A variant of the notion of a space of homogeneous type
In studying maximal functions and singular integral operators associated to certain curves or surfaces, one is led to the study of maximal functions and singular integral operators for which there is a natural family of balls, but such that these balls are not the balls of the space of homogeneous type. In order to overcome this difficulty, the authors give a double infinite sequence of nonnegative continuous functions \(\{\rho_j(x,y)\}\) on \(X\times X\), where \(X\) is a locally compact Hausdorff space equipped with a regular Borel measure \(\mu\), and corresponding balls \(B_j(x,r)= \{y\in X: \rho_j(y,x)< r\}\). Then the authors prove that if \(\{\rho_j\}\) and \(\{B_j\}\) satisfy certain conditions then the Hardy-Littlewood maximal function, associated to \(\{B_j\}\) is of strong type \((p,p)\), \(p>1\), and of weak type \((1,1)\). To the singular operator \[ Tf(x)= \int_X K(x,y) f(y) d\mu(y), \] the authors prove that if \(\{\rho_j\}\) and \(\{B_j\}\) satisfy certain conditions, \(Tf\) is bounded on \(L^2(X)\), and the kernel \(K(x,y)\) satisfies a Hörmander condition related to \(\{\rho_j\}\), then \(Tf\) is of weak type \((1,1)\) and of strong type \((p,p)\), \(1<p\leq 2\). As applications, the authors obtain weak type \((1,1)\) bounds and strong type \((p,p)\) bounds for some singular integral operators on a homogeneous Lie group. The authors also get similar results for maximal functions and singular integral operators on \(\mathbb{R}^n\) related to a continuous nonnegative pseudo-metric on \(\mathbb{R}^n\).
scientific article; zbMATH DE number 790376