A class of integral operators and Bessel Plancherel transform on \(L_\alpha^2\) (Q416321)

The author defines the measure \(\mu_{\alpha}\) on \([0,\infty)\) (\(\alpha>-1/2\)) by the formula NEWLINE\[NEWLINE d\mu_{\alpha}(x)=\frac{x^{2\alpha+1}}{2^{\alpha}\Gamma(\alpha+1)}\,dx, \quad x\geq0. NEWLINE\]NEWLINE Let \(L_{\alpha}^{2}\) be the corresponding weighted \(L^2\)-space on \([0,\infty)\). For a nonnegative measurable function \(\varphi\) on \([0,\infty)\) satisfying some additional conditions, there is defined the linear operator \(T_{\varphi}\) on \(L_{\alpha}^{2}\) by the formula NEWLINE\[NEWLINE (T_{\varphi}f)(x)=\frac{1}{x^{2\alpha+2}} \int_{0}^{\infty} \varphi \left( \frac{y}{x} \right) f(y)\,d\mu_{\alpha}(y). NEWLINE\]NEWLINE Under some additional assumptions, there is proved the equality \(T_{\varphi}\Phi_{\alpha}=\Phi_{\alpha}T_{\varphi}^{\ast}\), where \(\Phi_{\alpha}\) is the Bessel transform. There are given some applications of the obtained results for Riemann-Liouville and Weyl transforms.