Weighted inequalities for Hilbert transforms and multiplicators of Fourier transforms

• Main Authors: Kokilashvili V, Meskhi A
• Format: Article
• Language: English
• Published: SpringerOpen 1997-01-01
• Series: Journal of Inequalities and Applications
• Subjects: Hilbert transform; two-weight inequality; weak and strong type estimates; Fourier transform; multiplier
• Online Access: http://www.journalofinequalitiesandapplications.com/content/1/562605
• ISSN: 1025-5834; 1029-242X

<p/> <p>As is well known, invariant operators with a shift can be bounded from <inline-formula><graphic file="1029-242X-1997-562605-i1.gif"/></inline-formula> into <inline-formula><graphic file="1029-242X-1997-562605-i2.gif"/></inline-formula> only if <inline-formula><graphic file="1029-242X-1997-562605-i3.gif"/></inline-formula>. We show that the case <inline-formula><graphic file="1029-242X-1997-562605-i4.gif"/></inline-formula> might also hold for weighted spaces. We derive the sufficient conditions for the validity of strong (weak) <inline-formula><graphic file="1029-242X-1997-562605-i5.gif"/></inline-formula> type inequalities for the Hilbert transform when <inline-formula><graphic file="1029-242X-1997-562605-i6.gif"/></inline-formula>.</p> <p>The examples of couple of weights which guarantee the fulfillness of two-weighted strong (weak) type inequalities for singular integrals are presented. The method of proof of the main results allows us to generalize the results of this paper to the singular integrals which are defined on homogeneous groups.</p> <p>The Fourier multiplier theorem is also proved.</p>