Vol 4, 1994 cl CHANGES OF VARIABLE BEURLING-HELSON TYPE THEOREM 219

If v^ : T —^ T is nonlinear and belongs to C^, then

1к^ " 1л , ( т ) >Ф . 9^ ) И1^ " ^ n^T-' (3.1)

Thus (0.1) is unbounded for 1 < p < 2 [A].

Indeed if |^"(0I > p > 0 on the interval / Ç T then the van der Courput lemma gives

1 ( 1 / . e-^^^)^(A:)| < p-i/2|n|-i/2 , VA: G Z ,

and by interpolation between i^ ^^nd i^o for q = p/p — 1 we get

11 ( 1 / . e-^^^)^||,^ < 11(1/ • e-^^^)^||^^-2)/^ . 11(1/ . ^-гпч^^\^1ч <

<c ( / , p ) |n|^ 2 .

So ,

|J| / 27r= / е^^^(^^1/(^)е-^'^^(')^^27г I Jt

<

<5 ] | e-^(fc)|.|(l/e—^)^(fc)|<

fe€Z

and (3.1) follows.

We note that some modification of the van der Corput lemma (see [Kau], [Bj]) shows that if 99 6 C^ and all level sets {t : 9?'(t) = Ь\ are of measure zero, then

and as well as the above we obtain unboundedness of Це^^'^Ц^ .

However for an arbitrary nonlinear C^ mapping this approach is not fective: operator (0.1) in this case is always unbounded (see §4), but norms of e*^"^ in Ap{J) can be bounded even for some essentially nonlinear morphism (f of the circle. More exactly:

THEOREM 2. There exists a C^ diffeomorphism (p :J -^ J which is not hnear on any interval and at the same time Vp > 1

1|е^ " Л1л , ( т ) = 0(1) , InHoo, nez. (3.2)

( Obviously the only nontrivial case is 1 < p < 2.)