A generalized Kolmogorov inequality for the Hilbert transform (Q2759024)

In this paper, the author obtains an extended form of the Kolmogorov inequality NEWLINE\[NEWLINEm\{x:|Hf(x)|\geq \alpha\}\leq C\alpha^{-1} \int_{\mathbb{R}}|f(x)|dx,\quad \alpha> 0,\quad f\in L^1(\mathbb{R}),NEWLINE\]NEWLINE where \(m\) is the Lebesgue measure and \(H\) is the Hilbert transform defined by NEWLINE\[NEWLINEHf(x)= \text{p.v. }\pi^{-1} \int_{\mathbb{R}} y^{-1} f(x-y) dy.NEWLINE\]NEWLINE Introduce the norms NEWLINE\[NEWLINE\|f\|_{B_1}= \pi^{-1} \int_{\mathbb{R}}|xf(x)|(1+ x^2)^{-1} dx,\;\|f\|_{B_2}= \pi^{-1} \int_{\mathbb{R}}|f(x)|(1+ x^2)^{-1} dx,NEWLINE\]NEWLINE and define a weighted measure \(\mu\) by \(\mu(A)= \pi^{-1}\int_A(1+ x^2)^{-1} dx\).NEWLINENEWLINENEWLINEThe author proves the following result.NEWLINENEWLINENEWLINETheorem: Suppose that \(f\) satisfies \(\int_{\mathbb{R}}|f(x)|(1+|x|)^{-1} dx< \infty\). Then \(Hf(x)\) exists almost everywhere. If \(f\geq 0\) then for \(\alpha>\|f\|_{B_1}\), NEWLINE\[NEWLINE\mu\{x:|Hf(x)|\geq \alpha\}\leq{2\over \pi} \Biggl({\|f\|_{B_2}\over\alpha-\|f\|_{B_1}}+ {\|f\|_{B_2}\over \alpha+\|f\|_{B_1}}\Biggr).NEWLINE\]NEWLINE Also, the author shows how the classical Kolmogorov inequality can be obtained from the above inequality.