On weighted iterated Hardy-type operators
Let $\mathfrak{M}$ be the set of all Lebesgue measurable functions on $[0,\infty)$, and $\mathfrak{M}^+$ be the subset of $\mathfrak{M}$ consisting of all non-negative functions. Fix $\rho,v,u\in \mathfrak{M}^+$ and $0 < r,q < \infty$. The paper characterizes the estimate $$\Vert Tf\Vert_{L^r_\rho}\le C_T\Vert f\Vert_{L^1_v}\quad \left(f\in \mathfrak{M}^+\right),$$ where $T$ is the weighted iterated Hardy-type operator defined by $$Tf(x)=\left(\int_x^\infty u(y)\left(\int_0^yf(z)\,dz\right)^q\,dy\right)^{1/q}\quad (f\in \mathfrak{M}^+,\; 0 \le x< \infty),$$ $\Vert \cdot \Vert_{L^s_w}$ represents the (quasi)-norm of the $w$-weighted Lebesgue space $L^s_w$ $(w\in \mathfrak{M}^+, \; 0 < s < \infty)$, and $C_T$ is an optimal constant independent of $f$. To this end, the above estimate is reduced to weighted inequalities on the cone of all non-decreasing functions in $\mathfrak{M}^+$.
scientific article; zbMATH DE number 6909542
scientific article