Let [D.sub.r] = {z [member of] C(I, [E.sub.w], [parallel]z[parallel] [less than or equal to] r}, BV(I, R) denote the space of real

bounded variation functions with its classical norm [[parallel] x [parallel].sub.BV].

We say that u is of

bounded variation on the interval [a, b] whenever [[disjunction].sup.b.sub.a] u is finite.

Thereby [mathematical expression not reproducible] and BVH(G) is the space of functions on G with

bounded variation in the sense of Hardy.

In the centered setting, Kurka [12] showed that if f is of

bounded variation on R, then inequality (2) holds for M (with constant C = 240, 004).

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.

We start this section by recalling some pertinent concepts and key lemmas from the function of

bounded variation, fuzzy numbers, and fuzzy number equivalence classes which will be used later.

[xi] is a nondecreasing function of

bounded variation with real values and satisfies [[integral].sup.+[i nfinity].sub.0] [mu](t)d[xi](t) > 0; [[integral].sup.+[infinity].sub.0][mu](t)y'(t)d[xi](t) denotes the Riemann-Stieltjes abstract integral of y' with respect to [xi] and y is a weighted real function.

it is of

bounded variation with respect to the variables (t, x) and differentiable with respect to the third variable [lambda].

space of functions of

bounded variation. Therefore the space [W.sup.1,1.sub.[gamma],0]([R.sup.n]) is too small to discuss existences of extremal functions of the best constants.

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.e., we recall that: