A generalized Kolmogorov inequality for the Hilbert transform

Abstract:

If $f\in L^1(\mathbf {R}^1;(1+|x|)^{-1}dx)$ we can define the Hilbert transform $Hf$ almost everywhere (Lebesgue) and obtain an estimate for $\mu \{x: |Hf(x)|\ge \alpha \}$ where $\mu$ is a suitable finite measure. The classical Kolmogorov inequality for the Lebesgue measure of $\{x: |Hf(x)|\ge \alpha \}$ is obtained by a scaling argument.