The atomic decomposition of Besov-Bergman-Lipschitz spaces

Abstract:

Let $b$ denote a special atom, $b:[ - \pi ,\pi ) \to R,\;b(t) = 1/2\pi$ or, for any interval $I{\text { in }}[ - \pi ,\pi )\;$ $b(t) = - {\left | I \right |^{ - 1/p}}\mathcal {X}R(t) + {\left | I \right |^{ - 1/p}}\mathcal {X}L(t)$ $L$ is the left half of $I$, $R$ is the right half, $\left | I \right |$ 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 \[ {B^p} = \left \{ {f:[ - \pi ,\pi ) \to R;f(t) = \sum \limits _{n = 1}^\infty {{c_n}{b_n}(t),} \sum \limits _{n = 1}^\infty {\left | {{c_n}} \right | < \infty } } \right \}\]. We endow ${B^p}$ with the norm ${\left \| f \right \|_{{B^P}}} = {\text {Inf}}\sum \nolimits _{n = 1}^\infty {\left | {{c_n}} \right |}$, 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 \[ \Lambda (\alpha ,r,s) = \left \{ {f:[ - \pi ,\pi ) \to R,{{\left \| f \right \|}_{\Lambda (\alpha ,r,s)}} = {{\left \| f \right \|}_r} + {{\left ( {\int _{ - \pi }^\pi {\frac {{{{({{\left \| {f(x + t) - f(x)} \right \|}_r})}^s}}}{{{{\left | t \right |}^{1 + \alpha s}}}}dt} } \right )}^{1/s}} < \infty } \right \}\] where $||\;|{|_r}$ 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 \[ N{\left \| f \right \|_{{B^p}}} \leq {\left \| f \right \|_{\Lambda (1 - 1/p,1,1)}} \leq M{\left \| f \right \|_{{B^p}}}\].