Fractional Derivative Description of the Bloch Space

Abstract

AbstractWe establish new characterizations of the Bloch space $$\mathcal {B}$$ B which include descriptions in terms of classical fractional derivatives. Being precise, for an analytic function $$f(z)=\sum _{n=0}^\infty \widehat{f}(n) z^n$$ f ( z ) = ∑ n = 0 ∞ f ^ ( n ) z n in the unit disc $$\mathbb {D}$$ D , we define the fractional derivative $$ D^{\mu }(f)(z)=\sum \limits _{n=0}^{\infty } \frac{\widehat{f}(n)}{\mu _{2n+1}} z^n $$ D μ ( f ) ( z ) = ∑ n = 0 ∞ f ^ ( n ) μ 2 n + 1 z n induced by a radial weight $$\mu $$ μ , where $$\mu _{2n+1}=\int _0^1 r^{2n+1}\mu (r)\,dr$$ μ 2 n + 1 = ∫ 0 1 r 2 n + 1 μ ( r ) d r are the odd moments of $$\mu $$ μ . Then, we consider the space $$ \mathcal {B}^\mu $$ B μ of analytic functions f in $$\mathbb {D}$$ D such that $$\Vert f\Vert _{\mathcal {B}^\mu }=\sup _{z\in \mathbb {D}} \widehat{\mu }(z)|D^\mu (f)(z)|<\infty $$ ‖ f ‖ B μ = sup z ∈ D μ ^ ( z ) | D μ ( f ) ( z ) | < ∞ , where $$\widehat{\mu }(z)=\int _{|z|}^1 \mu (s)\,ds$$ μ ^ ( z ) = ∫ | z | 1 μ ( s ) d s . We prove that $$\mathcal {B}^\mu $$ B μ is continously embedded in $$\mathcal {B}$$ B for any radial weight $$\mu $$ μ , and $$\mathcal {B}=\mathcal {B}^\mu $$ B = B μ if and only if $$\mu \in \mathcal {D}=\widehat{\mathcal {D}}\cap \check{\mathcal {D}}$$ μ ∈ D = D ^ ∩ D ˇ . A radial weight $$\mu \in \widehat{\mathcal {D}}$$ μ ∈ D ^ if $$\sup _{0\le r<1}\frac{\widehat{\mu }(r)}{\widehat{\mu }\left( \frac{1+r}{2}\right) }<\infty $$ sup 0 ≤ r < 1 μ ^ ( r ) μ ^ 1 + r 2 < ∞ and a radial weight $$\mu \in \check{\mathcal {D}}$$ μ ∈ D ˇ if there exist $$K=K(\mu )>1$$ K = K ( μ ) > 1 such that $$\inf _{0\le r<1}\frac{\widehat{\mu }(r)}{\widehat{\mu }\left( 1-\frac{1-r}{K}\right) }>1.$$ inf 0 ≤ r < 1 μ ^ ( r ) μ ^ 1 - 1 - r K > 1 .