Differential Operators in Pairs of Sobolev Spaces

The most natural connection of the theory of multipliers with differential operators arises when one is looking for the bounds of the norms and essential norms of these operators mapping one Sobolev space into another.

In Sect. 10.1 we give estimates for the norms of general differential operators performing a mapping between two Sobolev spaces, formulated in terms of their coefficients as multipliers. These estimates involve multiplier norms of the coefficients, and for some values of integrability and smoothness parameters they are two-sided. We also describe a class of differential operators for which their continuity in pairs of Sobolev spaces is equivalent to the inclusion of the coefficients into classes of multipliers without any additional conditions on indices. We give a counterexample showing that in general the inclusion of the coefficients into the natural classes of multipliers is not necessary for the continuity of differential operators.

Estimates for the essential norms of general differential operators is the topic of Sect. 10.2. By the example of a Schrödinger operator in Rn considered in Sect. 10.3, we outline the role of the essential norm of a multiplier in the Fredholm theory of elliptic differential operators. The last Sect. 10.4 deals with a characterization of pairs of differential operators with constant coefficients which obey the dominance property between L2 and its weighted counterpart.