Non-Doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Carathéodory Spaces.

Intro -- Contents -- Chapter 1. Introduction -- 1.1. Carnot-Carathéodory spaces -- 1.2. The Chow-Rashevsky's accessibility theorem and CC metrics -- 1.3. The Nagel-Stein-Wainger polynomial and the size of the CC balls -- Chapter 2. Carnot groups -- 2.1. Carnot groups of step 2 -- 2.2. The Kaplan mapping -- 2.3. Groups of Heisenberg type -- Chapter 3. The characteristic set -- 3.1. A result of Derridj on the size of the characteristic set -- 3.2. Some geometric examples -- 3.3. Non-characteristic manifolds -- 3.4. Manifolds with controlled characteristic set -- Chapter 4. X-variation, X-perimeter and surface measure -- 4.1. The structure of functions in BV[sub(X,loc)] -- 4.2. X-Caccioppoli sets -- 4.3. X-perimeter and the perimeter measure -- Chapter 5. Geometric estimates from above on CC balls for the perimeter measure -- 5.1. A fundamental estimate -- 5.2. The X-perimeter of a C[sup(1,1)] domain is an upper 1-Ahlfors measure -- Chapter 6. Geometric estimates from below on CC balls for the perimeter measure -- 6.1. The relative isoperimetric inequality and Theorem 6.1 -- 6.2. A basic geometric lemma -- 6.3. Further analysis for Hörmander vector fields of step 2 -- 6.4. Second proof of Theorem 6.1 -- 6.5. Failure of the 1-Ahlfors condition for the X-perimeter of C[sup(1,α)] domains -- Chapter 7. Fine differentiability properties of Sobolev functions -- 7.1. Poincaré inequality fractional integrals and improved representation formulas -- 7.2. Fine mapping properties of fractional integration on metric spaces -- 7.3. Differentiation with respect to an upper Ahlfors measure -- 7.4. Upper Ahlfors measures and Hausdorff measure -- Chapter 8. Embedding a Sobolev space into a Besov space with respect to an upper Ahlfors measure -- 8.1. Some results from harmonic analysis -- 8.2. Two simple growth-estimates.

8.3. A key continuity estimate for a singular integral -- 8.4. The main theorem -- Chapter 9. The extension theorem for a Besov space with respect to a lower Ahlfors measure -- 9.1. Some auxiliary results -- 9.2. Proof of Theorem 9.1 -- Chapter 10. Traces on the boundary of (ε, δ) domains -- 10.1. The (ε, δ) condition is optimal for the existence of traces -- 10.2. Characterization of the traces on the boundary -- Chapter 11. The embedding of B[sup(p)][sub(β)](Ω, dμ) into L[sup(q)](Ω, dμ) -- Chapter 12. Returning to Carnot groups -- Chapter 13. The Neumann problem -- Chapter 14. The case of Lipschitz vector fields -- Bibliography.

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