Remainder terms of a nonlocal Sobolev inequality

Abstract

In this note, we study a nonlocal version of the Sobolev inequality where is the best constant, * denotes the standard convolution and denotes the classical Sobolev space with respect to the norm . By using the nondegeneracy property of the extremal functions, we prove that the existence of the gradient type remainder term and a reminder term in the weak ‐norm of above inequality for all with .