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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.

##### MSC:
 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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