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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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