ad 3.8, we have examles for the remaiig arameters i (3.8) as follows. Examle 3.9. Let 0 <, 0 ad α be real. Give δ > 1 ad γ 0, ut g = f γ χ Γδ. The the followig statemets hold for each r (0, 1): (a) Let 1/ > 1 α/. If 0 ad if 0 γ 1, the ĝ r K,α but g = o some oe set; (b) Let 1/ = 1 α/. (b1) If = 0 or 1 < ad if 0 < γ 1, the ĝ r K,α but g = o some oe set; (b2) If 0 < 1 ad if 1 1/ < γ 1 + 1/, the ĝ r K,α but g / K,α Refereces. [1] Ahlfors, L.V. Möbius trasformatios i several dimesios. Ordway Professorshi Lectures i Mathematics. Uiv. of Miesota, School of Mathematics, Mieaolis, Mi., MR (84m:30028), Zbl [2] Axler, S.; Bourdo, P.; Ramey, W. Harmoic fuctio theory. Secod Editio. Graduate Texts i Mathematics, 137. Sriger-Verlag, New York, MR (2001j:31001), Zbl [3] Choe, B.R. Note o the Berezi trasform o Herz saces, RIMS Kyokuroku 1519 (2006) [4] Choe, B.R.; Koo, H.; Lee, Y. J. Positive Schatte(-Herz) class Toelitz oerators o the half-sace, Potetial Aalysis 27 (2007) MR (2008k:31007), Zbl [5] Choe, B.R.; Koo, H.; Lee, Y. J. Positive Schatte class Toelitz oerators o the ball, Studia Mathematica 189 (2008) MR (2009j:47057), Zbl [6] Choe, B.R.; Koo, H.; Na, K. Positive Toelitz oerators of Schatte-Herz tye, Nagoya Math. J. 185 (2007) MR (2005b:47055), Zbl [7] Choe, B.R.; Lee, Y.J.; Na, K. Toelitz oerators o harmoic Bergma saces, Nagoya Math. J. 174 (2004) MR (2005b:47055), Zbl [8] Coifma, R.R.; Rochberg, R. Reresetatio theorems for holomorhic ad harmoic fuctios i L, Astérisue 77 (1980) MR (82j:32015), Zbl [9] Hereádez, E.; Yag, D. Iterolatio of Herz saces ad alicatios, Math. Nachr. 205 (1999) MR (2000e:46035), Zbl

13 SCHATTEN-HERZ CLASS TOEPLITZ OPERATORS 13 [10] Loaiza, M.; Lóez-García, M.; Pérez-Esteva, S. Herz classes ad Toelitz oerators i the disk, Itegral Euatios Oerator Theory 53 (2005) MR (2006g:47041), Zbl [11] Miao, J. Reroducig kerels for harmoic Bergma saces of the uit ball, Moatsh. Math. 125 (1998) MR (98k:46042), Zbl [12] Miao, J. Toelitz oerators o harmoic Bergma sace, Itegral Euatios Oerator Theory 27 (1997) MR (98a:47028), Zbl Deartmet of Mathematics, Korea Uiversity, Seoul , KOREA cbr@korea.ac.kr Deartmet of Mathematics, Korea Uiversity, Seoul , KOREA koohw@korea.ac.kr Deartmet of Mathematics, Choam Natioal Uiversity, Gwagju , KOREA leeyj@choam.ac.kr