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On weighted spaces of harmonic and holomorphic functions. (English) Zbl 0823.46025
This paper is concerned with the following weighted spaces of harmonic and holomorphic functions on the complex unit disc \(D\). Let \(v: [0, 1]\to [0, \infty[\) be a continuous, non-increasing function with \(v(1)= 0\) and \(v(r)> 0\) for all \(r\neq 1\). For \(f: D\to {\mathbb{C}}\) put \[ \| f\|_ v= \sup_{| z|< 1} | f(z)| v(| z|) \] and define \[ hv_ 0= \bigl\{f: D\to {\mathbb{C}}\text{ harmonic}: \lim_{| z|\to 1} | f(z)| v(| z|)= 0\text{ (uniform limit)}\bigr\}, \]
\[ Hv_ 0= \bigl\{f: D\to {\mathbb{C}}\text{ holomorphic }: \lim_{| z|\to 1} | f(z)| v(| z|)= 0\text{ (uniform limit)}\bigr\}. \] We deal with the question when these spaces are isomorphic to \(c_ 0\) and under which conditions on \(v\) the space \(Hv_ 0\) is complemented in \(hv_ 0\). It turns out that \(hv_ 0\) is always isomorphic to \(c_ 0\) provided \[ \inf_ n\;{v(1- 2^{- n- 1})\over v(1- 2^{- n})}> 0. \] Moreover, under this hypothesis we can determine precisely when \(Hv_ 0\) is isomorphic to \(c_ 0\). This is the case if and only if \(\limsup_{n\to\infty} {v(1- 2^{- n- k})\over v(1- 2^{- n})}< 1\) for some positive integer \(k\). Furthermore, we relate this condition to the growth of the trigonometric conjugate of a function in \(hv_ 0\). We obtain that the Riesz projection, regarded as an operator on \(hv_ 0\), is bounded if and only if this latter condition holds.

46E15 Banach spaces of continuous, differentiable or analytic functions
46B03 Isomorphic theory (including renorming) of Banach spaces
42A05 Trigonometric polynomials, inequalities, extremal problems
42A50 Conjugate functions, conjugate series, singular integrals
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