The dimension-free estimate for the truncated maximal operator

Abstract

We mainly study the dimension-free Lp{L}^{p}-inequality of the truncated maximal operator Mnaf(x)=supt>01∣Ba1∣∫Ba1f(x−ty)dy,{M}_{n}^{a}f\left(x)=\mathop{\sup }\limits_{t\gt 0}\frac{1}{| {B}_{a}^{1}| }\left|\mathop{\int }\limits_{{B}_{a}^{1}}f\left(x-ty){\rm{d}}y\hspace{-0.25em}\right|, where Ba1={x:a≤∣x∣≤1}{B}_{a}^{1}=\left\{x:a\le | x| \le 1\right\}. When 0≤an/(n−1)p\gt n\hspace{0.1em}\text{/}\hspace{0.1em}\left(n-1). When a=1a=1, we prove that ‖Mn1‖Lp(Rn)≤C(p)‖f‖Lp(Rn)\Vert {M}_{n}^{1}{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})}\le C\left(p)\Vert f{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})} for p≥2p\ge 2.

Keywords

• maximal function
• dimension-free estimate
• fourier multiplier
• 42b20
• 42b25