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Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces. (English) Zbl 1413.42040
Let $$\gamma:(0,\infty)\rightarrow(0,\infty)^d$$, $$\gamma=(\gamma_1,\dots,\gamma_d)$$, be a vector function such that:
1) $$\gamma_1(t)=t$$ $$(t>0)$$;
2) for each $$k$$ with $$2\leq k\leq d$$ the function $$\gamma_k$$ is increasing, continuous, $$\gamma_k(1)=1$$, $$\lim\limits_{t\rightarrow\infty}\gamma_k(t)=\infty$$, $$\lim\limits_{t\rightarrow 0}\gamma_k(t)=0$$ and there are constants $$c_{1,k},c_{2,k},\xi_k>1$$ for which $$c_{1,k}\gamma_k(t)\leq\gamma_k(\xi_kt)\leq c_{2,k}\gamma_k(t)$$ $$(t>0)$$.
Suppose $$\delta_1=1,$$ $$\delta_k\geq 1\;(2\leq k\leq d)$$ and $$\delta=(\delta_1,\dots,\delta_d)$$.
Denote the cone like set generated by $$\gamma$$ and $$\delta$$ as follows $\mathbb{R}^d_{\gamma,\delta}=\{(t_1,\dots,t_d)\in(0,\infty)^d:\delta_k^{-1}\gamma_k(t_k)\leq t_k\leq\delta_k\gamma_k(t_k)\;\;(1\leq k\leq d)\}.$ By $$\mathcal{I}^{\gamma,\delta}$$ denote the family of all $$d$$-dimensional intervals $$I=I_1\times\dots\times I_d$$ with $$(|I_1|,\dots,|I_d|)\in\mathbb{R}^d_{\gamma,\delta}$$ and having the centre at the origin.
The the cone-like maximal operator generated by $$\gamma$$ and $$\delta$$ is defined as follows $M^{\gamma,\delta}(f)(x)=\sup_{I\in\mathcal{I}^{\gamma,\delta}}\frac{1}{|x+I|}\int_{x+I}|f|\;\;\;\;\;(f\in L(\mathbb{R}^d),x\in\mathbb{R}^d).$ In the paper some results on the boundedness of classical Hardy-Littlewood maximal operator in variable Lebesgue spaces are extended on cone-like maximal operators. In particular, the following results are proved:
a) Let an exponent function $$p(\cdot):\mathbb{R}^d\rightarrow[1,\infty)$$ is such that $$1/p$$ is both locally log-continuous and log-continuous at infinity. Then the cone-like maximal operator $$M^{\gamma,\delta}$$ satisfies the following weak type estimation $\sup_{\lambda>0}\|\lambda\chi_{\{M^{\gamma,\delta}(f)>\lambda\}}\|_{p(\cdot)}\leq C \| f\|_{p(\cdot)}\;\;\;\;(f\in L^{p(\cdot)}(\mathbb{R}^d)).$ b) Let an exponent function $$p(\cdot):\mathbb{R}^d\rightarrow[1,\infty)$$ is such that $$1/p$$ is both locally log-continuous and log-continuous at infinity and $$\inf p>1$$. Then the cone-like maximal operator $$M^{\gamma,\delta}$$ satisfies the following strong type estimation $\| M^{\gamma,\delta}(f)\|_{p(\cdot)}\leq C\| f\|_{p(\cdot)}\;\;\;\;(f\in L^{p(\cdot)}(\mathbb{R}^d)).$ Note that a Besicovitch type covering result (see Theorem 3.1) proved in the paper is a particular case of the corresponding one proved by M. de Guzman in the work [Stud. Math. 34, 299–317 (1970; Zbl 0192.48804)] (see Corollary 1.7).

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42B35 Function spaces arising in harmonic analysis 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry)
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