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Embedding derivatives of Hardy spaces into Lebesgue spaces. (English) Zbl 0774.42011
Suppose $$0<p,q<\infty$$. Let $$u$$ be in the Hardy space $$H^ p(\mathbb{R}^ n)$$ and let $$U$$ be its harmonic extension to $$\mathbb{R}^{n+1}_ +$$. Let $$\beta$$ be a multi-index of length $$m$$ and $$D^ \beta$$ the corresponding differential monomial. For which positive measures $$\mu$$ on $$\mathbb{R}^{n+1}_ +$$ can we be sure that $$D^ \beta U$$ is in $$L^ q(\mathbb{R}^{n+1}_ +,\mu)$$?
For $$m=0$$ and $$p=q$$ this question is answered by a classical result of Carleson and for $$m=0$$, $$q>p$$ by Duren. Earlier results for general $$m$$ are due to Shirokov and the author. The remaining cases, $$0<q<p<\infty$$ and $$0<p=q<2$$ are settled here.
For fixed positive $$r$$ and for $$z\in \mathbb{R}^{n+1}_ +$$ let $$k(z)$$ be the $$\mu$$ measure of the hyperbolic ball centered at $$z$$ with radius $$r$$. Roughly, the embedding into the Lebesgue space will be bounded if and only if the function $$k$$ is in an appropriate weighted tent space.
One of the consequences of the proof is a result concerning interpolation of values at an $$\eta$$-lattice by functions of the form $$D^ \beta U$$.
In addition to the results being interesting, so are some of the technical aspects of the proof. Those include the development of some of the basics of the theory of weighted tent spaces and the use of Khinchine’s inequality to establish the necessity of the conditions.

##### MSC:
 42B30 $$H^p$$-spaces
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