Two more characterizations of Besov-Bergman-Lipschitz spaces (Q1084297)

For \(0<\alpha <1\), \(1<r,s<\infty\), let \(\Lambda (\alpha,r,s)=\{f: [0,2\pi]\to {\mathbb{R}}\); \(\| f\|_{\Lambda (\alpha,r,s)}=\| f\|_ r+(\int^{\pi}_{-\pi}\frac{(\| f(x+t)-f(x)\|_ r)^ s}{| t|^{1+\alpha s}}dt)<\infty \}\) where \(\| \|_ r\) is the Lebesgue \(L^ r\)-norm. Also define the spaces \(J^ p=\{g: D\to C\); analytic in D; \(\| g\|_{J^ p}=| g(0)| +(1/\pi)\int^{1}_{0}\int^{\pi}_{-\pi}| g'(re^{i\theta})| (1-r)^{(1/p)-1} d\theta dr<\infty \}\) for \(1\leq p<\infty\). The dash means derivative, D is the disc in the complex plane. These spaces are well known as Besov-Bergman-Lipschitz spaces. Define the spaces \(I^ p\) for \(1<p<\infty\) by \(I^ p=\{f: [-\pi,\pi]\to R\); \(f(t)=\sum^{\infty}_{i=1}f_ i(t)\) such that \(\sum^{\infty}_{i=1}\| f_ i\|_{L(p,1)}<\infty \}\) where \(f_ i\) is an even decreasing function and \(\| \|_{L(p,1)}\) is the Lorentz space. We endow \(I^ p\) with the norm \(\| f\|_{I^ p}=Inf\sum^{\infty}_{i=1}\| f_ i\|_{L(p,1)}\), where the infimum is taken over all possible representations of f. Similarly, we define the space \(G^ p\) as the space \(I^ p\) when we replace \(f_ i\) by odd-decreasing functions, then we have Theorem: The spaces \(I^ p\), \(G^ p\), \(J^ p\) and \(\Lambda\) (1- (1/p),1,1) are isomorphic as Banach spaces and these norms are equivalent.