Dimension-Free Estimates for Maximal Functions with Nonisotropic Dilations

We mainly study the dimension-free \(L^p\) $L^p$ -inequalities of the Hardy-Littlewood maximal functions with nonisotropic dilations. \( M_*^Gf(x,y)=\sup _{t>0}\frac{1}{|G|} \left| \int _{G}f(x-tu,y-t^2v)dudv \right| , \) $$M_{\ast }^{G}f(x,y)=\underset{t>0}{sup}\frac{1}{|G|}\left|{\int }_G,f,(x-tu,y-t^{2}v),d,u,d,v\right|,$$ where G is a bounded, closed and symmetric convex subset of \(\mathbb {R}^{n+1}\) ${\mathbb{R}}^{n+1}$ . We prove that there is a constant C(p, L, Q) such that \( \left\| M_*^{G}f\right\| _{L^p(\mathbb {R}^{n+1})}\le C_p\Vert f\Vert _{L^p(\mathbb {R}^{n+1})} \) $$∥M_{\ast }^G,f∥_{L^p({\mathbb{R}}^{n+1})}^{}\le C_p{\Vert f\Vert }_{L^p({\mathbb{R}}^{n+1})}$$ for \(1<p\le \infty \) $1<p\le \infty$ , where C(p, L(G), Q(G)) is a constant which may depend on p, L(G) and Q(G), but not explicitly on the dimension n.