QUASI-DIFFERENTIAL OPERATORS IN L^p SPACES

Abstract

For a very general class of ordinary quasi-differential expressions M with matrix-valued coefficients, the maximal operator T_M and the minimal operator T_⁰M are defined as operators of a subspace of L^p into L^q for arbitrary p, q∈[1, ∞], where the weak topologies induced by the dualities 〈L^p, L^p′〉 and 〈L^q, L^q′〉 with 1/p+1/p′=1 and 1/q+1/q′=1 are used instead of the norm topologies. It is shown that these operators and their adjoints possess the usual properties which are known for scalar differential expressions and the two cases p, q∈[1, ∞) and p, q∈(1, ∞].