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Let $n \in \mathbb N^* , \alpha \in \mathbb R$, and $w_i \in \mathbb R$ for all $i=1, \ldots, n$. Consider the map $f:\mathbb R_{\ge 0} \to \mathbb R$ defined by $$ f(x) := \sum_{i=1}^n w_i x^{\color{ …

Let $E := \mathbb R^d$. Let $f:E \to \mathbb R_{>0}$ be continuous and integrable. For $r>0$, we define $$ f_r (x) := \frac{f(x)}{ \frac{1}{|B(x, r)|} \int_{B(x, r)} f(y) \, \mathrm{d} y} \quad \foral …

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$ be the Bochner space of all $\mu$-integrable fun …

I'm reading a proof of below theorem from this paper. Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ass …

Let $\Omega$ be a metric space, $C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and $\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega …

Let $b:\mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R^d \to \mathcal M_{ d\times q} (\mathbb R)$ be Lipschitz. Let $(W_t, t\ge 0)$ be the standard $q$-dimensional Brownian motion. Then $$ d X_t = …