Discretization and antidiscretization of Lorentz norms with no restrictions on weights

TL;DR

This paper advances the discretization of weighted Lorentz norms by removing previous restrictions on weights, providing new estimates for the optimal constants in related inequalities, including cases with weights previously excluded.

Contribution

It introduces a restriction-free discretization method for weighted Lorentz norms and derives equivalent estimates for optimal constants in associated inequalities.

Findings

Eliminated non-degeneracy restrictions on weights in Lorentz norm discretization.

Provided new bounds for the optimal constant C in key inequalities.

Extended results to cases with weights previously excluded by restrictions.

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 $$\left( \int_0^L (f^*(t))^{p_2} w(t)\,\mathrm{d}t \right)^\frac 1{p_2} \le C \left( \int_0^L \left( \int_0^t u(s)\,\mathrm{d}s \right)^{-\frac {p_1}\alpha} \left( \int_0^t (f^*(s))^\alpha u(s) \,\mathrm{d}s \right)^\frac {p_1}\alpha v(t) \,\mathrm{d}t \right)^\frac 1{p_1}$$ holds for all relevant measurable functions, where $L\in(0,\infty]$, $\alpha, p_1, p_2 \in (0,\infty)$ and $u$, $v$, $w$ are locally integrable weights, $u$ being strictly positive. It 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 $p_1=\infty$ is also presented.