On weak type inequalities for dyadic maximal functions
Let \(M_df(x)\) be the dyadic maximal function on the euclidean space \(\mathbb R^n\). For \(p>1\) and a cube \(Q\in\mathbb R^n\), let \(\|M_df\|_{L^{p,\infty}(Q)} =\sup_{\lambda>0}\{\lambda \mid \{M_df(x)>\lambda\}\cap Q|^{1/p}\}\) be the local weak-\(L^p\) quasi-norm. The authors consider an extremum problem (or a Bellman function) \(\mathcal B_{p,\infty}(F,f,L)\) \(=\sup\Bigl\{\frac{1}{|Q|}\|M_d\phi\|_{L^{p,\infty}(Q)}^p; \roman{Av}_Q(\phi^p)=F,\, \roman{Av}_Q(\phi)=f,\,\sup_{R;Q\subset R} \roman{Av}_R (\phi)=L\Bigr\}, \) where \(\roman{Av}_Q\,g=\frac{1}{|Q|}\int_Q g(x)\,dx\). They determine the above function and discuss sharp refined weak type estimates for the dyadic maximal functions. To determine the above function, they treat more general families than the family of dyadic cubes, i.e. ``tree'' which has the fundamental properties of dyadic cubes. For these trees, the authors formulate extremum problems and solve them.
scientific article; zbMATH DE number 5349100