Mp{u,r) = j\u(rw)fdô{w),

Transcription

1 proceedings of the american mathematical society Volume 109. Number I, May 1990 CONVOLUTION IN THE HARMONIC HARDY CLASS h" WITH 0 < p < 1 MIROSLAV PAVLOVIC (Communicated by Paul S. Muhly) Abstract. It is proved that if u hp, 0 < p < 1, and v e hq, q > p, then Mq(u*v, r) = 0((1 -r)]~xi''\, r -* 1-, where u * v stands for the convolution of u and v. 1. Introduction Let h (A) denote the class of all (complex-valued) harmonic functions in the unit disc A of the complex plane C. The harmonic Hardy class hp, 0 < p < oo, is the subclass of h(a) consisting of those functions u for which w := sup M' (w, r) < oo. p 0<r<\ p Here M {u, r) is the pifx mean value of u : Mp{u,r) = j\u(rw)fdô{w), where T is the unit circle and dô is the normalized Lebesgue measure on T. The Hardy space Hp is the closed subspace of hp consisting of functions holomorphic in A. For p > 1, the structure of hp spaces is well known (see [1], Chapters 2, 3, and 4) and is similar to the structure of Hp and Lp spaces. The fundamental results concerning the case p < 1 were obtained by Hardy and Littlewood [2]; for further information and references we refer to [5]. Each u e h{a) has a unique series expansion oo u{rw) = Y û(k)r w, wet, 0 < r < 1, k = oo which converges uniformly and absolutely on compact subsets of A. The convolution of u and v(e h (A)) is defined by oo (u * v)(rw) = Y û(k)v(k)r w', w e T, 0 < r < 1, Received by the editors May 4, Mathematics Subject Classification (1985 Revision). Primary 30D American Mathematical Society /90 $1.00+ $.25 per page

2 130 MIROSLAV PAVLOVIC which can be written as (1.1) iu*v)ir2w)= Í u{rwqv{rqds{q, wet, 0<r<l. This identity along with Minkowski's inequality in continuous form is used to prove that Mp (u*v, r2) < Mp{u, r)mx(v, r), p>\. In particular, if u e hp, p > I, and v e h, then u * v e hp. This fact is extended by the following theorem. Theorem 1.1. If 0 < p < 1 and q > p, then there is a constant C Cp such that (1.2) Mq{u*v,r)<C il-r)l-x/p\\u\\p\\v\\q, 0 < r < 1, for all uehp, v ehq. < oo The analytic case of this theorem is considered in [3], and the proof is based on the inequality (1.3) MxiF,r)<Cpil-r)X-xlpMp(^F,{-^j {p < 1), which is valid for analytic functions. In 1 we use a certain maximal function to obtain a more general version of (1.3). However, the heart of our proof of Theorem 1.1 is the following fundamental result of Hardy and Littlewood [2]. Theorem HL. Let U be a harmonic function in a disc D centered at zq. Then, for each p > 0, \U{z0)\p < Cpm{D)-1 J \Uf dm, where C is a constant depending only on p, and dm stands for the Lebesgue measure in the plane. A short proof of Theorem HL is given in [4]. Theorem 1.1 is a consequence of a stronger result that will be proved in 3. Also, 3 contains a generalization of Theorem 1.1 to a class of weighted hp spaces. 2. A MAXIMAL FUNCTION For a complex-valued function F continuous in the unit disc A we define the function F+ by (2.1) F+irQ = sup{\f{rz)\ : \z - Q < l-r, z e T}, Ç 7\ r < 1. (Note that the supremum is taken over an arc.) This maximal function differs from (and is much simpler than) the radial maximal function of Hardy and Littlewood.

3 CONVOLUTION IN THE HARMONIC HARDY CLASS h" 131 Lemma 2.1. Let 0 < p < 1, and let F be a function continuous in A. Then (2.2) Mx{F,r) <C{l-r)X~X/pMp{F+,r), 0<r<l, where C depends only on p. Proof. First we prove that (2.3) MooiF,r)<C{l-r)-xlpMp(F+,r), 0<r<l, where M^iF, r) =sup{ F(rw) : w e T). For a fixed r, 0 < r < 1, let T, = {w e T : \w - Q < I - r}. It follows from (2.1) that if w e Tr, then Hence F+{rw) = sup {\F{rz)\ :wetz)> \F{rQ\. Mp{F+,r)= f F+irw)p dôiw) > j \F{rQ\p dô{w) = \F{rQ\pô{n). But 0{T ) is independent of Ç and equals \ arcsin(^jf-), whence Mpp(F\r)>\F{rÇ)\p{l-r)/K, which proves (2.3). We see that (2.3) holds for all p > 0. If p < 1, then and now (2.2) follows from (2.3). Mï{F,r)<Ml-p{F,r)Mpp{F,r) <Mxo:PiF,r)MppiF+,r), a When p > 1, there is a simple relation between the p\n mean values of a harmonic function U and c/+. Namely, it is easy to see that for the Poisson kernel P, we have P+ < CP, which implies that (2.4) Mp[u+,r)<CMp(u,l-^^j (p > 1) where C is an absolute constant. In the case p < 1 we must use some other mean values. For 0 < p < oo we let where ApiF,r) = i il - r)-2 ^ MPiF,<p) dtp r0= max{0,r-(l-r)/2} r,=r + (l-r)/2 = (l + r)/2, 0 < r < 1. Note that AAF, r) is "equivalent" to the ptix mean value of F over the annulus rn < \z\ < r.. liv

4 132 MIROSLAV PAVLOVIC Lemma 2.2. If p > 0 and if U is harmonic in A, then there is a constant C = Cp such that (2.5) Mp(u+,r) <CApiU,r), 0<r<l. This implies (2.4) because of the increasing property of MAU, r), p > 1. Proof. For fixed r and z e T let D = {w e C: \w - rz\ < (1 - r)/2}. By Theorem HL, we have (2.6) \U{rz)\p <C{\ -r)~2 [ \U{w)\p dm(w). JD If w e D, then 1 -rwz\ < (3/2)(l -r). From this and (2.6) we find that \U{rz)\p <C f \U{w)\"\l -rwzf2 Jd Since D is contained in the annulus we get dm{w). Air) = {wec: r - (1 -r)/2< \w\ <r+(l - r)/2} (2.7) C/(rz) " < C Í \U{w) l-rw;z ~dm{w), zet, JAtr) where C depends only on p. On the other hand, if z = = 1 and \z < 1 r, then for all wea,we have From this, (2.7), and (2.1) we see that U+ irc)" < C [ JAir) 1 rwç < rwz\ Hence, by integration over the circle = I, \U(w) 1 - rwç dmiw), C = 1 Mp (U+,r) <C [ \U(w)\p(\ -r\w\)~x dmiw). ^ ' JA[r) Integration in polar coordinates concludes the proof. D 3. Mean values of u*v Theorem 1.1 is an immediate consequence of the following: Theorem 3.1. If 0 < p < 1 and q > p, then there is a constant C = C that (3.1) Mq(u*v,r2) <C{l-r)X~ilpAp{u,r)Aq{v,r), 0<r<l, for all u,veh{a)., such

5 CONVOLUTION IN THE HARMONIC HARDY CLASS hf 133 Proof. By Lemma 2.2, it suffices to prove that (3.2) Mq (u*v, r2) < Cil - r)x~x,p Mp (u+, r) Mq (v+, r). Let 0 < p < 1 and q > p. For a fixed w e T let FirQ = u(rojvirwq, 0< r < 1, Ce7\ Then (m * w) (r'w) = {v * u) (r~w) = / F (rç) dô (C) (Here u and v are harmonic, but the proof of (3.2) is independent of this hypothesis.) By Lemma 2.1, iu*v) (r2w) " <MPiF, r) <Cil-r)p X Í F+irÇ)p ds{q. Since obviously F+irÇ) < u+irç)v+irwq, we get iu*v)(r2w^ <C{\ -r)p~x f u (r?)"'v+{rwqp dôiq Hence, by Minkowski's inequality for M, s = q/p > 1, i Mp (u * v, r~) = Ms ( u * v\p, r~\ <Cil-r)p ' f ( u+ (r^y v+ irwqps dôiw -, 1/5 dôio = C i 1 - r)p~ ' Í u+ (rt)" ds{q f v+ irwqq dö {w = Cil-r)p-XMpp(u+,r)Mp(v+,r), piq which was to be proved. D The following generalization of Theorem 1.1 follows directly from Theorem 3.1. Here hp (a) = {u h (A) : Mp {u, r) = 0 ((1 - / )""), r -> 1} and A (a) = {«/t(a):mp(m,r) = 0((l-r) B),r-»l}, where a is a real number and 0 < p < oo. Thus /*p(0) = hp, while /tg(0) coincides with the space h^ which was considered by Shapiro [5].

6 134 MIROSLAV PAVLOVIC Theorem 3.2. Let 0 < p < 1 and q > p. Then (i) If ue h"ia) and v e h"{ß), then u*ve hqia + ß + l/p - 1) ; (ii) If ue hpia) iresp. u e hpia)) and v e hq{ß) iresp. v e hqqiß)), then u*v e hiiaar ß + l/p - 1). In contrast to the case of analytic functions, the spaces hpia) (0 < p < 1) need not be trivial for a < 0. For example, we have the following interesting fact. Corollary 3.1. If 1/2 < p < 1, then the space hp{l - l/p) is an infinitedimensional algebra irelative to the operation *) with the Poisson kernel as unit. Proof. That hpil - l/p) is an algebra follows from Theorem 3.2(i). The rest follows from the fact that all the functions Pw(z)=t1~ z 2, W T> 2eA, 11 - wz\ (the rotates of the Poisson kernel P Pw, w 1) belong to hp{l l/p). [5] and [4]. o See References 1. P. L. Duren, Theory of Hp spaces, Academic Press, New York, G H. Hardy and J. E. Littlewood, Some properties of conjugate functions, J. Reine Angew. Math. 167(1932), M. Pavlovic, An inequality for the integral means of a Hadamard product, Proc. Amer. Math. Soc. 103 (1988) _, Mean values of harmonic conjugates in the unit disc, Complex Variables 10 ( 1988), J. H. Shapiro, Linear topological properties ofthe harmonic Hardy class hp with 0<p< 1. Illinois J. Math. 29 (1985), Prirodno-matematicki fakultet. Institut za matematiku, Beograd, Yugoslavia