The form boundedness criterion for the Laplacian operator

In this paper, we characterize the class of measurable functions (or, more generally, real- or complex-valued distributions) f such that the commutator operatorCf=[f,Δ] acting from H˙1(Rd) to H˙−1(Rd). In other words, we give a complete solution to the problem for the boundedness of the Laplacian −Δ, which is fundamental to quantum mechanics.