Abstract.

This paper explores the maximal regularity and separability properties of abstract differential operator equations in weighted spaces. Using Fourier multiplier theorems we will obtain the coercive properties of convolution differential-operator equations (CDOEs) with unbounded operator coefficients in weighted Lp spaces. Finally, these results are applied to establish well-posedness of the Cauchy problem for degenerate parabolic convolution differentialoperator equations. In particular we study the R−positivity of the corresponding convolutionelliptic operators.

Keywords:

convolution differential-operator equation, R -positivity, weighted spaces, operatorvalued functions, degenerate convolution equations, weighted multiplier condition, R− boundedness.