I have come across a nice result attributed to Ciesielski (Ciesielski, Z. (1960). On the isomorphisms of the spaces $H_{\alpha}$ and m. Bull. Acad. Pol. Sci. Ser. Sci. Math. Phys. 8, 217–222.), even if I can't find the original paper. It gives a sequential equivalent of $\alpha$-Hölder-norm for any function $f$ in $H_{\alpha}([0,1])$, defined by, \begin{aligned} \|f\|_\alpha^{seq}:=\sup _{j \geqslant 1} 2^{j \alpha} \max _{1 \leqslant k \leqslant 2^{j-1}} \mid f\left((2 k) 2^{-j}\right) & -2 f\left((2 k-1) 2^{-j}\right)+f\left((2 k-2) 2^{-j}\right)|+| f(0)|+|f(1)| \end{aligned} This is pretty handy to study some stochastic processes (such as Brownian Motion). As I am looking at random fields, I wonder if, somehow, somewhere, this result has been generalized for functions in $H_{\alpha}([0,1]^{d})$, $d>1$ ?
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$\begingroup$ The article I am currently reading states that $\|f\|_{\alpha}^{seq}$ is equivalent to $$\|f\|_{\alpha}=|f(0)|+|f(1)|+\sup _{\substack{s, t \in[0,1]}} \frac{|x(t)-x(s)|}{|t-s|^{\alpha}}$$ $\endgroup$– BabaUtahCommented Feb 14, 2023 at 12:59
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$\begingroup$ This is exactly this idea of discretized increments that I find attractive, but I don't know if it can be generalized well in higher dimensions than 1. $\endgroup$– BabaUtahCommented Feb 14, 2023 at 13:03
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$\begingroup$ I think that this article : math.univ-lille1.fr/~suquet/Articles/georgia1.pdf answer my question, thank you. $\endgroup$– BabaUtahCommented Feb 14, 2023 at 14:24
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$\begingroup$ isn’t the generalisation to higher dimension immediate if you apply the 1 estimate to each variable? $\endgroup$– Pietro MajerCommented Feb 14, 2023 at 14:49
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