Sobolev-Morrey spaces associated with evolution equations. (English) Zbl 1157.35022

In the Fifties Calderon and Zygmund proved the regularity of the solutions to parabolic equations, using deep estimates on the kernel of the operator in \(L^p\) spaces. In the Sixties Morrey, Stampacchia and Campanato developed an alternative approach to the regularity, using a different class of function spaces, the so called Morrey-Campanato function spaces [see for instance M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, 105. Princeton, New Jersey: Princeton University Press (1983; Zbl 0516.49003)]. These function spaces were used, in general, to prove regularity (and maximal regularity) for solutions to parabolic equations with smooth coefficients. In this paper the author studies the properties of these function spaces, in order to use them in [ibid., No. 9, 1031–1078 (2008; Zbl 1157.35023)] to study parabolic equations with non-smooth coefficients.


35D10 Regularity of generalized solutions of PDE (MSC2000)
35R05 PDEs with low regular coefficients and/or low regular data
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35K90 Abstract parabolic equations