About integral of Weber--Shafheitlin

I. S. Belov

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2003), pp. 481-489.

Résumé

Let $L_{\lambda}^{p}$ be the function space at half-line with the norm $\|f\|_{p,\lambda}^{p}= \int_{0}^{\infty}|f(x)|^{p}x^{-\lambda}\,dx$. In the work the operators $A_{\mu}$ of multiplicative convolution with Bessel function $ A_{\mu}f(x)=\int_{0}^{\infty}J_{\mu}(xt)f(t)t^{-\lambda}\,dt$ are considered and their following propeties are proved. The operators $A_{\mu}$, $\mu \geq 0$, are bounded on $L^{2}(\lambda)$, $-1\leq \lambda\leq 1$. $A_{\mu}$, $\mu>0$, are bounded on $L_{\lambda}^{p}$, $1\leq p\leq\infty$, but $A_{0}$ is unbounded on $L_{1}^{p}$, $1\leq p\leq \infty$. The operators $A_{\mu}$ are unbounded on $ L_{\lambda}^{p}$ $p\not= 2$, $1\leq \lambda 1$. With some relations between values $(\mu, \nu, \lambda, p)$ the products $A_{\nu}A_{\mu}$ are bounded on $L_{\lambda}^{p}$.