The strong Lefschetz property in codimension two

Every artinian quotient of $K[x,y]$ has the strong Lefschetz property if $K$ is a field of characteristic zero or is an infinite field whose characteristic is greater than the regularity of the quotient. We improve this bound in the case of monomial ideals. Using this we classify when both bounds are sharp. Moreover, we prove that the artinian quotient of a monomial ideal in $K[x,y]$ always has the strong Lefschetz property, regardless of the characteristic of the field, exactly when the ideal is lexsegment. As a consequence we describe a family of non-monomial complete intersections that always have the strong Lefschetz property.