Abstract:
We consider the space $M_p(\mathbb R^d)$ of $L^p$-Fourier
multipliers and give a detailed proof of the following result
announced by the authors in $\lbrack10\rbrack$: if $\varphi\colon\mathbb
R^d\to \lbrack0, 2\pi\lbrack$ is a measurable function and
$\|e^{in\varphi}\|_{M_p}=O(1)$, $n\in\mathbb Z$, for some $p\ne
2$, then the function $\varphi$ is linear in domains complementary
to some closed set $E(\varphi)$ of Lebesgue measure zero, and the
set of values of the gradient of $\varphi$ is finite. We also
consider the question of which sets can appear as $E(\varphi)$. We
study the behaviour of the norms of the exponential functions
$e^{i\lambda\varphi}$ in the case when the frequency $\lambda$ tends
to infinity along a sequence of real numbers. In particular, we
construct a homeomorphism $\varphi$ of the line $\mathbb R$ which is
non-linear on every interval and satisfies
$\|e^{i2^n\varphi}\|_{M_p(\mathbb R)}=O(1)$, $n=0, 1, 2,\dots$,
for all $p$, $1<p<\infty$.
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