Let . Since

is a control, the estimate

easily implies that for ,

We stress that it does not imply a bound on the 1-variation of the path . What we can get for this path, are bounds in -variation:

**Proposition:*** Let . There exists a constant , depending only on , such that for every and ,
where
*

**Proof:** This is an easy consequence of the Chen’s relations. Indeed,

and we conclude with the binomial inequality

We are now ready for a second major estimate which is the key to define iterated integrals of a path with -bounded variation when .

**Theorem:*** Let , and such that
and
Then there exists a constant depending only on and such that for and
where is the control
*

**Proof:** We prove by induction on that for some constants ,

For , we trivially have

and

.

Not let us assume that the result is true for with . Let

From the Chen’s relations, for ,

Therefore, from the binomial inequality

where

We deduce

with . A correct choice of finishes the induction argument