Integral Operators with Homogeneous Kernels in Grand Lebesgue Spaces

Sufficient conditions on the kernel and the grandizer that ensure the boundedness of integral operators with homogeneous kernels in grand Lebesgue spaces on $\mathbb R^n$ as well as an upper bound for their norms are obtained. For some classes of grandizers, necessary conditions and lower bounds for the norm of these operators are also obtained. In the case of a radial kernel, stronger estimates are established in terms of one-dimensional grand norms of spherical means of the function. A sufficient condition for the boundedness of the operator with homogeneous kernel in classical Lebesgue spaces with arbitrary radial weight is obtained. As an application, boundedness in grand spaces of the one-dimensional operator of fractional Riemann–Liouville integration and of a multidimensional Hilbert-type operator is studied.