On boundedness and compactness of a certain class of kernel operators
\noindent The author considers the integral operator \[ Kf(x) = w(x) \int_{a(x)}^{b(x)} k(x,y) f(y) v(y) \, dy \] \noindent on \(\mathbb R^+ = [0, \infty)\), where \(a(x), b(x)\) are differentiable and strictly increasing in \((0, \infty)\), \(a(0) = b(0) = 0\), and \(a(x) < b(x)\) for \(0 < x < \infty\), \(a(\infty) = b(\infty) = \infty\). The kernel \(k > 0\) on \(\{(x,y)\mid x>0\), \( a(x) < y < b(x) \}\), and satisfies Oinarov's conditions \(\mathcal{O}_a\) and/or \(\mathcal{O}_b\), where \(k \in \mathcal{O}_b\) if there exists a constant \(D \geq 1\), independent of \(x, y, z\), such that \[ \frac1D k(x,y) \leq k(x, b(z)) + k(z,y) \leq D k(x,y),\;z \leq x, \;a(x) \leq y \leq b(z), \] \noindent and where \(k \in \mathcal{O}_a\) if there exists a constant \(D \geq 1\), independent of \(x, y, z\), such that \[ \frac1D k(x,y) \leq k(x, a(z)) + k(z,y) \leq D k(x,y),\;z \geq x,\;a(z) \leq y \leq b(x). \] \noindent These operators have been studied and sufficient conditions that \(K: L_p \mapsto L_q\) have been given in a paper of Stepanov and Ushakova, which involve a double supremum of the kernel, e.g., one needs the finiteness of terms like \[ A_{b,0} = \sup_{t > 0} A_{b,0}(t) \] \[ = \sup_{t > 0} \sup_{b^{-1}(a(t)) \leq s \leq t} \left( \int_s^t k^q(x, b(x))w^q(x) \, dx \right)^{1/q} \left( \int_{a(t)}^{b(s)} v^{p^{\prime}}(y)\, dy \right)^{1/p^{\prime}} , \] \noindent and other terms like it to show that \(K\) is bounded above and below in terms of the given constants, with other conditions on the constants implying that \(K\) is compact. \noindent The author gives a criterion that only involves one supremum, but requires the introduction of new functions related to \(a, b\) called fairway functions that express boundedness of \(K\) satisfying one or both of Oinarov's conditions. The conditions are as long as the above sample; we refer the interested reader to the paper for details. \noindent The author gives examples of several cases to which the result applies, and gives explicit computations of the fairway functions for these examples.
scientific article; zbMATH DE number 5974888