<p>We obtain estimates for the norm of convolution operators in generalized local Morrey type spaces. We find sufficient boundedness conditions on convolution operators and characterize the corresponding associated spaces. In this setting, we prove Young–O’Neil type inequalities. The obtained inequalities provide new estimates for convolution operators with kernels in weak Lebesgue spaces. The results obtained generalize known results for the Lebesgue, Lorentz, and Morrey spaces and refine the boundedness criteria for integral operators. Applications include potential operators and singular integrals in Morrey type spaces.</p>