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We have the obvious embeddings, for $s_1 \leq s_2$ and $r_1 \leq r_2$, \begin{align} W^{s_2,r} \subseteq W^{s_1,r}, \\ W^{s,r_2} \subseteq W^{s,r_1}. \end{align} Now, is the following result true? … \end{equation} Of course, due to the embeddings above, $W^{\max(s_1,s_2),\max(r_1,r_2)}$ is included in $W^{s_1,r_1}$ and $W^{s_2,r_2}$ and therefore in their intersection. …

Set $w(x) = (1 + |x|^2)^{1/2}$ with $|\cdot|$ the Euclidian norm on $\mathbb{R}^n$. For $s,\mu \in \mathbb{R}$, we define the Sobolev space $$H_2^{s}(\mathbb{R}^n) = \left\{f : \lVert f \rVert_{s} := …

Let $D(\mathbb{R})$ be the space of functions from $\mathbb{R}$ to $\mathbb{R}$ that are right continuous with left limits (also referred to as càdlàg functions). $D(\mathbb{R})$ is often called the …

Question: Are there embeddings results from $B_{p_1,q_1}^{s_1}(\mathbb{R}^d;\mu_1)$ into $B_{p_2,q_2}^{s_2}(\mathbb{R}^d;\mu_2)$ for $p_1 > p_2$? …