Reproducing kernel Hilbert spaces from Sobolev spaces with weight/density functions

Question

I would like to understand which of the statements about the Sobolev space $H^1(\mathbb{R})$ remain true if one introduces a density/weight function in the definition.

Details

The Sobolev space $H^1(\mathbb{R})$ are those square integrable functions whose first weak derivatives exist almost everywhere and are square integrable or briefly $f^2\in L^1$ and $(f')^2\in L^1.$ This space $H^1(\mathbb{R})$ has the following properties

1. It is a reproducing kernel Hilbert space with inner product $$\langle f,g\rangle=\int_{\mathbb{R}} f(x)g(x)\,dx + \int_{\mathbb{R}} f'(x)g'(x)\,dx.$$
2. The functions in $H^1(\mathbb{R})$ are continuous.

Let $w:\mathbb{R}\rightarrow\mathbb{R}$ be a weight function or density, which is a strictly positive function with $\int w(x)\,dx=1.$ Now define the weighted $L^1$ space as $L^1(w):=\left\{ f:\mathbb{R}\rightarrow\mathbb{R} \mid fw\in L^1\right\}$ with norm $\lVert f\rVert_w=\int_{\mathbb{R}} |f(x)| w(x)\,dx$ and the weighted Sobolev space $H^1_w(\mathbb{R})$ by replacing $L^1$ with $L^1(w)$ in the definition above. This means the inner product of $H^1_w(\mathbb{R})$ would be $$\langle f,g\rangle_w=\int_{\mathbb{R}} f(x)g(x)w(x)\,dx + \int_{\mathbb{R}} f'(x)g'(x)w(x)\,dx.$$

Question

Does $H^1_w(\mathbb{R})$ still have the two properties? I.e. is it still a reproducing kernel Hilbert space consisting of continuous functions?

EDIT: The literature for weighted Sobolev spaces seems to focus on weights which are of "Muckenhoupt class" (see this related question) or "doubling measures". But finite measures are never doubling measures.

EDIT2 I would like to use the weights in applications to control the asymptotic behaviour of the functions in the RKHS. E.g. I would like to have spaces containing constant functions, polynomials (up to a certain degree) or exponential functions. This means I am quite relaxed about the properties of $w$. $w$ may be assumed to be continuous or even differentiable, if this helps. Typical examples for $w$ would be functions such as $\exp(-x^2)$, $\frac{1}{\cosh x}$ or $\frac{1}{(1+x^2)^k}$.

Answer 1

Suppose $w$ is continuous and strictly positive.

Let $n\in\Bbb N$ be given. We will show that functions in $H_w^1(\Bbb R)$ continuous on $(-n,n)$ (which implies that they are continuous).

Let $c_n>0$ be the minimum of $w$ over $[-n,n]$ and let $f\in H_w^1(\Bbb R)$. Then we have $$ \|f\|_{H^1((-n,n))}^2 = \int_{-n}^n |f(t)|^2 + |f(t)'|^2 \,\mathrm dt \leq \int_{-n}^n |f(t)|^2 + |f(t)'|^2 c_n^{-1} w(t) \,\mathrm dt = c_n^{-1} \|f\|_{H_w^1((-n,n))}^2 \leq c_n^{-1} \|f\|_{H_w^1(\Bbb R)}^2. $$ Thus $\|f\|_{H^1((-n,n))}$ is finite and therefore $f\in H^1((-n,n))$ (if we restrict $f$ to $(-n,n)$). Since functions in $H^1((-n,n))$ are continuous, there exists a continuous representative of $f$ on $(-n,n)$.

Can the continuous representative depend on $n$?: Suppose $n<m$ and $f_n$, $f_m$ are continuous representatives on $[-n,n]$, $[-m,m]$. Then we have $f_n = f_m$ a.e. on $[-n,n]$, and since both functions are continuous on $[-n,n]$, this means that $f_n=f_m$ everywhere on $[-n,n]$. Thus, one can always extend the continuous representative to a larger interval, while still being continuous. In the end, we can define $f(x) = f_n(x)$, where $n$ is such that $x\in[-n,n]$, and similar to the above arguments, the function $f$ has to be continuous.

We can also obtain a second thing from the inequality: If for a given $x\in \Bbb R$ we choose $n$ such that $x\in (-n,n)$, we can use the RKHS-Property of the unweighted $H^1((-n,n))$ to obtain $$ |f(x)| \leq C\|f\|_{H^1((-n,n))} \leq Cc_n^{-1} \|f\|_{H_w^1(\Bbb R)}. $$ This inequality shows that the point evaluation functionals $\delta_x$ are continuous, which implies that $H_w^1$ is a RKHS (see Theorem 2.28 in the linked document) (note that the constants $C,c_n$ can depend on $x$, but this is ok).