Maximal functions associated with curves and the Calderón-Zygmund method of rotations

Abstract:

Let ${\delta _t}(t > 0)$ be a dilation in ${{\mathbf {R}}^n}(n \geqslant 2)$ defined by \[ {\delta _t}x = ({t^{{\alpha _1}}}{x_1},{t^{{\alpha _2}}}{x_2}, \ldots ,{t^{{\alpha _n}}}{x_n})\qquad (x = ({x_1},{x_2}, \ldots ,{x_n})),\] where ${\alpha _i} > 0(i = 1,2, \ldots ,n)$ and ${\alpha _i} \ne {\alpha _j}$ if $i \ne j$. For $\nu \in {{\mathbf {R}}^n}$ with $|\nu | = 1$, let ${\Gamma _\nu }:(0,\infty ) \to {{\mathbf {R}}^n}$ be a curve defined by ${\Gamma _\nu }(t) = {\delta _t}\nu (0 < t < \infty )$. Using maximal functions associated with the curves ${\Gamma _\nu }$, we define an operator $M$ which is a nonisotropic analogue of the one studied in R. Fefferman [2]. It is proved that $M$ is a bounded operator on ${L^p}({{\mathbf {R}}^n})$ for some $p$ with $1 < p < 2$. As its application we prove the ${L^p}$ boundedness of operators of the form ${T^{\ast }}(f)(x) = {\sup _{\varepsilon > 0}}|{T_\varepsilon }(f)(x)|$, where ${T_\varepsilon }$ is an integral operator associated with a variable kernel with mixed homogeneity.