Maximal averages over linear and monomial polyhedra
Let \(\mathfrak P=\{\mathbf p_1,\dots,\mathbf p_d\}\) be a spanning set of vectors in \(\mathbb R^n\), let \(F_p(\mathbf x)=x_1^{p_1}\dots x_n^{p_n}\), where \(\mathbf p=(p_1,\dots,p_n)\), and consider the set of monomials \(\mathcal P=\{F_{\mathbf p_1}, \dots ,F_{\mathbf p_d}\}\). Let \(\mathcal O_n=\{\mathbf x=(x_1,\dots ,x_n)\in \mathbb R^n\mid x_j >0,1\leq j \leq n\}\). For every \(\mathbf x\in \mathcal O_n\) and every \(\overline \delta =(\delta_1,\dots ,\delta_d)\) with \(\delta_j>0, 1\leq j \leq d\), define \(B_{\mathcal P}(\mathbf x;\overline \delta)\) be the closure of the connected component of the interior of the set \(\{\mathbf y\in \mathcal O_n \mid |F_{p_j}(\mathbf y)-F_{pj}(\mathbf x)|\leq \delta_j, 1\leq j\leq d\}\) which contains \(\mathbf x\). Define the maximal operator \[ \mathcal M_{\mathcal P}[f](\mathbf x)=\sup_{\overline \delta}|B_{\mathcal P}(\mathbf x;\overline \delta)|^{-1}\int_{B_{\mathcal P}(\mathbf x;\overline \delta)}|f(\mathbf y)|d\mathbf y. \] The goal of this paper is to establish the \(L^p\) boundedness of this maximal operator. In fact, the authors also consider the \(L^p\) boundedness of the following more general maximal operators: Let \[ \mathbb W_{\mathfrak P}(\overline \alpha, \overline \beta)=\{\mathbf x \in \mathcal O_n \mid \alpha_j \leq F_{\mathbf p}(x) \leq \beta_j,1 \leq j \leq d\} \] and define \[ \mathcal Mf(\mathbf x) = \sup _{\mathbb W_{\mathfrak P}}\mu_{\mathbf e}(\mathbb W_{\mathfrak P})^{-1}\int_{\mathbb W_{\mathfrak P}}|f(\mathbf {xy})|d\mu_{\mathbf e}(\mathbf y). \] The idea is to construct a superset \(\mathcal N\) of \(\mathbb W\) such that \(\mathcal N\) and \(\mathbb W\) have comparable measures, and the maximal operator defined by the \(\mathcal N\)'s are comparable with the classical strong maximal operators. The construction requires a detailed study of the geometric properties of these monomial polyhedra. With some restriction on the monomial polyhedra, the authors establish the following main results of the paper. Theorem 7.1. Fix a spanning set of vectors \(\mathfrak P=\{\mathbf p_1,\dots,\mathbf p_d\}\) in \(\mathbb R^n\), fix a vector \(\mathbf e \in \mathbb R^n\), and fix small constants \(0 < \kappa_0,\epsilon_0 <1\), Let \(\mathcal G\) denote the collection of all monomial \(\mathfrak P\)-polyhedra in \(\mathcal O_n\) that are \((\kappa_0,\epsilon_0)\)-admissible. Define \[ \mathcal Mf(\mathbf x) = \sup _{\mathbb W_{\mathfrak P}\in \mathcal G}\frac 1 {\mu_{\mathbf e}(\mathbb W_{\mathfrak P})}\int_{\mathbb W_{\mathfrak P}}|f(\mathbf {xy})|d\mu_{\mathbf e}(\mathbf y). \] Then for every \(1 < p\leq \infty\), there exists a constant \(C=C(p,\mathbf e, \mathfrak P,\kappa_0,\epsilon_0)>0\) such that \[ \int_{\mathcal O_n}|f(\mathbf x)|^pd\mu_{\mathbf e}(\mathbf x) \leq C\int_{\mathcal O_n}|f(\mathbf x)|^pd\mu_{\mathbf e}(\mathbf x) \] for all \(f\in L^p(\mathcal O_n,d\mu_{\mathbf e})\). Since the \(B_{\mathcal P}(\mathbf x;\overline \delta)\) satisfies the \((\kappa_0,\epsilon_0)\)-admissibility conditions above, they also have the following results. {Corollary 7.2} Let \(\mathfrak P=\{\mathbf p_1,\dots,\mathbf p_d\}\) be a spanning set of vectors in \(\mathbb R^n\), and let \(\mathcal P = \{F_{\mathbf p_j} \mid \mathbf p_j\in \mathfrak P\}\). Given \(\mathbf e \in \mathbb R^n\) we define the maximal operator \(\mathcal M_{\mathcal P,\mathbf e}\) by \[ \mathcal M_{\mathcal P,\mathbf e}[f](\mathbf x)=\sup_{B_{\mathcal P} (\mathbf x;\overline \delta)}\frac 1{\mu_{\mathbf e}(B_{\mathcal P}(\mathbf x;\overline \delta))}\int_{B_{\mathcal P} (\mathbf x;\overline \delta)}|f(\mathbf y)|d\mu_{\mathbf e}(\mathbf y) \] where \(B_{\mathcal P}(\mathbf x, \overline \delta)\) is the monomial ball defined above. Then \(\mathcal M_{\mathcal P,\mathbf e}\) is bounded on \(L^p\) for all \(1<p\leq \infty\). In addition, if \(\mathbf e\) does not lie in the linear span of any \((n-1)\) of the vectors in \(\mathfrak P\), the \(\mathcal M_{\mathcal P,\mathbf e}\) has the same behavior near \(L^1\) as the strong maximal operator.
scientific article; zbMATH DE number 5598217
scientific article