A supercritical estimate for Bessel potentials on Lorentz spaces

In this paper we give a simple proof of an estimate for Bessel potentials acting on Lorentz spaces in the supercritical exponent: let 1<p=dα-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p=\frac{d}{\alpha -1}$$\end{document} and 1≤q≤+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le q \le + \infty $$\end{document}. If f∈Lp,q(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in L^{p,q}({\mathbb {R}}^d)$$\end{document}, then there exists a constant C=C(α,d,p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C =C(\alpha ,d , p,q)$$\end{document} such that |gα∗f(x)-gα∗f(z)|≤C|x-z||ln(|x-z|)|+11q′‖f‖Lp,q.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \vert g_{\alpha }*f(x)- g_{\alpha }*f(z) \vert \le C \vert x-z \vert \left(|\ln (|x-z|)| +1 \right)^{\frac{1}{q'}} \Vert f \Vert _{L^{p,q}}. $$\end{document}