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Are there any known function spaces where the gradients are uniformly bounded? For a problem I’m working on, I’ve been able to show my functions are bounded in a ball within an RKHS (reproducing kernel Hilbert space), but I can’t show the gradients are bounded.

Just wondering if there are any such spaces that constrain the size of the gradient as well.

The reason I ask is because in this paper https://arxiv.org/pdf/1905.11882.pdf, the authors rely on showing the optimal "potential" functions lie in a set with a bounded Holder norm. They're able to show specific bounds on the Holder norms (lines 11 and 12 in that paper).

However, in my setting, I have only been able to show that the norm of my optimal potential function is bounded in an RKHS. But I can't find any way of controlling the norms of the gradients similar to what the authors did there. Or if I can't do that, are there standard bounds for the covering numbers if the functions are bounded and lie in an RKHS?

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  • $\begingroup$ On what kind of domain? Sobolev embedding should say that an appropriate Sobolev space would do the trick. As a trivial example, on an interval $I \subset \mathbb{R}$, a function in $H^1(I)$ is bounded (even absolutely continuous), so a function in $H^2(I)$ has bounded derivative (and is even $C^1$). $\endgroup$
    – Nate Eldredge
    Commented Mar 29, 2021 at 4:18
  • $\begingroup$ Just a real valued domain. $\endgroup$
    – Kashif
    Commented Mar 29, 2021 at 4:30
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    $\begingroup$ Why not $C^1_b(\Omega)$? (=bounded functions on the open set $\Omega$ with continuous bounded partial derivatives) $\endgroup$
    – Pietro Majer
    Commented Mar 29, 2021 at 8:08
  • $\begingroup$ Added edits to the question to give more detail. $\endgroup$
    – Kashif
    Commented Mar 29, 2021 at 11:49

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Maybe I'm misinterpreting your question but no nontrivial vector space of differentiable functions has uniformly bounded gradients. If $\inf_x \|\nabla f(x)\|= M>0$, then for $c>1$, $\inf_x\|\nabla cf(x)\|= cM>M>0$.

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