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Two power-weight inequalities for the Hilbert transform on the cones of monotone functions. (English) Zbl 1237.44005

Authors’ abstract: Our main goal is to obtain the characterization of the weighted \[ \Biggl(\int_{\mathbb{R}} |{\mathcal H}f(x)|^q|x|^udx\Biggr)^{1/q}\leq C\Biggl(\int_{\mathbb{R}}|f(x)|^p|x|^\beta dx\Biggr)^{1/p},\tag{1} \] restricted to the cones of the odd or even monotone functions defined as \[ {\mathfrak M}^{\downarrow}_0= \{f: f(x)= -f(-x),\,f\downarrow\text{ on }\mathbb{R}_+\} \] and \[ {\mathfrak M}^{\uparrow}_0= \{f: f(x)= f(-x), f\uparrow\text{ on }\mathbb{R}_+\}, \] respectively. Then (1) is reduced either to the inequality \[ \Biggl(\int_{\mathbb{R}_+} |{\mathcal H}_0 f(x)|^q x^u dx\Biggr)^{1/q}\leq C\Biggl(\int_{\mathbb{R}_+} |f(x)|^p x^\beta dx\Biggr)^{1/p},\;f\uparrow, \] where \[ {\mathcal H}_0 f(x)=\text{p.v. }\int_{\mathbb{R}_+} {tf(t)\over x^2- t^2}\,dt, \] or to the inequality \[ \Biggl(\int_{\mathbb{R}_+} |{\mathcal H}_0 f(x)|^q x^u dx\Biggr)^{1/q}\leq C\Biggl(\int_{\mathbb{R}_+} |f(x)|^p x^\beta dx\Biggr)^{1/p},\;f\uparrow, \] where \[ {\mathcal H}_0 f(x)= \text{p.v. }\int_{\mathbb{R}_+} {xf(t)\over x^2- t^2}\,dt. \]

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
26A48 Monotonic functions, generalizations
42A50 Conjugate functions, conjugate series, singular integrals
26D10 Inequalities involving derivatives and differential and integral operators
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