A Note on Endpoint Sobolev Regularity of a Class of Bilinear Maximal Functions

摘要

We consider the endpoint Sobolev regularity of a class of bilinear restricted maximal function Ma(f ->)(x):=supx is an element of:= sup x.B(z,a|x|) 1 |B(z, a|x|)| 2 B(z,a|x|) |f1(y)|dy B(z,a|x|) |f2(y)|dy. We show that the map Ma : W 1,1 (Rn) x W 1,1 (Rn). W 1,1/2 (Rn) is bounded if a. (0, 1/2) and not bounded if a is an element of [1/2,+infinity).