Hp(Bn) = {f: f e H(Bn),\\f\\p < œ},

Transcription

1 proceedings of the american mathematical society Volume 103, Number 1, May 1988 THE SPACES Hp(Bn), 0 < p < 1, AND Bpq(Bn), 0 < p < q < 1, ARE NOT LOCALLY CONVEX SHI JI-HUAI (Communicated by Irwin Kra) ABSTRACT. In this note, we use the Ryll-Wojtaszczyk polynomials to prove that the spaces HP(Bn), 0 < p < 1, and Bpq{Bn), 0 < p < q < 1, fail to be locally convex. Let Bn denote the unit ball of complex vector space C", dbn its boundary and an the positive rotation-invariant measure on dbn with an(dbn) = 1. By H(Bn) we denote the class of all functions holomorphic in Bn. The Hardy space Hp, 0 < p <, is defined on Bn by where Hp(Bn) = {f: f e H(Bn),\\f\\p < œ}, (1) / p= sup Mp(r,f), Mp(r,f)=\[ \f(rç)\p dan(ç)\. 0<r<l UdBn ) where The space Bpq, 0 < p < q <, is defined on Bn by Bpq(Bn) = {/:/ H(Bn), / Bp, < }, (2) / bm = j^u-rr^-i/^-im^r,/)*} For p > 1, Hp(Bn) is a Banach space with norm (1) and for q > 1, Bpq(Bn) is a Banach space with norm (2). For 0 < p < 1 and 0 < p < q < 1, we introduce the metrics d(f,g) = \\f-g\\p for Hp(Bn) and d{f,g) = \\f-9\\%pq for Bpq(Bn). These metrics turn Hp(Bn) and Bvq(Bn) respectively into F-spaces (complete, metrizable linear topological spaces). Livingston first noticed that HP(U) fails to be locally convex for 0 < p < 1 and U a unit disc, [2]. In this note we will prove that Hp(Bn), 0 < p < 1, and Bpq(Bn), 0 < p < q < 1, are not locally convex. Received by the editors December 9, Mathematics Subject Classification (1985 Revision). Primary 32A35; Secondary 46A American Mathematical Society /88 $ $.25 per page

2 70 SHI JI-HUAI Let / G H(Bn) with the homogeneous expansion f(z) = YLkLo Ek(z) and ß > 0; the /3th fractional derivative and fractional integral of / are defined respectively by and fc=0 v '. ^ r(fc + i) F, In the above, T denotes the Gamma function. It is known that /^ and /[^j are holomorphic on Bn [3]. In 1983 Ryll and Wojtaszczyk [5] proved the following theorem: For every n > 1, there is a sequence pi, pa, with pk G P(k,n), such that Pfc = l, llpfclb > v^/2n, k = \,2,..., where P(k, n) is the vector space of homogeneous polynomials of degree A: in the complex variables Z\,...,zn regarded as functions on dbn and llpfcll = sup pfc(ç), \\Pkh = \ \Pk(ç)\2dan(ç)\ ç dbn UdBn ) {pk}, k = 1, 2,..., are called the Ryll-Wojtaszczyk polynomials, for which we have LEMMA 1. Let {nk} be a lacunary sequence of positive integers and {pnk} the Ryll-Wojtaszczyk polynomials. If f(z) J2'kLiakPnk(z) 'with Yl'kLi \ak\2 < > then for any p, 0 < p < 1, there is a constant A independent { M2 i V2 <a / p. of f, such that PRF. Since p t(ç) < 1, c G dbn, p J i > p J i > 7r/4n. Proposition 1.6 of [5] gives i» y/2 (3) Em2 <^n/iii- Here, and later, A denotes a finite constant, not necessarily the same at each occurrence, independent of /. It is easy to see (4) / ß < m2hp"j2<em2- Ic=l Combining (3) and (4) we obtain (5) \\fh<a\\f\\i. On the other hand, for any p, 0 < p < 1, Holder's inequality gives (6) / i< /?/(2-p) / l2(1^)/(2"p)- From (5) and (6), we have (7) / 2<A / P. Now (3) and (7) give the desired inequality.

3 THE SPACES H"{Bn) ARE NOT LOCALLY CONVEX 71 A function /, holomorphic in Bn and continuous on Bn, belongs to Lipschitz class AQ, 0 < a < 1, if it satisfies the Lipschitz condition \f(ï)-f(v)\<a\ç-v\a, c, n G dbn, A independent of ç, r. \ç-v\-0, LEMMA 2. Let f(z) = J2'k'=oPk(z) be the homogeneous expansion of f on Bn. If f e Aa, 0 < a < 1, then J2T=i ks\\fk\\i < for anv s < 2a- PRF. Since / G AQ, by Theorem 6 of [3], (8) \fw(rç)\ < A(l - r)a~l, 0<r<l, çedbn, where A is independent of r and c. Inequality (8) and /JdB \fw(ri)\2dan(ç) = f^(k + l)2r2k\\fk\\ k=0 give r2n 2k=i k2\\fk\\l < A(l - r)2^"1). Hence N sn = j2k2\\fk\\l<an2{1~a)- fc=i Let r = 1 - TV"1; we have N N N-l E fcsn^ii2 = E ^ii^ii2^8-2 = E 5*[^-2 - (fc+1)s_2i+^^ k = l A:=l iv-1 < A ^ [F"2" - (fc + 1)5"2q] + 0(1) = 0(1) k=l for s < 2a. The lemma is proved. THEOREM l. Prf. Let Hp(Bn), 0 < p < 1, is not locally convex. Í ^ f(z) = Y^akpnk(z): ^ ofc 2<l, k=l ) where {nk} is a lacunary sequence of natural numbers and {p fc} are the Ryll- Wojtaszczyk polynomials. It is clear that M is a subspace of Hp(Bn), 0 < p < 1. Take a sequence {ck} such that Yl'kLi \ck\2 < an(^ f r anv > 0, X^fcLi ^ lcfc 2. Define a linear functional <p on M as follows: >(/) = EafcCfc' /(^) = E13^"*^) M The Schwarz Lemma and Lemma 1 give / y/2 M/) <AÍ K 2J <A / P, / M. s-2

4 72 SHI JI-HUAI This proves that p is a continuous linear functional on M. If Hp(Bn), 0 < p < 1, is locally convex, then there exists a continuous linear functional $ on Hp(Bn), such that $(/) = <p(f) on M [4, Theorem 3.6]. By Theorem 9 of [3], there is some g e Aa, 0 < a < 1, such that (9) (/) = lim / f(rç)g^)dan(c) r^ljdbn for all / ehp(bn). Let g(z) = Y^T=0 Fk(*)\ Lemma 2 gives Take / = pnk in (9); since $(/) = <p(f) = ck, we have cfc < F J 2, k = 1,2,... Now (10) gives *!<&!* < n2 FBJ <, s<2a. This contradiction proves that Hp(Bn) fails to be locally convex. THEOREM 2. Bpq(Bn), 0 <p <q <1, is not locally convex. Before proving the theorem, we first prove the following lemma. LEMMA 3. Suppose 0 < p < q < 1 and ß = n(l/p - 1/q). If fw G Bpq(Bn), then f e Hq(Bn) and there exists a constant A independent of f, such that \\f\\q < (io) EFHFfcll2<00' s<2qfc=i M\î[0]\\Bvq- PRF. A direct computation gives for 0 < r < 1 and ß > 0, and. -Pk f/(rf)l < fvtrrxi / {P) fc=i ypfc-i ^-P)0'l\f[0]{rpç)\dp, where p* = 1-2~k. Let Hk(rç) = mp{\fw(rpi)\: pk-i <P< Pk}- Then (11) /(rc) *<A^//fc'(rc)2-^. fc=i Here we have used the condition 0 < q < 1. Now Theorem 3 of [1] and the monotonicity of the mean give M^(r,f) < A E 2-«"fcsup{M«(rp,/M): pfc_! < p < Pk} fc=i -1 < A 2-'**M«(rp*,/[0)<A / (l-p^-^^p,/^)^. '0 Taking /3 = n(l/p 1/g) in the above inequality, the desired result follows.

5 THE SPACES H"(B ) ARE NOT LOCALLY CONVEX 73 Prf of Theorem 2. Let ß = n(l/p - l/q) and E = {h: h e Hp(Bn), h[ß] e M}. Taking k = q in Theorem 4 of [3], we have (12) [ /HflM <A / P, 0<p<<7<. This shows that E is a subspace of Bpq(Bn). It is clear that for every he E, there exists an / G M with /'^ = h, hence h can be written as,o\ ut \ V^ T(nk + ß+l) (13) h(z) = g T{nk + 1) akpnk (z). Take a sequence {bk} such that Yl'kLi n\ l^fc 2 <! and X^Li n1z for any e > 0. Define a linear functional on E as follows: \ k\2 (H) The series in (14) is convergent ^h) = ET{rk(n+0+V)akbk^ hek ->* since T(nk + ß + 1)/T(nk + 1) =s npk and ( \!/2 EKI2), hee. Since /l^l = Zi G Hp(Bn) C Bpq(Bn), according to Lemma 3, there is a constant A such that (16) 11/11, <^II/I/?1IIbm=A A bp,. (15), (16) and Lemma 1 give \ip(h)\ < A\\h\\Bpq for hee. This proves that ib is a continuous linear functional on the subspace E of Bpq(Bn). If Bpq(Bn) is locally convex, then there is a continuous linear functional \f' on Bpq(Bn) with 9(h) = ip(h) on E. By (12), 9 is also a continuous linear functional on Hp(Bn). Suppose p satisfies n/(n +1) < p < n/(n + Z 1), Z a natural number. By Theorem 9 of [3], there exists a v with i)i'-1' G AQ, a = n(l/p - 1) - Z + 1, such that (17) 9(h) = lim f ft(rc)^dcrn(ç) r^1/as for every h G Hp(Bn). Let v(z) = X^tlo ^fcí*); Lemma 2 tells us (is) EfcS+2(i_1)iiGfcii2<o ' s<2a- Take A = (T(nk + ß + 1)/T(nk + l))p t in (17); since we have 9(h) = rp(h) = (T(nk +ß+ (19) 6fc < GnJ 2, k = l,2,... l)/t(nk + l))bk, Chse n > 0 such that n + 2/3 < 2n(l/p - I) n and let s = 2a rj. Then s + 2(1-1) = 2n(l/p - 1) - rf > r + Iß. Now (18) and (19) give E-r2^i2^E<+2(i_1)n^ii2<-- k=\ This is a contradiction. The theorem is proved. J

6 74 SHI JI-HUAI References 1. S. Bochner, Classes of holomorphic functions of several variables in circular domains, Proc. Nat. Acad. Sei. U.S.A. 46 (1960), A. E. Livingston, The space Hp, 0 < p < 1, is not normable, Pacific J. Math. 3 (1953), J. Mitchell and K. T. Hahn, Representation of linear functionals in Hp spaces over bounded symmetric domains m Cn, J. Math. Anal. Appl. 56 (1976), W. Rudin, Functional analysis, McGraw-Hill, J. Ryll and P. Wojtaszczyk, On homogeneous polynomials on a complex ball, Trans. Amer. Math. Soc. 276 (1983), Department of Mathematics, University of Science and technology of China, Hefei, Anhui, People's Republic of China