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

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^* (τ)]^p\,dτ\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 $f^{\ast }$ 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 $$(T_{u,b}g)(t):=\underset{t\le \tau <\infty }{\sup }\frac{u(\tau )}{B(\tau )}{\int }_0^{\tau }g(s)b(s)ds,t\in (0,\infty ),$$ 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 $M_{\varphi ,{\Lambda }^{\alpha }(b)}$, defined with $0<\alpha <\infty$ and functions $b,\varphi :(0,\infty )\to (0,\infty )$ for all measurable functions $f$ on ${\mathbb{R}}^n$ by \begin{equation*} M_{ϕ,Λ^α(b)}f(x) : = \sup_{Q \ni x} \frac{\|f χ_Q\|_{Λ^α(b)}}{ϕ(|Q|)}, \qquad x \in {\mathbb R}^n, \end{equation*} from $\mathrm{G\Gamma }(p_1,m_1,v)$ into $\mathrm{G\Gamma }(p_2,m_2,w)$. Here ${\Lambda }^{\alpha }(b)$ and $\mathrm{G\Gamma }(p,m,w)$ are the classical and generalized Lorentz spaces, respectively.

41 pages