On an analog of the Muckenhoupt condition in domains with a quasiconformal boundary. (Russian. English summary) Zbl 0706.30015

Chikvinidze, L. N.

Tr. Tbilis. Mat. Inst. Razmadze 88, 41-58 (1989).

[For the entire collection see Zbl 0676.00013.]
Let w be a nonnegative measurable function in a bounded Jordan domain G with quasiconformal boundary \(\Gamma\). Let g be a quasiconformal reflection in \(\Gamma\). Let T denote the operator defined by \[ Tf(z)=- \frac{1}{\pi}\iint_{G}\frac{f(\zeta)g_{\xi}(\zeta)}{(g(\zeta)-z)^ 2}d\sigma (\zeta),\quad z\in G, \] where \(f\in L_ p(G,w):=\{f\) is measurable in G and \(\iint_{G}| f|^ pw d\sigma <\infty \}\), \(d\sigma\) denoting the Lebesgue measure on \({\mathbb{C}}.\)
Main result: If \(p>1\) then there is a positive constant \(c_ p\) such that \[ \iint_{G}| Tf(z)|^ p d\sigma (z)\leq c_ p\iint_{G}| f(z)|^ p w(z)d\sigma (z),\quad f\in L_ p(G,w) \] if and only if \[ (\frac{1}{| Q|}\iint_{Q\cap G}w d\sigma)(\frac{1}{| Q|}\iint_{Q\cap G}w^{-1/(p-1)} d\sigma)^{p-1}<c const \] for every square \(Q=[x-a,x+a]\times [y- a,y+a]\) with \(z=x+iy\in \Gamma\) and \(a>0\).

Reviewer: J.Siciak

MSC:

30C62

Quasiconformal mappings in the complex plane

Keywords:

weighted norm inequalities; quasiconformal boundary; quasiconformal reflection

Citations:

Zbl 0676.00013