Projections on hardy spaces in the lie ball

Publicacions Matemàtiques, Vol 38 (1994), 57-68 . PROJEOTIONS ON HARDY SPACE S IN THE LIE BALL DAVID BEKOLL É Abstract   on the Lie ball w of e n , n > 3, we prove that for all p E [1, ce) , p 2, the Hardy space H P (w) is an uncomplemented subspac e of the Lebesgue space L P (áow, da), where aow denotes the Shilo v boundary of w and da is a normalized invariant measure on aow . 1 . Introductio n Let w = w n denote the Lie ball of e n , n > 3, defined b y c~= zEc n :l—21zi 2 + > 0 an d 2 } C 1 The Shilov boundary 49 0 w of w is a 0 w = {e i e x : 8 E [0, 27r} and x E S n _ 1 } , wher e Sn _ 1 denotes th e unit spher e o f R n . Denote by G th e compac t grou p consistin g o f th e followin g transformation s in c n : g belong s to G i f ther e are h E S4 (n) an d B E [o, 27r} suc h tha t g(z ) = e ie h(z ) fo r al l z E e n . Th e grou p G is subgrou p o f th e automorphis m grou p o f w an d G act s transitivel y on aow . Moreover , a normalize d G-invarian t measur e on á0 w is do-( e ie x ) = 4 ( ti ¡+ 1 d8 dp,(x) , wher e d p denotes th e Lebesgu e surface measure o n Recali tha t th e Hard y spac e H P (w), 1 ç p < oo , is define d as consistin g of all functions f holomorphic in w and such tha t sup J ~ 0<r<1 ao w Ever y f E H P (w) has radial limit s a--a .e . on a 0 w an d whe n w e identif y f wit h it s boundar y valu e function , th e spac e H P (w) can be viewe d as a closed subspace of L P (aow, do . ) . For this fact, c .f . e .g . [VI . The main result of the present paper is the following : 58  D . B EKoLL É Theorem 1 .1 . For all p E [1, oo), p  2, H P (w) is an uncomplemented subspace of L P (áow, dms) . In Chapter 12 of [R1], the same result is proved for p = 1 in the uni t ball of e n . It turns out that the result for the Lie ball (Theorem 1 .1 ) can be deduced exactly in the same way from the following theorem : Theorem 1 .2 [BeBo] . In the Lie ball w of e n , n > 3, the Szegó . projectionis L p -unbounded for all p E (1, oo) p 2 . We recall that the Szegd projection for w is the orthogonal projectio n of L 2 (aow, do-) onto H2 (w) . Let U denote the unit disc in the complex plane . Then the followin g question is open : are the Hardy spaces H P (w) and H P (U) isomorphi c for p E (1, oo), p 2? P . Wojtaszczyk [W] answered this questio n affirmatively in the unit ball B n of e n , n > 2 . His proof uses the fac t that H P (B n ) is a complemented subspace of L p (aB n ) , by a theore m of [KV], and hence it cannot be applied to the Lie ball case . On th e other hand, T . Wolniewicz [Wo] proved that for classical domainsD o f class I (in Hua's standard notation [H]), the Hardy space Hl (D) is no t isomorphic to H 1 (U) except when D is a unit ball . In particular, H 1 (D ) is not isomorphic to H l (U) when D is the Lie ball w4 in C 4 . It seem s unlikely that H 1 (wn) is isomorphicto H1 (U) when n = 3 and n > 5 (i .e . for domains of class IV) . Finally, it seems to be certain that Theorems 1 .1 and 1 .2 can be ex - tended to all bounded symmetric domains of tube type . To do this, a n appropriate reference would be [K] . In the second section, we recall a decomposition ofL 2 (áow, da) into a direct sum of G -invariant subspaces(c .f . e .g . [BCKD . We denote by N the polynomial on c n defined by n (1)  N(z) _ E z 3 j= 1 and by q the differential polynomial defined b y n a 2 q z _E a i= 1 z Denote by H k , k E 7G+, the space consisting of restrictions to aow o f homogeneous holomorphic polynomials p of degree k such that q p = O . (2) PROJECTIONS ON HARDY SPACES IN THE LIE BALL  59 Then L 2 (Dow, da) =  E : : N~ H k and the spaces N 3 H k are pairwis e (j,k)EZXZ + orthogonal with respect to the usual inner product in L 2 (aow, da) . In the third section, we study the orthogonal projection of L 2 (aow, da ) into N j H k and linear maps T : H k1 —> Nj2 Hk2 that commute wit h G . In the fourth section, we characterize the G -invariant closed subspace s of L 2 (aow, d a), 1 Ç p < oc and of C (ao , w) . This extends a theorem du e to A . Nagel and W . Rudin (Theorem 12 .3 .6 in [R 1 D for the unit ball B n of e n . Denote by H (p, g) , (p, q) E Z+ x Z + the space of homogeneou s harmonic polynomials on ~ n that have total degree p in the variable s z 1 , . . . , zn , and total degree q in the variables Z1, . . . , z n . Then L 2 (aB n ) is the direct sum of the spaces H (p, q) and the theorem of Nagel an d Rudin in B n is stated in terms of the H (p, q) spaces . Their proof stil l works in the Lie ball case, when the spaces H(p, g) are replaced by th e spaces N j H k . Finally, in the fifth section, we prove Theorem 1 .1 . 2 . Decomposition of L 2 (Dow, da ) When we identify ~ n with R 2n , we let P denote the space of al l complex -valued polynomials on e n . By the Stone-Weierstrass Theorem , the space of restrictions to aow of elements of P is dense in C(aow) , and hence in L 2 (aow, d a) . Denote by P ( k ) the space of restrictions to aow o f homogeneous complexvalued polynomials of degreek ; the n +o o (3)  L 2 (aow, da) = E P (k) . k= 0 Let N be the polynomial on ~ n defined by (1) ; the n ;¡a ° , = N(z)-1z ; Furthermore, when Pk denote the space of restrictions to aow of hornogeneous holomorphic polynomials of degree k án tC n , we obtain : F(k) _ k E EDN i P k ' ( 4 ) j= o For q E Pk, q(z) =E b a z a , define q by q ( z) =E baz° e and equip P k cxEA  aE A with the differential inner produc t ( p 4) = k~ p ( D )4, 60  D . BEKOLL É r7 ci  a a ~ where p(D) = E aa az,  . . . a zn  and A = {a = (a b . . . , a n ) : aE A ictii Denote by H k , k > 2, the orthogonal of NPk_ 2 in Pk relative to th e inner product (i) and set H } = P I , Ho = Po . We have the following lemma : Lemma 2 .1 . The space H k consists of all restrictions to 8ow of homogeneous holomorphic polynomials of degree k that are annihilated b y the differential polynomial q defined in (2) . Moreover, [ k / 2 ] (s)  P k = E ED N2 Hk-2j . ~ — o Proof : This is just Lemma 3 .2 in [Co We] . ■ In the sequel, the letter Q will stand for Z x Z . W e prove next th e following lemma : Lemma 2 .2 . Por all (j, k) E Q, the space N j H k is G -invariant . Proof : We must prove that (Np)oq5 E N i H k for all ~ E G and p E H k . There is a constant cí(q5) . such that '(N j p) o çb — c(4i)N . (p o O), an d hence it suffices to prove that p o 0EH k . But q commutes with G, i .e . q (p o O) = (Op) o ~ for all E G and p E Pk . Then, q p = o implie s q (p o O) = o and the desi-red conclusion foliows from the first assertio n of Lemma 2 .1 . ■ In the rest of the paper, L 2 (a7ow, dcr ) will be equipped with the usua l inner product : ( 7 )  [f, g ] _  f 9 do- . f p L J We prove the following proposition : Proposition 2 .3 . The spaces N j Hk, (j, k) E Q, are pairwise orthogonal with respect to the inraer product [ , ] . Proof For all (j ' , k 1 ) and (j 2 , k2) in Q such that ( j i, k l )  Cj2,k 2 } and for allp EH kx and g EH k2 , we have : 27 r [N1 p, N32 q ] =  e 2i e( j 1 +1~ 1 — j2 — k2 )d0 (fd) . o  n _ i PROJECTIONS ON HARDY SPACES IN THE LIE BALL  6 1 Assume first that k l k 2 . On R n , p and q are homogeneous harmoni c polynomials of respective degrees k l and k 2 ; it is known (cf . e .g . [R1, p . 254] } that the integral over Sn_ 1 is then zero . Assume next that k i = k 2 and j 1 h . In this case, the integral over [o, 27r} is zero . This prove s the proposition . ■ We deduce next the following corollary : Corollary 2 .4 . The space L Z (áow, do-) is the direct sum of the pairwise orthogonal subspaces N 3 H k , (j, k) E Q . Proof : In view of Proposition 2 .3, it is enough to prove tha t L 2 (aow, d o-) is contained in the sum of the spaces N jH k , (j, k) E Q . Combining (4) and (6) yields : p(k) = ~ e N-3+i Hk-2z • j=0y1, . . .,j C 2=~,~, . . .,[ 2 ] The conclusion then follows from(3) . This completes the proof . ■ 3 . The spaces N j H k Denote by S the representation of G in P defined by S o p(z) =p(0-1 .(z) ) aád denote by  the restriction of S to , N jHk . We have , the followin g theorem :  . Theorem 3 .1 . The representation S jk of G in N j H k is irreducible , i . e . N 2 H k has no proper G -invariant subspace . Proof : Since SO(n) c G, it is known (cf . Theorem 3 .3 in [Co We] ) that the representation 8,9k in H k is irreducible . On the other hand , N -j Y is a G-invariant subspace of H k whenever Y is a G-invariant subspace of N j H k . The conclusion follows . ■ Fix (j, k) E Q and denote by 7~ j k the orthogonal pro j ection o f L 2 (Dow, do-) onto its closed (since it is finite-dimensional) subspac e N i H k . We prove first the following lemma : Lemma 3 .2 . The pro jection  commutes with G, i . e . for all c~ E G and f E L 2 (8 0 co, dQ )y 7r j k (f oO) = (nkf)oçb . Proa : This is an immediate consequence of the G -invariance of N j H k (Lemma 2 .2) and (NH k )' . ■ We prove next the following theorem : 62  D . BEKOLL É Theorem 3 .3 . 1 . To every z E óow corresponda a unique K z E N j H k that satisfies (8) (7rkf)(z) _ {f, K]  (f E L2 ( a oW , dms )) • 2 . These kernel functions K z have the following addtional properties : (9) K z (w) = K,,, (z)  (z,wEaow) , (lo)  7r,k .f = f f (C) K C d a (C)  (fEL 2 (8 0 w,da ) ao w Ko(z) = Kz o 0 -1 (O E G) , (12)  K z =K z o I I / for all xlf E G that fix z, an d (13)  K z (z) = K w (w) > 0  (z,w E áow) . P roo f : W e write 7r for 7rjk . 1 . In the finite -dimensional space N j H k , the norms max IN 3 p l and ao W ii N j pl j are equivalent . The correspondence f --3 (irf)(z) is then a bounded linear functiona l on L 2 (5 0 w, do. ) . Thus there is a unique K z E L 2 (aow, do-) that satisfie s (8) . Since 7r f = 0 for every f E (NH k )', it follows that K z E N j H k . 2. For f = K w , (8) become s K wl z 1 = [ K w, K z] = [K Z ,K W ] = K z (w) . This proves (9) and (9) shows that (10) is just another way of writin g (8) . Since 7r commutes with G (Lemma 3 .2), (nf)((z)) = 7r( f o O)(z) = [ f o O, K z ] -- [f, K z o 0 —1 ] for every f E .L 2 (aow, do . ) . Combined with (8) , this proves (11), hence also its special case (12) . Finally, Ko(z)(0(z)) = (K z o 0 —1 )(Oz)) = K z (z ) is another consequence of (11) . It proves (13), becaus e K z (z) = [K z ,K z ] > O . E Let us now prove the following proposition : PROJECTIONS ON HARDY SPACES IN THE LIE BALL  6 3 Proposition 3 .4 . For each z E 5 0 w, the space N 3 H k contains a unique f such that f(z) = 1 and f = f o ■ P for every x lf E G tha t fixes z . Proo f : It follows from (12) and (13) that f = K(z } satisfies the required properties . To prove the uniqueness, denote by e 1 the poin t (1, o, . . . , 0) and assume z = e 1 without loss of generality . The invarianc e f = f o ■ P for every ~ E SO(n) that fixes e l shows then, for f = N jp , p E H k and for each w1, that (W2, . . . , wn) ~ p(w i ,w 2 , . . . , w n ) is a polynomial in w1 + • • • + w~ . Since p is a homogeneous holomorphi c polynomial of degree k on ~ n that satisfies p(el) = 1,p therefore has th e form r (14)  p(w) E c i( w2 + . . . + w ñ)i w i-z i i= l where r = [ti and e0 , . . . , c n are constants, co = 1 . Differentiation o f (14) gives T— L (15)  Ep(w) = E b i (11)1  2 i k—2i— 2 z- o wher e (16)  b i = (k - 2i)(k - 2i - 1)c 2 + (2i + 1)(n + 2 i)ci+l > for all i = 0, . . . , r - 1 . Since Ep = o, (15) vanishes, so that b i = o for all i and (16) successively determines e 1 , . .. , c T . This proves th e proposition . ■ Our next result is the following : Theorem 3 .5 . The irreductible representations Sj1 k1 and S i2 k2 o f G in N31 Hk1 and N 32 H k2 respectively are inequivalent uihen (j 1 ,k i) 7 4 (j 2 ,k 2 ) . More precisely, suppose T : N31 Hk1 --> N32 H k 2 is linear an d commutes with G . The n 1. T = Q when ( j i, k 1 ) ~(j 2 , k 2 } , 2. when (j i ,k i) _ (j 2 ,k 2 ), there is a constant c such that T = c l where 1 is the identity operator . Proof : The assertion 2 follows from Schur's Lemma, as well as th e following assertion : either T = o or T is invertible when k 1 } ~ (j 2 ,k 2 ) . Let us prove that T = 0 in the latter case . 64  D . BEKoLL É Let Kz E N a l Hk1 and Kz E N i2 H k2 be as in Theorem 3 .3 . If ~ E G fixes z, the n (17)  (TK)oW=T(Ko)=TK z by (12), since T commutes with G . By Proposition 3 .4, (17) shows tha t to every z E áow corresponds a constant c(z) such that TKz = c(z)Kz . Hence TKz (z) = c(z)KZ (z) . By (13), Kz (z) is independent of z . If w = O(z), E G, then by (11), (TK)(w) = T(Kz o O -1 )(0(z)) = TKz (z) . We conclude from this that c(z) = c, the saure for all z E óo w . If f E N~1H~ 1 , (10) shows tha t (is)  f  do-(() . ao w Apply T to (18) and use TK~ = clq . : T f = c faa,, f (()K dcr(() = c7ra2k2 f, for every f E N ar H k ~ . I f ( j 1, k1 ) (j2, k2 ), 7ri1 k 1 f = o and hence T f = O ., This proves th e theorem . ■ 4 . G- invariant spaces on a o w For S~ c Q the algebraic sum of all N a H k with (j, k) E SZ will b e denoted by En . We adopt the convention that E~ = {o} when SZ i s empty . The letter X will stand for any of the Banach spaces C(a 0 w) o r L p (aow, da), 1 < p < oc . The X -closure of E s- 1 will be denoted b y )(o . Trivially, every X 2 is a G -invariant closed subspace in X . Th e proof of the converse is the main topic of the section . In the sequel, w e write C for C (a q w ) and L p for L P (aow, da) . We begin withthe case X = L 2 : Theorem 4 .1 . If Y is a G-invariant closed subspace of L 2 (áow, dQ ) and if S2 is the set of all (j,k) E Q such, that 'nr jk Y {0}, then Y = (L 2 ) . n Proof Pick (j, k) E SZ . Since Y is G- invariantand 7rak commutes wit h G, 7r ik Y is a nontrivial invariant subspace of N3Hk . By Theorem 3 .1 , we have therefore 7ra k Y = N a H k . Let Yo = { f E Y : 7rj k f = 0}, and let Y 1 be the orthogonal complemen t of Yo in Y . Then Yo is G -invariant, hence so is Y 1 . Moreover, Ira k - 4 N a H k is an isomorphism whose inverse we denote by A . PROJECTIONS ON HARDY SPACES IN THE LIE BALL  6 5 Choose (r, s) E Q, (r, s)  (j, k), and consider the linear map T = 7r rs o A : N 3 H k - -} N T H s . It is clear that T commutes with G ; hence , by Theorem 3 .5, T = 0 and 7 r TS Y I = {0} for every (r, s) (j, k) . B y Corollary 2 .4, Y 1 is a nontrivial subspace of N a .I k , isomorphic to N a H k ; since Y 1 is G -invariant, then Y 1 = N a H k by Theorem 3 .1 . Furthermore , N j .FI kc Y, for every (j,k) E S2 ; in other words, (L 2 ) 0 C Y . Sinc e 7rTS Y = {O} for every (r,$)  (j, k), a second look at Corollary 2 . 4 completes the proof of the theorem . ■ We shall need the following three lemmas : Lemma 4 .2 . If f E X, then ~-> f o is a continuous map of G int o X . Proof . If E > O, then ii f -  < e for some g E C . There is a neigh - bourhood N of the identity in G such that llg - g o OII,, 0 c e for ever y E N . Since f - f o Ol  f - g l -}- lg - g o 01 -}- (g - f) o 01, we obtai n f - f o < 3e for every ~ E N . ■ Lemma 4 .3 . If Y is a G -invariant closed subspace of X, then Y n c is dense in Y . Proof : Pick f E Y, choose N as in the proof of Lemma 4 .2, let A : G - > [0, oo} be continuous, with support in N and f G A d O = 1 . Define g =  A(0) ,f o do- . G The integrand is a continuous Y -valued function . Hence g E Y . If T E G is such that T(e l ) = (, where e 1 = (1, 0, . . . , 0), the invariance of th e Haar measure shows tha t 9(Ç) = IG n (01) .Í((ei)) dte . Thus, g E Y f1 C . Finally, the relatio n A(0)(f - f o 0) d O f proves that 11 f -  < 3e, since II f - f o 011 < 3E if E N . ■ Lemma 4 .4 . If Y c C, Y is a G-invariant space, and someg E C i s not in the uniform closure of Y, then g is not in the L 2 -closure of Y . Proof : There is a regular complex Borel measure m on r7ow such tha t f f dm = 0 for all f E Y, but f g dm = 1 . There is a neighbourhood