Limiting weak–type behavior for singular integral and maximal operators

Abstract:

The following limit result holds for the weak–type (1,1) constant of dilation-commuting singular integral operator $T$ in $\mathbb {R}^n$: for $f\in L^1(\mathbb {R}^n)$, $f\geq 0$, \[ \lim _{\lambda \rightarrow 0} \lambda \hspace {1mm}m\{x\in \mathbb {R}^n: |Tf(x)|>\lambda \} = \frac {1}{n} \int _{S^{n-1}}|\Omega (x)|d\sigma (x)\|f\|_1.\] For the maximal operator $M$, the corresponding result is \[ \lim _{\lambda \rightarrow 0} \lambda \hspace {1mm}m\{x\in \mathbb {R}^n: |Mf(x)|>\lambda \} = \|f\|_1.\]