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The atomic decomposition of Besov-Bergman-Lipschitz spaces


Author: Geraldo Soares De Souza
Journal: Proc. Amer. Math. Soc. 94 (1985), 682-686
MSC: Primary 46E35; Secondary 42C15
DOI: https://doi.org/10.1090/S0002-9939-1985-0792283-5
MathSciNet review: 792283
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Abstract: Let $ b$ denote a special atom, $ b:[ - \pi ,\pi ) \to R,\;b(t) = 1/2\pi $ or, for any interval $ I{\text{ in }}[ - \pi ,\pi )\;$ $ b(t) = - {\left\vert I \right\vert^{ - 1/p}}\mathcal{X}R(t) + {\left\vert I \right\vert^{ - 1/p}}\mathcal{X}L(t)$ $ L$ is the left half of $ I$, $ R$ is the right half, $ \left\vert I \right\vert$ denotes the length of $ I$ and $ \mathcal{X}E$ the characteristic function of $ E$. For $ 1/2 < p < \infty $, let $ ({b_n})$ be special atoms and $ ({c_n})$ a sequence of real numbers; then we define the space

$\displaystyle {B^p} = \left\{ {f:[ - \pi ,\pi ) \to R;f(t) = \sum\limits_{n = 1... ...sum\limits_{n = 1}^\infty {\left\vert {{c_n}} \right\vert < \infty } } \right\}$

. We endow $ {B^p}$ with the norm $ {\left\Vert f \right\Vert _{{B^P}}} = {\text{Inf}}\sum\nolimits_{n = 1}^\infty {\left\vert {{c_n}} \right\vert} $, where the infimum is taken over all possible representations of $ f$.

In the early 1960s, the following spaces were introduced, now known as Besov-Bergman-Lipschitz spaces. For $ 0 < \alpha < 1$, $ 1 \leq r$, $ s \leq \infty $, let

$\displaystyle \Lambda (\alpha ,r,s) = \left\{ {f:[ - \pi ,\pi ) \to R,{{\left\V... ...\vert t \right\vert}^{1 + \alpha s}}}}dt} } \right)}^{1/s}} < \infty } \right\}$

where $ \vert\vert\;\vert{\vert _r}$ is the Lebesgue space $ {L^r}$-norm.

Now we write down the main theorem of this paper which is as follows.

THEOREM $ f \in {B^P}$ for $ 1 < p < \infty $ if and only if $ f \in \Lambda (1 - 1/p,1,1)$. Moreover, there are absolute constants $ M$ and $ N$ such that

$\displaystyle N{\left\Vert f \right\Vert _{{B^p}}} \leq {\left\Vert f \right\Vert _{\Lambda (1 - 1/p,1,1)}} \leq M{\left\Vert f \right\Vert _{{B^p}}}$

.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0792283-5
Keywords: Besov-Bergman-Lipschitz spaces, equivalence of Banach spaces, analytic functions, atomic decomposition
Article copyright: © Copyright 1985 American Mathematical Society