On a class of linear operators.
We consider elliptic nonlinear equations in a separable Hilbert space and their solutions in spaces of Sobolev type.
Let and , where a(s) is a positive continuous function such that and b(s) is quasi-increasing and . Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants and such that for all ; (jj) there exist positive constants and such that for all .