Some Problems Of Convex Analysis In Carnot Groups

The main aim of this thesis is to extend some results of classical convex analysis to notions of Carnot groups, which contain the inequalities of Hadamard type, the Lipschitz continuity, the comparison principle and the second order derivatives for convex functions on Carnot groups G and regularity of quasiconvex functions on G.In Chapter 2 we recall the basic facts about Carnot groups G and some main notions of convexity on G.In Chapter 3, for a Carnot group G, we introduce the concept of r-convex functions and obtain some inequalities of Hadamard type for if-r-convex functions on G, our results are entirely new even for Euclidean spaces.In Chapter 4, we give a new proof of the local Lipschitz continuity of H-convex functions on the Heisenberg group. For a Carnot group of step two, we obtain a characterization of H-convex functions and prove that H-convex functions are locally Lipschitz continuous.In Chapter 5, for a Carnot group, we deal with the comparison principle for σ2(if)-convex functions and establish an estimate for sub-Laplacian operators, applying the main results in Chapter 4 and Chapter 5 we derive that the non-symmetrized distributional second derivatives are signed Radon measures, and obtain the existence a.e. of second order horizontal derivatives for H-convex functions in a Carnot group of step two.In Chapter 6, we introduce the concept of H-quasiconvex functions in Carnot groups G, and give some interesting properties similar to H-convex functions on G.