Brownian martingales and some new results on interpolation of Hardy spaces
The following result is proved: If \(f\in H_1(\mathbb{R}^n) +L_\infty (\mathbb{R}^n)\), then there exist two linear operators \(U\) and \(V\) such that \(Uf=Nf\), \(U\) is contractive from \(H_1(\mathbb{R}^n)\) to \(L_1(\mathbb{R}^n)\) and also from \(L_\infty (\mathbb{R}^n)\) to \(L_\infty (\mathbb{R}^n)\) and, moreover, \(V(Nf)=f\) and \(V\) is bounded from \(L_1(\mathbb{R}^n)\) to \(H_1(\mathbb{R}^n)\) and from \(L_\infty(\mathbb{R}^n)\) to \(L_\infty (\mathbb{R}^n)\), where \(Nf\) denotes the nontangential maximal function of \(f\). From this it easily follows that the couple \((H_1(\mathbb{R}^n)\), \(L_\infty (\mathbb{R}^n))\) is a Calderón couple. These results are proved with the help of Brownian motion and Hardy spaces containing continuous martingales. Similar results are proved for the couple \((L_p(\mathbb{R}^n)\), BMO\((\mathbb{R}^n))\) \((1<p<\infty)\).
scientific article; zbMATH DE number 1385173