Degenerate matrix-weighted Sobolev spaces are shown to embed compactly into weighted L^q spaces locally, under local Poincaré and Sobolev inequalities with gain sigma>1, with an application to p-admissible weights.
Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces
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abstract
We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound \mu(B(x,r)) \le Cr^d. Our spaces are only assumed to satisfy the geometric doubling property: every ball of radius r can be covered by at most N balls of radius r/2. A key ingredient is the construction of random systems of dyadic cubes in such spaces.
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An Improved Compact Embedding Theorem for Degenerate Sobolev Spaces
Degenerate matrix-weighted Sobolev spaces are shown to embed compactly into weighted L^q spaces locally, under local Poincaré and Sobolev inequalities with gain sigma>1, with an application to p-admissible weights.