A multiplier theorem for the Hankel transform
The author uses a Riesz function technique to prove certain multiplier theorems for the Hankel transform, analogous to the classical Hörmander-Mihlin theorem for the Fourier transform. Fix \(\alpha\geq 1/2\) and let \(k_0\) denote the least integer greater than \(\alpha+1\). For a bounded function \(m\) on \(\mathbb{R}_+= (0,\infty)\) define the multiplier operator \(T_m\) by \((T^{\widehat{\;}}_m g)= m\widehat g\), where \(\widehat{\;}\) denotes the Hankel transform \[ \widehat g(y)= 2^\alpha\Gamma(\alpha+ 1) \int^\infty_0 g(x) (yx)^{-\alpha} J_\alpha(xy) x^{2\alpha+1}dx \] and \(J_\alpha\) is the Bessel function of the first kind of order \(\alpha\). The author proves the following two theorems. Theorem 1. If \(m\) satisfies \(\sup_{R>0} R^{-1} \int^{2R}_R| x^k m^{(k)}(x)|^2 dx<\infty\), where \(k= 0,1,\dots, k_0\), then \(T_m\) is of weak type \((1,1)\) and, consequently is bounded on \(L^p(\mathbb{R}_+, x^{2\alpha+1}dx)\) for all \(p\in (1,\infty)\). Theorem 2. If \(m\) satisfies \(\sup_{R>0} R^{-1} \int^{2R}_R| x^\nu m^{(\nu)}(x)|^2 dx<\infty\), where \(\nu>\alpha+ 1\) and \(m^{(\nu)}\) is the Weyl fractional derivative of order \(\nu\), then the result in Theorem 1 also holds.
scientific article; zbMATH DE number 1267963