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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 $$\mathbb R^n$$. 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 $$L^p(\mathbb R)$$. More recently this definition has been generalised to $$\mathbb R^n$$ by introducing generalised conjugate harmonic functions in $$\mathbb R_+^{n+1}=\{(x,t): x\in\mathbb R^n$$, $$t>0\}$$. 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 $$L^1(\mathbb R^n)$$ such that $$R_jf\in L^1(\mathbb R^n)$$, $$j=1,\dots,n$$, where $$R_j$$ is the $$j$$-th Riesz transform. (In terms of Fourier transforms, $$(R_jf)^\wedge(y)= y_j\hat f(y)/|y|$$. When $$n=1$$ the definition says that the Hilbert transform of $$f$$ is in $$L^1$$, or equivalently $$\int f(x)\,dx=0$$ and $$f=g+\bar h$$ 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 $$\varphi\to\int f\varphi$$ (suitably interpreted if $$f\varphi\not\in L^1$$), where $$\varphi$$ is a function of bounded mean oscillation (BMO), which means that there is a constant $$C>0$$ (depending an $$\varphi$$) such that $$\int_Q |f-f_Q|\leq C|Q|$$ for any cube $$Q$$ in $$\mathbb R^n$$, where $$f_Q= |Q|^{-1}\int_Q f$$. 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 $$L^\infty$$. If $$T$$ is a convolution operator (i.e. $$Tf= K*f$$ for some distribution $$K$$ an $$\mathbb R^n$$) mapping $$L^\infty$$ 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 $$1<p<\infty$$. The condition that $$T$$ map $$L^\infty$$ 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 $$p$$ $$(0<p<\infty)$$. They define these first as spaces of harmonic functions an $$\mathbb R_+^{n+1}$$, without reference to boundary values. Specifically, a harmonic function $$u_0$$ is in $$H^p$$ if there exist harmonic functions $$u_1,\dots,u_n$$ satisfying
$\partial u_j/\partial x_i=0, \quad \sum_{i=0}^n \partial u_i/\partial x_i \quad\text{and}\quad \sup_{t>0} \int_{\mathbb R^n}|u(x,t)|^p\,dx<\infty,$
where $$|u|^2= \sum_{i=0}^n |u_i|^2$$. 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 $$p\geq (n-1)/n$$). The main result is as follows: let $$u$$ be harmonic in $$\mathbb R_+^{n+1}$$ and define $$u^*(x)= \sup_t |u(x,t)|$$. Then $$u\in H^p$$ if and only if $$u^*\in L^p$$. (For $$n=1$$ this was proved by Burkholder, Gundy and Silverstein in 1971). Finally the authors consider boundary values. If $$u\in H^p$$ then $$u(x,t)\to f(x)$$ as $$t\to0$$, in the distribution sense, where $$f$$ is a tempered distribution on $$\mathbb R^n$$. 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 $$\varphi$$ on $$\mathbb R^n$$, decreasing rapidly at $$\infty$$, with $$\int\varphi=1$$. Put $$\varphi_t(x)= t^{-n} \varphi(x/t)$$, and for any tempered disribution $$f$$ write $$f^*(x)= \sup_{t>0} |\varphi_t*f(x)|$$. Then (for $$0<p<\infty$$) $$f\in H^p$$ if and only if $$f^*\in L^p$$. (“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 $$\mathbb R^n$$. The paper concludes with a proof that certain singular integral operators map $$H^p$$ to itself.
Reviewer: Alexander M. Davie

##### MSC:
 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B25 Maximal functions, Littlewood-Paley theory 30D55 $$H^p$$-classes (MSC2000) 42A50 Conjugate functions, conjugate series, singular integrals 44A35 Convolution as an integral transform 46F10 Operations with distributions and generalized functions
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##### References:
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