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## Bound the value of function by integration of derivatives

Given $f(x,y) \in C^2([0,1]^2)$ (by which I mean $C^2$ in some open neighborhood), with $f_x, f_y, f_{xy} \in L^1([0,1]^2, dxdy)$ (which is sure case since they are continuous), does the following hold? $$\sup |f| \le \iint |f|+|f_x|+|f_y|+|f_{xy}|\, dxdy$$ I think it is safe to discretise this function under the assumptions, divide $f$ into […]

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## Let $A$ be a finite set in a metric space. Prove that $H_p(A)=0$ for all $p>0$.

Suppose that $(X,d)$ is a metric space and let $p\geq 0, \delta >0$. Let $\mathcal{E}$ be the collection of those subsets of $X$ with diameter $\leq \delta$ together with the set X, and define $d(A) = (\text{diam }A)^p$. We define: \begin{equation*} H_p^{\delta}(A)=\inf\,\{\sum_{j=1}^\infty d(E_j) : E_j \in \mathcal{E} \text { and } A \subset \bigcup_{j=1}^{\infty} E_j […]

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## Tightness of Hilbert-space-valued arrays

Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n$ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}}^2 < \infty$ that is strictly stationary (which means that $W_{n,1},\ldots , W_{n,n}$ is strictly stationary for every $n$). After Remark 3.3 in “Central Limit and Functional Central […]