Motivated by Shuhao Cao's answer in Weighted Poincare Inequality, I checked out Kufner's book weighted Sobolev spaces.
Question Is the Jacobi weight either a power-type weight or a general weight in his sense?
Here I mean Jacobi weight by the weight function $(1-x)^\alpha(1+x)^\beta$ on $(-1,1)$.
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Kufner on p. 12 motivates the notion of weighted spaces as
The usefulness of the space $L^p(\Omega;\sigma)$ ($L^p$ space with the measure weighted by a function $\sigma$)...is...evident: it suffices to consider the special case $p=2$, $\Omega$ an interval $(a,b)$ and the weight of the form $(x-a)^\alpha(b-x)^\beta$, $\alpha,\beta\in\mathbb{R}$ and to recall the application of the corresponding weighted spaces in the theory of orthogonal polynomials.
So it seems the Jacobi weight is an important example that motivates the theory.
There are two types of weights he considers. Let me introduce them briefly. (See p. 17)
Let $$ d_{\partial\Omega}(x)=\inf_{y\in \partial\Omega}|x-y| $$ be the distance of the point $x$ from the set $\partial\Omega$ (boundary).
Power-type weight A weight function $\sigma$ is called power-type weight if it is of the form $$ \sigma(x)=[d_{\partial\Omega}(x)]^\epsilon $$ for some real number $\epsilon$.
General-type weights Let $s=s(t)$ be a continuous positive function defined for $t>0$ and such that either $$ \lim_{t\to 0}s(t)=0 $$ or $$ \lim_{t\to 0}s(t)=\infty. $$ A weight function $\sigma$ is called general-type weight if it is of the form $$ \sigma(x)=s(d_{\partial\Omega}(x)). $$
Going back to my question, is the Jacobi weight at least general-type weight? The book only considers either of the types, and it sounds weird that the weight he considers in the "Motivation" section, i.e., essentially the Jacobi weight, is not included in theory. I can see that the Jacobi weight is a product of two functions of distance to boundary points (end points) but I cannot see if it can be seen as power-type or general-type.
In case of $\Omega=(-1,1)$, the distance function is $d_\Omega(x)=\min(1-x,1+x)$, so this Jacobi weight is not of one of the types you mention.
However, close to the boundary, this weight behaves like a power-type weight. For instance, if $x$ is close to $1$, there is the lower bound $$ (1-x)^\alpha(1+x)^\beta \ge d_\Omega(x)^{\alpha} \ x\ge 1, $$ and this is all that matters for the use of such weights (in my experience).
Usually a partition of unity argument is involded, that localizes everything near a corner/end-point of $\Omega$. Then the behaviour of the weight away from the corner is not important. Also factors of the weight that are 'large' near the corner can be safely ignored in such circumstances.