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Given $\Omega$ as $[0,1]^n$ or the closed unit ball in $\mathbb{R}^n$, we can consider the algebra of complex valued polynomials with pointwise multiplication and its closure with respect to the norm

$$ \|p\| = \|p\|_\infty + \sum_{k=1}^n\left\| \frac{\partial p}{\partial x_k} \right\|_1. $$

I am wondering if this Banach algebra has been studied. If so, does it have a common name, and what are some resources that I can read about it?

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  • $\begingroup$ What multiplication operation? Pointwise multiplication? What is your bound for $\|\nabla(pq)\|_1$ ? $\endgroup$
    – Gerald Edgar
    Commented Mar 1, 2021 at 12:08
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    $\begingroup$ As you probably know, for $n=1$ this is (possibly up to some renorming) the Banach algebra of absolutely continuous functions AC[0,1]. Some years ago I was looking for higher-dimensional analogues and I think one candidate was the example you describe, but I don't recall seeing it explicitly defined and studied as a Banach algebra $\endgroup$
    – Yemon Choi
    Commented Mar 1, 2021 at 13:07
  • $\begingroup$ @GeraldEdgar The definition seems to make sense to me: is your point that the norm defined above might not be submultiplicative? $\endgroup$
    – Yemon Choi
    Commented Mar 1, 2021 at 13:12
  • $\begingroup$ Sorry, I have corrected the norm. It should be submultiplicative now. As for the operation, it is pointwise multiplication. I'll edit that in too. $\endgroup$
    – Alan
    Commented Mar 6, 2021 at 0:57
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    $\begingroup$ @DCM the background of my research consists of operators that have an $AC[a,b]$ functional calculus and their integral representations. Trying to generalise the theory to compact subsets of $\mathbb{C}$ with an appropriate class of functions is why I was wondering if this object has been studied. $\endgroup$
    – Alan
    Commented Mar 7, 2021 at 11:58

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