Let $U\subset\mathbb{R}^n$ be open, and let $V\subsetneq W\subsetneq U$ be open sets with $\overline V$ compact and contained in $W$ and $\overline W$ compact and contained in $U$. Let $u\in H^1(U)$ satisfy \begin{align*} -\Delta u=f \end{align*} in the weak sense on $U$, where $f\in L^2(U)$. Then $u\in H^2(V)$ and there exists $C>0$, depending on $n$ and on the cutoff geometry needed to localize from $W$ to $V$, in particular on separation distances such as $\operatorname{dist}(V,\partial W)$, such that \begin{align*} \|u\|_{H^2(V)}\le C\left(\|f\|_{L^2(W)}+\|u\|_{L^2(W)}\right). \end{align*}

Let $U\subset\mathbb{R}^n$ be bounded and open. Let $u,v\in C^2(U)\cap C(\overline{U})$ satisfy \begin{align*} -\Delta u\le -\Delta v \quad \text{in } U, \end{align*} and \begin{align*} u\le v \quad \text{on } \partial U. \end{align*} Equivalently, $-\Delta(u-v)\le 0$ in $U$ and $u-v\le 0$ on $\partial U$. Then $u\le v$ in $U$.

Let $U\subset\mathbb{R}^n$ be open, and let $A\in L^\infty(U;\mathbb{R}^{n\times n})$ be a real symmetric matrix field uniformly elliptic with constants $\theta$ and $\Theta$. Let $u\in H^1_{\mathrm{loc}}(U)$ be a weak solution of \begin{align*} -\sum_{i,j=1}^n \partial_{x_i}\left(A_{ij}\partial_{x_j}u\right)=0 \end{align*} in $U$. If $V\Subset W\Subset U$ are open sets, then there exist $\gamma\in(0,1)$ and $C>0$, depending on $n$, $\theta$, $\Theta$, and the compact-containment geometry of $V\Subset W$, such that $u$ has a representative in $C^{0,\gamma}(V)$ and \begin{align*} \|u\|_{L^\infty(V)} + \sup_{\substack{x,y\in V\\x\ne y}}\frac{|u(x)-u(y)|}{|x-y|^\gamma} \le C\|u\|_{L^2(W)}. \end{align*}

Let $U\subset\mathbb{R}^n$ be bounded and open. Let $A\in L^\infty(U;\mathbb{R}^{n\times n})$ be uniformly elliptic with constants $\theta$ and $\Theta$. Let $u\in H_0^1(U)$ satisfy \begin{align*} \int_U \sum_{i,j=1}^n A_{ij}\partial_{x_j}u\,\partial_{x_i}v\,d\mathcal{L}^n = f(v) \end{align*} for every $v\in H_0^1(U)$, where $f\in H^{-1}(U)$. Then there exists $C>0$, depending only on $U$ and $\theta$, such that \begin{align*} \|u\|_{H_0^1(U)}\le C\|f\|_{H^{-1}(U)}. \end{align*}

Let $X,Y: (\Omega, \mathcal F) \to (\mathbb R, \mathcal B(\mathbb R))$ be independent real-valued random variables with laws $\mu$ and $\nu$. Then $X+Y$ has law $\mu * \nu$.

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $A,B \in \mathcal F$ with $\mathbb P(B)>0$. Then $A$ and $B$ are independent if and only if \begin{align*} \mathbb P(A \mid B)=\mathbb P(A). \end{align*} If also $\mathbb P(A)>0$, this is equivalent to \begin{align*} \mathbb P(B \mid A)=\mathbb P(B). \end{align*}

Let $X_1, \ldots, X_n$ be random variables taking values in countable sets $E_1, \ldots, E_n$. Then $X_1, \ldots, X_n$ are independent if and only if for every $(x_1, \ldots, x_n) \in E_1 \times \cdots \times E_n$, \begin{align*} \mathbb P(X_1=x_1, \ldots, X_n=x_n) = \prod_{i=1}^n \mathbb P(X_i=x_i). \end{align*}