Weak type estimates for averaging operators (Q2712731)

The present paper is concerned with the study of boundedness on a non-normable weak type space of the average operator of a family of functions with quasinorm uniformly bounded in a fixed space \(E\). The main result is the following:NEWLINENEWLINENEWLINELet \((N,P)\) be a probability measure space and consider a measurable function \(f: N\times M\to\mathbb{R}\). Define the average operator NEWLINE\[NEWLINETf(x)= \int_N f(x,\theta) dP(\theta).NEWLINE\]NEWLINE Let \(({\mathcal M},\mu)\) be a \(\sigma\)-finite measure space and \(W\) a nonnegative function on \((0,\infty)\) such that \(W(0)= 0\). Define NEWLINE\[NEWLINEL_{W,\infty}(\mu)= \Biggl\{f:\|f\|_{L_{W,\infty}(\mu)}= \sup_y W(y)\lambda_f(y)< \infty\Biggr\}NEWLINE\]NEWLINE and \(\widetilde W(R)= \sup_{x\leq R} {R-x\over \int^R_x{1\over W(u)} du}\).NEWLINENEWLINENEWLINEIf there exists \(C> 0\) such that \(\|f(\theta,\cdot)\|_{L_{W,\infty}(\mu)}\leq C\), then \(\|Tf\|_{L_{\widetilde W,\infty}}\leq C\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00028].