We consider the bilinear estimates of a product of functions in homogeneous Besov spaces showed by Bony . It is seen that if we change the condition of indices denoting differential orders, then we can find examples of functions that never satisfy the bilinear estimates. Such examples can be constructed due to those used in the ill-posedness problem of the Navier-Stokes equations, such as Bourgain-Pavlović  and Yoneda .
"Counter-examples of the Bilinear Estimates of the Holder Type Inequality in Homogeneous Besov Spaces." Tokyo J. Math. 43 (2) 295 - 304, December 2020. https://doi.org/10.3836/tjm/1502179307