For $u(x) = |x|^\alpha$ on the unit ball $B_1(0) \subset \mathbb{R}^n$:

• $u \in L^p(B_1)$ if $\alpha > -n/p$
• $u \in W^{1,p}(B_1)$ if $\alpha > 1 - n/p$
• $u \in C^0(\overline{B_1})$ if $\alpha \geq 0$

The function $|x|$ is in $H^1$ in 1D but $|x|^{1/2}$ is not. In 3D, $1/|x|$ is in $L^2$ locally but not $H^1$ near the origin.

Mollifiers create smooth approximations: For $\eta_\epsilon(x) = \epsilon^{-n}\eta(x/\epsilon)$ where $\eta \in C_c^\infty$ with $\int \eta = 1$, define: $u_\epsilon = \eta_\epsilon * u$

If $u \in W^{k,p}(\Omega)$, then $u_\epsilon \in C^\infty$ and $u_\epsilon \to u$ in $W^{k,p}_{\text{loc}}$ as $\epsilon \to 0$.

This proves density of smooth functions and justifies treating Sobolev functions as limits of smooth ones.

For $\Omega = \mathbb{R}^n_+ = \{x_n > 0\}$, the extension: $Eu(x', x_n) = \begin{cases} u(x', x_n) & x_n > 0 \\ \sum_{k=1}^m a_k u(x', -kx_n) & x_n < 0 \end{cases}$ with appropriately chosen $a_k$ maps $W^{k,p}(\mathbb{R}^n_+) \to W^{k,p}(\mathbb{R}^n)$ boundedly.

This reduces problems on half-spaces to problems on whole space, where Fourier methods apply.

On a Riemannian manifold $(M, g)$, define $H^1(M)$ using: $\|u\|_{H^1}^2 = \int_M (u^2 + |\nabla_g u|^2)\,dV_g$

where $\nabla_g$ is the covariant derivative and $dV_g$ the volume form. Sobolev theory extends to this setting, essential for geometric PDEs.

For studying equations with singular coefficients, use weighted spaces: $\|u\|_{H^1_w}^2 = \int_\Omega (w_0 u^2 + w_1|\nabla u|^2)\,dx$

where $w_0, w_1$ are weight functions. These arise in problems with degeneracies or infinite domains (e.g., $w_1(x) = e^{-|x|}$ for exponentially weighted spaces).