Stepanov, Vladimir D.; Tikhonov, Sergey Yu. Two power-weight inequalities for the Hilbert transform on the cones of monotone functions. (English) Zbl 1237.44005 Complex Var. Elliptic Equ. 56, No. 10-11, 1039-1047 (2011). 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 \[ {\mathcal H}_0 f(x)= \text{p.v. }\int_{\mathbb{R}_+} {xf(t)\over x^2- t^2}\,dt. \] Reviewer: Frank Stenger (Salt Lake City) Cited in 6 Documents 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 Keywords:Hilbert transform; weighted norm inequalities; monotone functions PDFBibTeX XMLCite \textit{V. D. Stepanov} and \textit{S. Yu. Tikhonov}, Complex Var. Elliptic Equ. 56, No. 10--11, 1039--1047 (2011; Zbl 1237.44005) Full Text: DOI References: [1] DOI: 10.1090/S0002-9947-1973-0312139-8 [2] DOI: 10.1215/S0012-7094-36-00228-4 · Zbl 0014.21402 [3] DOI: 10.1112/plms/s3-8.1.135 · Zbl 0082.28101 [4] DOI: 10.1090/S0002-9939-1976-0399739-2 [5] Liflyand E, C. R. Acad. Sci. Paris, Ser. I 348 pp 1253– (2010) [6] Paley REAC, Note II, Trans. Amer. Math. Soc. 35 pp 354– (1933) [7] DOI: 10.1090/S0002-9947-1934-1501757-2 [8] DOI: 10.1016/j.aim.2007.05.022 · Zbl 1129.42007 [9] Sawyer E, Stud. Math. 96 pp 145– (1990) [10] Hardy GG, Inequalities (1934) [11] Johannson M, Math. Inequal. Appl. 11 pp 393– (2008) [12] DOI: 10.4153/CJM-1993-064-2 · Zbl 0798.42010 [13] DOI: 10.1112/jlms/s2-48.3.465 · Zbl 0837.26011 [14] Stein EM, Introduction to Fourier Analysis on Euclidean spaces (1971) [15] DOI: 10.4153/CJM-1993-006-3 · Zbl 0796.26008 [16] Liflyand, E and Tikhonov, S. 2009.The Fourier Transforms of General Monotone Functions, Analysis and Mathematical Physics, 373–391. Basel, Switzerland: Trends in Mathematics, Birkhäuser, Verlag AG. · Zbl 1297.42009 [17] DOI: 10.1016/j.jmaa.2006.02.053 · Zbl 1106.42003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.