  This note is devoted to a simple proof of the generalized Leibniz rule in
bounded domains. The operators under consideration are the so-called spectral
Laplacian and the restricted Laplacian. Equations involving such operators have
been lately considered by Constantin and Ignatova in the framework of the SQG
equation \cite{CI} in bounded domains and by two of the authors \cite{HungJuan}
in the framework of the porous medium with nonlocal pressure in bounded
domains. We will use the estimates in this work in a forthcoming paper on the
study of Porous Medium Equations with pressure given by Riesz-type potentials .
