A generalization of an inequality by N. V. Krylov

In Krylov (Journal of the Juliusz Schauder Center 4 (1994), 355–364), a parabolic Littlewood–Paley inequality and its application to an L ^p -estimate of the gradient of the heat kernel are proved. These estimates are crucial tools in the development of a theory of parabolic stochastic partial differential equations (Krylov, Mathematical Surveys and Monographs vol. 64 (1999), 185–242). We generalize these inequalities so that they can be applied to stochastic integrodifferential equations.