Dear Colleagues,

I have a question concerning smoothness classes defined in terms
of decay of Fourier coefficients. I formulate my question as a
conjecture. I believe that the conjecture is true, but I was not able
neither to prove, nor to disprove it. I hope somebody will be able to do
this, or, at least, to give me a hint.

Best wishes, Alexei
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\documentclass[12pt]{article}
\newtheorem{conjecture}{Conjecture}
\newcommand{\ind}{\mathrm{ind}\,}
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\begin{document}
Let ${\bf T}$ be the unit circle. For a complex-valued function $a\in
L1({\bf T})$, let
$\{a_k\}_{k=-\infty}^\infty$ be the sequence of the Fourier coefficients
of $a$,
\[
a_k:=\frac{1}{2\pi}\int_0^{2\pi}a(e^{i\theta})e^{-ik\theta}d\theta.
\]
Let $W$ be the Wiener algebra of all functions $a$ on ${\bf T}$ of the form
\[
a(t)=\sum_{k=-\infty}^\infty a_k t^k
\quad (t\in{\bf T})
\]
for which
\[
\|a\|_W:=\sum_{k=-\infty}^\infty |a_k|<\infty.
\]
It is well known that $W$ is a Banach algebra under the norm $\|\cdot\|_W$
and that $W$ is continuously imbedded into $C({\bf T})$, the Banach algebra
of all complex-valued continuous functions with the maximum norm.
We shall denote the Cauchy index of a continuous function $a$
by $\ind a$.

Let $p:[0,\infty)\to[0,\infty)$ be a right-continuous non-decreasing
function such that $p(0)=0$, $p(t)>0$ for $t>0$, and
$\lim\limits_{t\to\infty} p(t)=\infty$.
Then the function $q(s)=\sup\{t :\ p(t)\le s\}$
(defined for $s\ge 0$) has the same
properties as the function $p$. The convex functions $\Phi$ and $\Psi$
defined by the equalities
\[
\Phi(x):=\int_0^x p(t) dt,
\quad
\Psi(x):=\int_0^x q(s)ds
\quad (x\ge 0)
\]
are called \textit{complementary $N$-functions}.
An $N$-function $\Phi$ is said to satisfy the $\Delta_20$-condition
if
\[
\limsup_{x\to 0}\frac{\Phi(2x)}{\Phi(x)}<\infty.
\]
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\begin{conjecture}
There exist sequences $\{\varphi_k\}_{k=0}^\infty,
\{\psi_k\}_{k=0}^\infty$ of positvie numbers and constants
$C_\varphi,C_\psi,M\in(0,\infty)$ such that
\begin{enumerate}
\item[{\rm (a)}]
$\varphi_0=\psi_0=1$;
\item[{\rm (b)}]
for all $k\in{\bf N}$,
\[
\varphi_{k-1}\le\varphi_k,
\quad
\psi_{k-1}\le\psi_k;
\quad
\varphi_{2k}\le C_\varphi \varphi_k,
\quad
\psi_{2k}\le C_\psi\psi_k,
\quad
k\le M \varphi_k\psi_k;
\]
\end{enumerate}
there exist complementary $N$-functions $\Phi,\Psi$ both satisfying
the $\Delta_20$-condition, and there exists a function $a\in W$ such that
\[
a(t)\ne 0\ \mbox{for all}\ t\in{\bf T},
\quad\quad
\ind a=0;
\]
for all $p\in(1,\infty)$ and all $\alpha\in[0,1]$,
\[
\sum_{k=1}^\infty \Big(|a_{-k}|(k+1)^\alpha\Big)^p
+
\sum_{k=0}^\infty \Big(|a_k|(k+1)^{1-\alpha}\Big)^{p/(p-1)}=\infty,
\]
but
\[
\sum_{k=1}^\infty \Phi(|a_{-k}|\varphi_k)
+
\sum_{k=0}^\infty \Psi(|a_k|\psi_k)<\infty.
\]
\end{conjecture}
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\end{document}

--
Alexei Yu. Karlovich (Oleksiy Karlovych)
Departamento de Matematica
Instituto Superior Tecnico
Av. Rovisco Pais
1049-001 Lisboa,
Portugal

Phone: +351-21 8417037
Fax: +351-21 8417598
E-mail: akarlov@...
http://www.math.ist.utl.pt/~akarlov