Weighted Hardy inequalities, real interpolation methods and vector measures
The authors obtain a relationship between some conditions on the pair \((\rho, q)\) and the \(K\)-spaces obtained when the real interpolation method defined by a parameter function is applied to the pairs \((L^1, L^{\infty})\) and \((L^{1,\infty}, L^{\infty})\) of function spaces associated to the semivariation of a vector measure, where \(\rho\) is merely a positive measurable function defined on \((0,\infty)\). The approach is based on the relationship of the pair \((\rho, q)\) with a weighted Hardy-type inequality for non-increasing functions \(f\) \[ \int_0^{\infty}\left(\frac{1}{t}\int_0^tf(s)ds\right)^q w(t)\, dt\leq C\int_0^{\infty} f(t)^q w(t)\, dt, \;1\leq q<\infty, \] where the weights \(w\) belong to the Ariño-Muckenhoupt class, that is, \[ \int_r^{\infty}\frac{w(t)}{t^q} \, dt\leq \frac{C}{r^q}\int_0^{r}w(t)\, dt, \] for all \(r>0\) (see [\textit{M. A. Ariño} and \textit{B. Muckenhoupt}, Trans. Am. Math. Soc. 320, No. 2, 727--735 (1990; Zbl 0716.42016)] and [\textit{E. Sawyer}, Stud. Math. 96, No. 2, 145--158 (1990; Zbl 0705.42014)]).
scientific article; zbMATH DE number 6483912
scientific article