On the Riemann--Liouville operators with variable limits (Q5938484)

Under consideration is the integral operator \[ Tf= v(x)\chi_{(a,b)}(x)\int_{\psi(x)}^{\varphi(x)}\frac{f(y) dy}{(x-y)^{1-\alpha}},\quad x\in (a,b), \] where \(0\leq a< b\leq \infty\), \(\alpha>0\), the weight function \(v(x)\) is measurable, and \(\varphi\) and \(\psi\) are absolutely continuous nondecreasing functions such that \(0\leq \psi(x)\leq \varphi(x)\leq x\). The article contains necessary and sufficient conditions for the operator \(T\:L_p(0,\infty)\to L_q(0,\infty)\), \(p,q\geq 1\), to be bounded and compact. Estimates for various norms of this operator are also given. For instance, if \(\psi(x)=\varphi(a)\) and \(\max(1/\alpha,1)<p\leq q<\infty\) then \(T\: L_p(0,\infty)\to L_q(0,\infty)\) is bounded if and only if \[ A=\sup_{a'<t<b} \Bigl(\int_t^b |(\Omega v)(x)|^q dx\Bigr)^{1/q} \bigl(\varphi(t)-\varphi(a)\bigr)^{1/p'}< \infty, \] where \(p'=p/(p-1)\), \((\Omega v)(x)=v(x)(x-\varphi(a))^{\alpha-1}\), and \(a'=\varphi_r^{-1}(\varphi(a))\) with \(\varphi_r^{-1}(x)=\sup\{s:\;\varphi(s)=x,\;s\in[a,b]\}\). In this case, \(\frac{1}{M}A\leq \|T\|_{L_p\to L_q} \leq M A\), with some constant \(M\) depending on \(p,q,\alpha\).