On some restricted inequalities for the iterated Hardy-type operator involving suprema and their applications

TL;DR

This paper characterizes certain inequalities involving iterated Hardy-type operators with suprema, providing conditions for their validity and applying results to compute norms of generalized maximal operators in Lorentz spaces.

Contribution

The paper offers new characterizations of inequalities for iterated Hardy-type operators involving suprema and applies these to determine norms of generalized maximal operators in Lorentz spaces.

Findings

Derived necessary and sufficient conditions for the inequalities to hold.

Calculated the norm of a generalized maximal operator between Lorentz spaces.

Extended classical results to more general weighted and rearrangement-invariant settings.

Abstract

In this paper we characterize the inequality \begin{equation*} \bigg( \int_0^{\infty} \bigg( \int_0^x \big[ T_{u,b}f^* (t)\big]^r\,dt\bigg)^{\frac{q}{r}} w(x)\,dx\bigg)^{\frac{1}{q}} \le C \, \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (\tau)]^p\,d\tau \bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} \end{equation*} for $1 < m < p \le r < q < \infty$ or $1 < m \le r < \min\{p,q\} < \infty$, where $w$ and $v$ are weight functions on $(0,\infty)$. The inequality is required to hold with some positive constant $C$ for all measurable functions defined on measure space $({\mathbb R}^n,dx)$. Here $f^*$ is the non-increasing rearrangement of a measurable function $f$ defined on ${\mathbb R}^n$ and $T_{u,b}$ is the iterated Hardy-type operator involving suprema, whish is defined for a measurable non-negative function $f$ on $(0,\infty)$ by $$ (T_{u,b} g)(t) : = \sup_{t \le \tau < \infty} \frac{u(\tau)}{B(\tau)} \int_0^{\tau} g(s)b(s)\,ds,\qquad t \in (0,\infty), $$ where $u$ and $b$ are two weight functions on $(0,\infty)$ such that $u$ is continuous on $(0,\infty)$ and the function $B(t) : = \int_0^t b(s)\,ds$ satisfies $0 < B(t) < \infty$ for every $t \in (0,\infty)$. At the end of the paper, as an application of obtained results, we calculate the norm of the generalized maximal operator $M_{\phi,\Lambda^{\alpha}(b)}$, defined with $0 < \alpha < \infty$ and functions $b,\,\phi: (0,\infty) \rightarrow (0,\infty)$ for all measurable functions $f$ on ${\mathbb R}^n$ by \begin{equation*} M_{\phi,\Lambda^{\alpha}(b)}f(x) : = \sup_{Q \ni x} \frac{\|f \chi_Q\|_{\Lambda^{\alpha}(b)}}{\phi (|Q|)}, \qquad x \in {\mathbb R}^n, \end{equation*} from ${\operatorname{G\Gamma}}(p_1,m_1,v)$ into ${\operatorname{G\Gamma}}(p_2,m_2,w)$. Here $\Lambda^{\alpha}(b)$ and ${\operatorname{G\Gamma}}(p,m,w)$ are the classical and generalized Lorentz spaces, respectively.