Balayage of Carleson Measures and Hankel Operators on Generalized Hardy Spaces

Abstract

IntroductionLet D denote the unit disc in the complex plane and T the unit circle with normalized HAAR measure do. For 0 < p < co, H P denotes the usual HARDY space of analytic functions in D. Let P be the orthogonal projection from L2( T) onto H 2 . If b is a holomorphic function in D, we define the HANKEL operator with symbol b as a formal FOURIER series by(We are defining Hb as an anti-linear operator only as a matter of convenience). NEHARI (see [S]) proved that Hb extends to a bounded operator on H 2 iff b E BMO. This result is a consequence of duality and RIESZ factorization and also holds for H P if 1 < p < co. JANSON, PEETRE and SEMMES [7] proved that Hb extends to a continuous operator on H P , 0 < p < 1 iff b belongs to the LIPSCHITZ space A'/,-'; and when p = 1, iff b E LMO (Logarithmic Mean Oscillation; see [lo]). This was independently proven by TOLOKONNIKOV [12], who also showed, using methods of functional analysis, that, if b E A",-', then Hb extends to a continuous operator from H P to Hieak and that Hb is bounded from H P (0 < p < 1) to H ' iff b belongs to a generalized LIP~CHITZ Space A, with e(t) = t",-'/log (llt) (for the definition of A, see [9] or [5]). Another proof of these results, which extends to some of the HARDY spaces H, defined by JANSON in [5], was given in [l]. Our aim in this paper is to give a new proof of these results which generalizes easily to all JANSON'S H, spaces and relies on balayage of CARLESON measures. More precisely, we first prove (for the definitions, see section 2) the following theorem, where e denotes a growth function of upper type less than 1. Theorem 1. Let H , be a generalized HARDY space with dual space BMO,. Then Hb extends to a continuous operator from H , into Hkeak iff b E BMO,. Moreover, i f b E BMO,, there exists a continuous operator %from H, into L' such that Hb = P Q T.The operator & is obtained as a balayage with respect to the POISSON kernel for the disc of a generalized CARLESON measure, using a representation theorem for f E BMO, due to SMITH [lo] and Theorem 1 follows easily from the generalized CARLESON property. More generally, we find necessary and sufficient conditions on the symbol b so that the HANKEL operator Hb maps one H,, space to another HV2. This gives Theorem 2 (section 3). The key

On the Product of Functions in BMO and H^\text{1}

“…These spaces have previously been dealt with by many authors, see [6,35,57] and further references given there. We shall make substantial use of the following weighted Hardy-Orlicz space:…”

On the Duality of Hardy Spaces NearH1

“…Theories of harmonic analysis, and in particular Hardy spaces, which use norms more general than those of the Lebesgue L p spaces have a long history [19,3,14,15,7,8,16,23,20,11,12,21,1,2]. Although we are concerned here with spaces which behave like L 1 under the action of maximal operators and singular integrals there would be nothing to gain by restricting the remarks which follow to that setting.…”

“…Smith made a regularity assumption on the growth function \ in order to obtain his representation theorem. In [2] (final remark) it is observed that this restriction can be removed, thereby describing the dual of still more nonconvex Hardy spaces as comprising the balayages of a class of measures.…”

“…This particular description of the dual of the Hardy space H 1 has received a number of very different treatments from a variety of authors [5,10,13]. The previous work concerned with generalisations of this theorem [2,11,21] has treated only the case of nonconvex Hardy spaces. Here we are able to extend Fefferman's theorem to an important class of Banach spaces in a manner which sheds light upon the original result at the level of the function norm itself.…”

“…Since each of these two conditions is characterised in terms of the behaviour of the complementary Young's function it will make matters more straightforward in what follows if we work backwards from the corresponding associated spaces. Consider the examples of the following Young's functions 9(x) x p , p< ; e log x log (n) x , n 3; e log x(log (2) x) p , p<1; e (log x log (2) xÂlog (n) x) , n 3; e $ log x log (2) x , $>0; e (log x) p , p>1; e x : , 0<:< ; exp (2) x, ..., exp (n) x, ..., higher forms, ..., where log (n) x=log log } } } log x and exp (n) x=exp exp } } } exp x (n times). Note that only the values of these functions for large x are relevant.…”