Cordes–Nirenberg's imbedding and restricting with application to an elliptic equation

This paper not only presents a criterion for the Cordes–Nirenberg space [Formula: see text] imbedding between the associate Morrey space [Formula: see text] and the Morrey space [Formula: see text], but also characterizes a Radon measure [Formula: see text] such that the Cordes–Nirenberg potential space [Formula: see text] is restricted to the [Formula: see text]-based Campanato space [Formula: see text] — thereby giving an application of the discovered characterization to the regularity for a class of the elliptic equations with symmetric [Formula: see text]-coefficients.

Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants

For [Formula: see text] and [Formula: see text], let [Formula: see text] be the space of harmonic functions [Formula: see text] on the upper half-space [Formula: see text] satisfying [Formula: see text] and [Formula: see text] be the Campanato space on [Formula: see text]. We show that [Formula: see text] coincides with [Formula: see text] for all [Formula: see text], where the case [Formula: see text] was originally discovered by Fabes, Johnson and Neri [E. B. Fabes, R. L. Johnson and U. Neri, Spaces of harmonic functions representable by Poisson integrals of functions in BMO and [Formula: see text], Indiana Univ. Math. J. 25 (1976) 159–170] and yet the case [Formula: see text] was left open. Moreover, for the scaling invariant version of [Formula: see text], [Formula: see text], which comprises all harmonic functions [Formula: see text] on [Formula: see text] satisfying [Formula: see text] we show that [Formula: see text], where [Formula: see text] is the collection of all functions [Formula: see text] such that [Formula: see text] are in [Formula: see text]. Analogues for solutions to the heat equation are also established. As an application, we show that the spaces [Formula: see text] unify naturally [Formula: see text], [Formula: see text] and [Formula: see text] which can be effectively adapted and applicable to suit handling the well/ill-posedness of the incompressible Navier–Stokes system on [Formula: see text].

Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces

In this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of a large class of parabolic rough operators on them. The author also establishes the parabolic local Campanato space estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respectively. At last, parabolic Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.