The generalized Hardy operator with kernel and variable integral limits in Banach function spaces
The authors consider the integral operator \[ Kf(x):=v(x)\int_{a(x)}^{b(x)} k(x,y) u(y) f(y)dy \] for \(x>0\), where \(a\) and \(b\) are nondecreasing functions, \(u\) and \(v\) are non-negative and finite functions, and \(k(x,y)\geq 0\) is nondecreasing in \(x\), nondecreasing in \(y\) and \(k(x,z)\leq D[k(x,b(y))+k(y,z)]\) for \(y\leq x\) and \(a(x)\leq z \leq b(y)\). It is shown that the integral operator \(K: X\rightarrow Y\), where \(X\) and \(Y\) are Banach function spaces with \(l\)-condition is bounded if and only if \[ \sup_{x\leq y, a(y)\leq b(x)}\|\chi_{(x, y)}(\cdot)v(\cdot)k(\cdot, b(x))\|_Y\|\chi_{a(y), b(x)}u\|_{X'}+ \] \[ \sup_{x\leq y, a(y)\leq b(x)}\|\chi_{(x, y)} v\|_Y\|\chi_{(a(y), b(x))}(\cdot)K(x,\cdot)u(\cdot)\|_{X'}<\infty . \]
scientific article; zbMATH DE number 1374886