Estimates of oscillations of the Hardy transform (Q1810199)

The author considers the linear transform \((Pf) (x) = {1\over x}\int_{0}^{x} f(t) dt\), \(x > 0\), of locally Lebesgue integrable functions on \(R_{ + }= [0, + \infty)\). He calls it Hardy transform. Let us set \(\|f\|_{\text{BLO}}=\sup\limits_I L(f;I)\), where \(L(f;I) = {1\over\text{ mes} I}\int_{I} f(x) dx - \text{ess}\inf\limits_{x \in I} f(x) \) and \(I\) is an interval, \(I \subset R_{ + } \).The class \(\text{BLO}\) consists of all functions \(f\) for which \(\|f\|_{\text{BLO}}< + \infty \). This class is nonlinear. Theorem 1. If the function \(f\) is nonincreasing on \(R_{ + } \) then \({1\over e}\|f\|_{\text{BLO}}\leq \|Pf\|_{\text{BLO}}\leq \|f\|_{\text{BLO}} \) and the constants in both sides of these inequalities are best possible. In Theorem 2 the best possible value of the constant \(C > 0\) in the inequality \(\|Pf\|_{\text{BMO}}\geq C\|f\|_{\text{BLO}}\) for nonincreasing functions \(f\) on \(R_+\) is obtained. Also, the inequality \(\|Pf\|_{\text{BMO}}\geq{e\alpha _{0}\over 4}\|f\|_{\text{BMO}}\) for such functions is proved, where \(\alpha_0\approx 0.52\).