Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations

Source abstract

In this paper, we show that a pseudo-differential operator associated to a symbol a ∈ L ∞ ( R × R , L ( H ) ) a\in L^{\infty }(\mathbb {R}\times \mathbb {R},\mathcal {L}(H)) ( H H being a Hilbert space) which admits a holomorphic extension to a suitable sector of C \mathbb {C} acts as a bounded operator on L 2 ( R , H ) L^{2}(\mathbb {R},H) . By showing that maximal L p L^{p} -regularity for the non-autonomous parabolic equation u ′ ( t ) + A ( t ) u ( t ) = f ( t ) , u ( 0 ) = 0 u’(t) + A(t)u(t) = f(t), u(0)=0 is independent of p ∈ ( 1 , ∞ ) p\in (1,\infty ) , we obtain as a consequence a maximal L p ( [ 0 , T ] , H ) L^{p}([0,T],H) -regularity result for solutions of the above equation.