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Derivatives of Hardy Functions

Boo Rim Choe
Proceedings of the American Mathematical Society
Vol. 110, No. 3 (Nov., 1990), pp. 781-787
DOI: 10.2307/2047921
Stable URL: http://www.jstor.org/stable/2047921
Page Count: 7
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Abstract

Let B be the open unit ball of $\operatorname{C}^n$, and set S = ∂ B. It is shown that if $\varphi \in L^p(S), \varphi > 0$, is a lower semicontinuous function on S and $1/q > 1 + 1/p$, then, for a given $\varepsilon > 0$, there exists a function f ∈ Hp(B) with f(0) = 0 such that |f*| = φ almost everywhere on S and $\int_B|\nabla f|^q dV < \varepsilon$ where V denotes the normalized volume measure on B.

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