Newtonian space

Remarks

Example: on $X = \R$ with Lebesgue measure, $u(x) = \max(0, 1 - \abs{x})$ lies in $\widetilde{N}^{1,p}(\R)$. Indeed $u \in L^p$, and $g = \mathbf{1}_{[-1, 1]}$ is an upper gradient in $L^p$, since $\abs{u(s) - u(t)} \le \abs{\int_s^t g}$ for real $s, t$ and a curve from $\gamma(a)$ to $\gamma(b)$ covers at least the interval between them.

Non-example: on $X = [0, 1]$ with Lebesgue measure, $u = \mathbf{1}_{[1/2, 1]}$ is in $L^p$ but has no upper gradient in $L^p$. An upper gradient $g$ would need $\int_{1/2 - \varepsilon}^{1/2 + \varepsilon} g \ge 1$ for every $\varepsilon > 0$, and these integrals tend to $0$ when $g$ is integrable.

The space $N^{1,p}(X)$ is a Banach space. Its elements are classes of functions defined at every point, and two functions in one class agree outside a set of $p$-capacity zero, not only almost everywhere. The definition is often given with $p$-weak upper gradients, which may fail the curve inequality on a family of curves of $p$-modulus zero. This gives the same space and the same norm. If $X$ contains no nonconstant rectifiable curves, for example a Cantor set, then $g = 0$ is an upper gradient of every function and $N^{1,p}(X) = L^p(X, \mu)$. For an open set $\Omega \subset \R^n$ with Lebesgue measure and $1 < p < \infty$, $N^{1,p}(\Omega)$ agrees with the Sobolev space $W^{1,p}(\Omega)$, each class of $W^{1,p}$ being represented by suitable pointwise defined functions.