Remarks on bilinear estimates in the Sobolev spaces (Harmonic Analysis and Nonlinear Partial Differential Equations)

"Harmonic Analysis and Nonlinear Partial Differential Equations". June 30～July 2, 2014. edited by Hideo Kubo and Mitsuru Sugimoto. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.The boundedness of integral operators of convolution type in the weighted Lebesgue spaces and the Leibniz rule of order one in one dimensional case are studied. As a byproduct, a simple proof of the fact that the standard Sobolev space H^{s}(mathbb{R}^{n}) forms an algebra for s>n/2 is given. Moreover, a simple proof of the Leibniz rule is also given, where a remarkable cancellation property is observed in the standard Littlewood-Paley argument