We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in [Formula: see text] with respect to upper Ahlfors regular measures [Formula: see text], that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure [Formula: see text] and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented.
In this paper we prove an existence and uniqueness theorem for the Dirichlet problem for linear elliptic equations (in unbounded domains) whose leading coefficients are locally VMO and satisfy a suitable condition at infinity
In this paper we prove a uniqueness and existence theorem for the Dirichlet problem inW2,pfor second order linear elliptic equations in unbounded domains of the plane. Here the leading coefficients are locally of classVMOand satisfy a suitable condition at infinity.
Abstract. A version of the Lebesgue differentiation theorem is offered, where the L p norm is replaced with any rearrangement-invariant norm. Necessary and sufficient conditions for a norm of this kind to support the Lebesgue differentiation theorem are established. In particular, Lorentz, Orlicz and other customary norms for which Lebesgue's theorem holds are characterized.
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