Criteria for the boundedness and compactness of integral transforms with positive kernels (Q2747920)

Let \(-\infty< a< b\leq\infty\) and \(\nu\) be a nonnegative \(\sigma\)-finite Borel measure on \((a,b)\). Denote by \(L^q_\nu(a,b)\) \((0< q<\infty)\) the class of all \(\nu\)-measurable functions \(g: (a,b)\to\mathbb{R}\) for which NEWLINE\[NEWLINE\|g\|_{L^q_\nu(a,b)}= \Biggl(\int_{(a,b)}|g(x)|^q d\nu\Biggr)^{1/q}< \inftyNEWLINE\]NEWLINE (if \(\nu\) is absolutely continuous, i.e. \(d\nu= v(x) dx\) with \(v\) a positive Lebesgue-measurable function on \((a,b)\), the class \(L^q_\nu(a,b)\) is denoted by \(L^q_v(a,b)\)).NEWLINENEWLINENEWLINELet \(k\) be a positive Lebesgue-measurable function on \(\{(x,y): a< y< x< b\}\) satisfying the following conditions:NEWLINENEWLINENEWLINE(a) \(k\) belongs to \(V\) \((k\in V)\), i.e. there exists a positive constant \(d_1\) such that for all \(x\), \(y\), \(z\) with \(a< y< z< x< b\) NEWLINE\[NEWLINEk(x,y)\leq d_1 k(x,z);NEWLINE\]NEWLINE (b) \(k\) belongs to \(V_p\), \(1< p< \infty\) \((k\in V_p)\), i.e. there exists a positive constant \(d_2\) such that for all \(x\in (a,b)\) NEWLINE\[NEWLINE\int^x_{a+(x- a)/2} k^{p'}(x,y) dy\leq d_2(x- a)k^{p'}(x,a+ (x- a)/2),NEWLINE\]NEWLINE where \(p'= p/(p-1)\).NEWLINENEWLINENEWLINEIn the paper under review, the author gives necessary and sufficient conditions for the boundedness and for the compactness of the integral operator NEWLINE\[NEWLINEK(f)(x)= \int^x_a k(x,y) f(y) dyNEWLINE\]NEWLINE from \(L^p(a,b)\) to \(L^q_\nu(a,b)\) with \(1< p\leq q<\infty\) or \(0< q< p<\infty\) and \(p>1\), and with \(k\in V\cap V_p\).NEWLINENEWLINENEWLINEAs pointed out in this paper, examples of such integral operators are given by the Riemann-Liouville operators and by the integral operators with power-logarithmic kernels for suitable parameters.NEWLINENEWLINENEWLINEIn the non-compact case the author also gives the upper and the lower bound for the distance of \(K\) from the subspace of compact operators from \(L^p(a,b)\) to \(L^q_v(a,b)\) when \(1< p\leq q<\infty\).