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H p spaces of several variables. (English) Zbl 0257.46078

This paper is a major contribution to the study of H p spaces, singular integrals, and harmonic analysis an n . Classically the theory of H p spaces arose from analytic function theory. H p was defined as the space of analytic functions in the upper half plane with boundary values in L p (). More recently this definition has been generalised to n by introducing generalised conjugate harmonic functions in + n+1 ={(x,t):x n , t>0}. The authors present several intrinsic descriptions of H p , of a real variable nature, not involving conjugate functions. These results greatly clarify the meaning of H p , as well as throwing new light on the behaviour of convolution operators an L p . The following is a summary of some of the main results. The first main result is the description of the dual of H 1 . H 1 is the Banach space of all functions f in L 1 ( n ) such that R j fL 1 ( n ), j=1,,n, where R j is the j-th Riesz transform. (In terms of Fourier transforms, (R j f) (y)=y j f ^(y)/|y|. When n=1 the definition says that the Hilbert transform of f is in L 1 , or equivalently f(x)dx=0 and f=g+h ¯ where g and h are in the classical “analytic” H 1 ). The authors prove that the dual of H ' is the space of all functionals of the form φfφ (suitably interpreted if fφL 1 ), where φ is a function of bounded mean oscillation (BMO), which means that there is a constant C>0 (depending an φ) such that Q |f-f Q |C|Q| for any cube Q in n , where f Q =|Q| -1 Q f. The proof of this remarkable and deep result depends an an inequality of Carleson and some clever manipulations of Littlewood Paley functions. The hard part is to show that a BMO function defines a bounded functional on H 1 . The essential difficulties are already present in the case n=1. This duality leads to a new approach to convolution operators which brings out the usefulness of BMO as a substitute for L . If T is a convolution operator (i.e. Tf=K*f for some distribution K an n ) mapping L into BMO, then the authors show, using the duality and a description of L p related to BMO, that T maps H 1 into H 1 , BMO into BMO, and L p into L p for 1<p<. The condition that T map L into BMO is relatively easy to verify for singular integral operators of Calderón-Zygmund type. A more refined version of this result, involving Calderón’s complex method of interpolation, enables the authors to prove new results on L p multipliers. The authors then turn to H p spaces for general p (0<p<). They define these first as spaces of harmonic functions an + n+1 , without reference to boundary values. Specifically, a harmonic function u 0 is in H p if there exist harmonic functions u 1 ,,u n satisfying

u j /x i =0, i=0 n u i /x i andsup t>0 n |u(x,t)| p dx<,

where |u| 2 = i=0 n |u i | 2 . This definition is appropriate if p>(n-1)/n. In general a more elaborate version (here omitted) is needed (the point is that |u| p is subharmonic only if p(n-1)/n). The main result is as follows: let u be harmonic in + n+1 and define u * (x)=sup t |u(x,t)|. Then uH p if and only if u * L p . (For n=1 this was proved by Burkholder, Gundy and Silverstein in 1971). Finally the authors consider boundary values. If uH p then u(x,t)f(x) as t0, in the distribution sense, where f is a tempered distribution on n . Denote the set of such f also by H p . Then H p =L p for p>1 and for p=1 this definition is consistent with the earlier one. The last result above characterises H p in terms of Poisson integrals (as u is the Poisson integral of f). The authors show that the Poisson kernel can be replaced by any smooth approximate identity – more precisely, fix a smooth function φ on n , decreasing rapidly at , with φ=1. Put φ t (x)=t -n φ(x/t), and for any tempered disribution f write f * (x)=sup t>0 |φ t *f(x)|. Then (for 0<p<) fH p if and only if f * L p . (“Non-tangential” versions of this and the preceding result are also given). This result implies for example that one could define H p in terms of the wave equation rather than Laplace’s and get the same space of functions on n . The paper concludes with a proof that certain singular integral operators map H p to itself.


MSC:
46J15Banach algebras of differentiable or analytic functions, H p -spaces
46E30Spaces of measurable functions
42B25Maximal functions, Littlewood-Paley theory
30D55H (sup p)-classes (MSC2000)
42A50Conjugate functions, conjugate series, singular integrals, one variable
44A35Convolution (integral transforms)
46F10Operations with distributions (generalized functions)
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