Weak-type estimates for the modified Hankel transform (Q2773359)

Fix \(\alpha>-1/2\), and for \(1< p<\infty\), consider the space \(L^p_{(\alpha)}\) of measurable functions \(f\) on \(\mathbb{R}_+\) satisfying \(\|f\|_p= (\int^\infty_0|f(x)|^p x^{2\alpha+1} dx)^{1/p}<\infty\). For \(f\in L^1\), one defines the modified Hankel transform of order \(\alpha\) by NEWLINE\[NEWLINEH_\alpha f(y)= \int^\infty_0 J_\alpha(xy)(xy)^{- \alpha} f(x) x^{2\alpha+1} dx,NEWLINE\]NEWLINE where \(J_\alpha\) is the Bessel function of the first kind of order \(\alpha\). For any bounded function \(m\) on \(\mathbb{R}_+\), the multiplier operator \(T_m\) is defined by \(H_\alpha(T_m f)= mH_\alpha f\). Using the Galé and Pytlik representation of Riesz functions the author proves the following theorem:NEWLINENEWLINENEWLINETheorem. Fix \(\alpha>-1/2\), \(1\leq s\leq 2\). Assume that a bounded function \(m\) satisfies NEWLINE\[NEWLINE\sup_{j\in\mathbb{Z}} \Biggl(\int^{2^{j+1}}_{2^j}|x^\lambda m^{(\lambda)}(x)|^sx^{-1}dx\Biggr)< \infty,NEWLINE\]NEWLINE where \(m^{(\lambda)}\) denotes the fractional derivative of order \(\lambda\) for \(\lambda> \alpha+1/2+ 1/s\). Then the operator \(T_m\) is of weak type \((1,1)\), and consequently is bounded on \(L^p_{(\alpha)}\).