Discretization and antidiscretization of Lorentz norms with no restrictions on weights

Abstract

We improve the discretization technique for weighted Lorentz norms by eliminating all “non-degeneracy” restrictions on the involved weights. We use the new method to provide equivalent estimates on the optimal constant C such that the inequality

$$\begin{aligned} \left( \int _0^L (f^*(t))^{q} w(t)\,\mathrm {d} t\right) ^\frac{1}{q} \le C \left( \int _0^L \left( \int _0^t u(s)\,\mathrm {d} s\right) ^{-p} \left( \int _0^t f^*(s) u(s) \,\mathrm {d} s\right) ^p v(t) \,\mathrm {d} t\right) ^\frac{1}{p} \end{aligned}$$

holds for all relevant measurable functions, where $L\in (0,\infty ]$, $p, q \in (0,\infty )$ and u, v, w are locally integrable weights, u being strictly positive. In the case of weights that would be otherwise excluded by the restrictions, it is shown that additional limit terms naturally appear in the characterizations of the optimal C. A weak analogue for $p=\infty$ is also presented.