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If you are interested, we have given an intrinsic definition of a weak derivative for maps between manifolds A. Convent et J. Van Schaftingen, emphasized Intrinsic colocal weak derivatives and Sobolev …

Your question can be rephrased by asking whether one has a Hölder estimate $$ |u|_{W^{s, p}} \le C |u|_{W^{s, q}}, $$ when $p < q$ or whether $W^{s, q} \subset W^{s, p}$. There is no such embedding …

Yes they are. Step 1 There exists measurable sections $e_1, e_2, \dotsc, e_m$, where $m = \dim M$, of $TM$ (measurable functions mapping a point $x$ to a vector of its tangent plane $T_xM$) such that …

Here is a variational argument to prove that the maximizers do not change sign. If $f \in H^1 (\mathbb{R}^N)$ be a maximizer, $u$ can be written as $$ f = f_+ - f_-, $$ with $f_+ \ne 0$ and $f_- \ …

This seems to be a consequence of the Cauchy–Schwarz inequality: $$ \begin{split} \bigg\vert \sum_{\vert \alpha \vert = k} \sum_{\vert \beta \vert = k} \int_{\mathbb{R}^m} D^\alpha f D^\beta \phi \big …

One way to reformulate this is to consider a good extension $\bar{g}$ to the whole $\Omega$ of the function $g$, and then set $v = u - \bar{g}$. The function $v$ solves then the problem $$ \left\{ \b …

If $k \in (0, 2]$, we define the multiplier $$ m (\xi) = (1 + \vert \xi \vert^2)^\frac{k}{2} - \vert \xi \vert^k. $$ We observe that if $\vert \xi \vert \ge 2$, then by differentiability $$ \big …

This follows from two facts: $(H^s \cap L^\infty) (\mathbb{T})$ is a Banach algebra (see for example in the framework of fractional spaces $W^{s, p}$ ($W^{s, 2} = H^s$) Bourgain, Brezis, Mironescu, …

You can take as an example $$ u(x) = x_1^{k - 1} (\log \lvert x \rvert)^\beta: $$ if $\beta < 1 - \frac{1}{n}$, $u \in W^{k,n} (B_1)$ and if $\beta > 0$ then $D^{k - 1} u \not \in L^\infty (B_1)$.

You can represent on every simplex $\sigma$ of your triangulation the derivative $D f_k$ by $$ D f_k = \frac{1}{\lvert \sigma \rvert} \sum_{i = 0}^n \int_{\sigma} \Bigl(\frac{1}{(1 - \beta_i (x))^n} …

If $V_1 \ne \mathfrak{g}$, then $H W^{1, p} \ne W^{1, p}$. The construction is based on the notion of dilation on the Carnot group $\mathbb{G}$. The dilation $\delta_r$ is defined by for $X \in V_i$ …

Partial answer: according to Triebel (Theory of function spaces, 1983, Remark 2.7.5, p. 139), the trace of the Besov space $B^{1/p, p}_1 (\Omega)$ is $L^p (\partial \Omega)$, but the linear extension …

The property is proved in the litterature (together with its $W^{1, p}$ counterpart): Augusto C. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Different …

The map $(-\Delta + 1)^{-\frac{1}{2}}: L^p (\Omega) \to W^{1, p}_0 (\Omega)$ is a linear bijection when $\Omega$ is smooth and $1 < p < +\infty$, where $\Delta$ is the Laplacian with Dirichlet boundar …

The problem here is with the definition of $H^\ell (\partial D)$. Typically, $\partial D$ is a manifold and you can define this set by local charts. In order to define $H^\ell (\partial D)$ with $0 < …