Double-weighted estimates of the integration operator for classes \(\Phi(L)\) (Q1326084)

Let \(P\) be the integration operator \(Px(t)= \int^ t_ 0 x(s)ds\), where \(x\) is a measurable function. A subset \(M\) of a set of measurable functions is said to be an ideal if it follows from the fact that \(x(t)\in M\) and \(| q(t)|\leq 1\) almost everywhere that \(x(t) q(t)\in M\). Suppose that \(\Phi: \mathbb{R}\to \mathbb{R}_ +\) is an even function which is increasing on \(\mathbb{R}_ +\) such that \(\Phi(0)= 0\). The following problems are solved: for an ideal set \(M\) find necessary and sufficient conditions on a pair of weights \((w, v)\) under which the following estimates are always fulfilled \[ \int^ \infty_ 0 \Phi(Px(t)) w(t)dt\leq c_ 0\int^ \infty_ 0 \Phi(x(t)) v(t)dt\text{ or }\int_{\{t: Px(t)> \lambda\}} w(t)dt\leq {c_ 1\over \Phi(\lambda)} \int^ \infty_ 0 \Phi(x(t)) v(t)dt \] with constants \(c_ 0\) and \(c_ 1\) independent of \(x\) and \(M\), \(\lambda\in \mathbb{R}_ +\).