English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 weightfunctions on $(0,\infty)$. The inequality is required to hold with somepositive constant $C$ for all measurable functions defined on measure space$({\mathbb R}^n,dx)$. Here $f^*$ is the non-increasing rearrangement of ameasurable function $f$ defined on ${\mathbb R}^n$ and $T_{u,b}$ is theiterated Hardy-type operator involving suprema, whish is defined for ameasurable 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 weightfunctions on $(0,\infty)$ such that $u$ is continuous on $(0,\infty)$ and thefunction $B(t) : = \int_0^t b(s)\,ds$ satisfies $0 < B(t) < \infty$ for every$t \in (0,\infty)$.  At the end of the paper, as an application of obtained results, we calculatethe 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 Lorentzspaces, respectively.

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