The five selections that make up the main body of the text are devoted to geodesics in sub-Riemannian geometry, the geometry of subelliptic diffusions, the geometric foundations of rough paths, Sobolev and

bounded variation functions on metric measure spaces, and singularities of vector distributions.

In the future we shall study existences of extremal functions using suitable spaces of functions of

bounded variation (c.

1] : f [member of] C(J x R, R), A is a function of

bounded variation and the measure dA is non-negative.

There are no infinite dimensional closed subspaces of C [0,1] composed by just functions of

bounded variation.

for each x [member of] [a, b], provided f is of

bounded variation on [a, b], while u: [a, b] [right arrow] R is r-H-Holder continuous, i.

Hardy (1917) introduced the notions of regular convergence of double sequences and the notion of

bounded variation of double sequences as follows:

Let TV[0,1], respectively AC[0,1], denote the class of all functions of

bounded variation, respectively the absolutely continuous functions on [0,1].

Among the topics are variation on the p-Laplacian, extremal functions in Poincare-Sobolev inequalities for functions of

bounded variation, homocline type solutions for a class of differential equations with periodic coefficients, the cooperative case of quasilinear and singular systems, weighted asymmetric problems for an indefinite elliptic operator, multiple non-trivial solutions of the Dirichlet problem for the prescribed mean curvature equation, and existence of nodal solutions for some nonlinear elliptic problems.

As our main results will demonstrate, the perimeter which can be computed from the covariogram is the one from the theory of functions of

bounded variation (Ambrosio et al.

The results are stronger than that in paper [5], meanwhile the results are essential generalization of continuous dependence of

bounded variation solutions on parameters for Kurzweil equations in paper [6].

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (where the sum is taken over {b'} which form a finite Borel partition of b, for all Borel sets bin B; then [mu] is said to be of

bounded variation.

By definition for [PHI] (u) = u we have the ordinary

bounded variation V (f, [DELTA]) of the function f on a segment [DELTA].