Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int \_{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space,
 $$A(\mathbb T):= \{f\in L^{1}(\mathbb T): \hat{f}\in \ell^{1}(\mathbb Z), \ \text {that is,} \ \sum\_{n\in \mathbb Z} |\hat{f}(n)| < \infty \}.$$
$A(\mathbb T)$ is normed by the $L^{1}-$ norm on $\mathbb Z$:
$$||f||= \sum\_{n\in \mathbb Z} |\hat{f}(n)|; \ \text {for} \ f\in A(\mathbb T). $$
We also note that $A(\mathbb T)$ is a Banach algebra under pointwise addition and multiplication.
Let $f\_{0}\neq 0 \in A(\mathbb T)$ and fix it; and take $M= 2||f\_{0}||$ and put $B\_{M}= \{f\in A(\mathbb T): ||f||\leq M \}.$


Let $f, g \in B\_{M}.$



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> **My Question**: Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ , where $C$ is some constant ? If yes, what can we say about $C$ ?
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I guess, the trivial relation: $|f|^{2}f-|g|^{2}g=(f-g)|f|^{2}+g(|f|^{2}-|g|^{2})$; may be useful.


Thanks,



Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ?
What's your motivation?


The Fourier transform is a \*-homomorphism for the natural adjoint on $\ell^1(\mathbb Z)$ and so if $f$ is the transform of $a\in\ell^1(\mathbb Z)$ then $|f|^2f$ is the transform of $a^\*aa$. Similarly let $g$ and $b$ be related. As $\|a\|,\|b\| \leq M$ we perform a simply triangle-inequality argument, which is valid in any Banach $\*$-algebra:
\begin{align\*} \| a^\*aa - b^\*bb\| &= \| (a^\*-b^\*)aa + b^\*(a-b)a + b^\*b(a-b) \|
\\ &\leq \|a-b\|\|aa\| + \|b\| \|a\| \|a-b\| + \|b^\*b\| \|a-b\|
\leq 3M^2\|a-b\|. \end{align\*}


