Riesz Transform for the One-Dimensional -Generalized Fourier Transform

Abstract: For the <inline-formula id="IEq2"><alternatives><tex-math id="IEq2_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k,1)$$\end{document}</tex-math><inline-graphic href="11006_2023_2582_Article_IEq2.gif"></inline-graphic></alternatives></inline-formula>-generalized Fourier transform on the real line <inline-formula id="IEq3"><alternatives><tex-math id="IEq3_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R}$$\end{document}</tex-math><inline-graphic href="11006_2023_2582_Article_IEq3.gif"></inline-graphic></alternatives></inline-formula>, the Riesz transform is defined. For this transform, <inline-formula id="IEq4"><alternatives><tex-math id="IEq4_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}</tex-math><inline-graphic href="11006_2023_2582_Article_IEq4.gif"></inline-graphic></alternatives></inline-formula>-inequalities with power and piecewise power weights are proved for <inline-formula id="IEq5"><alternatives><tex-math id="IEq5_TeX">\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1&lt;p&lt;\infty$$\end{document}</tex-math><inline-graphic href="11006_2023_2582_Article_IEq5.gif"></inline-graphic></alternatives></inline-formula>.