Let $X$ be a topological space and $\mu$ be the Borel measure on $X$. Suppose $W\_1$ and $W\_2$ are continuous, non-negative functions from $X$ into the real numbers such that, for all integers $p > 0$,
$$
\int\_X W\_i(x)\ d\mu(x) < +\infty, \ \ i = 1,2.
$$
Define the weighted $\mathcal{L}^p$-space $\mathcal{L}\_W^p(X)$ as the space of measurable functions from $X$ to the reals for which the integral
$$
\left(\int\_X f(x)^pW(x)\ d\mu(x)\right)^{1/p} 
$$
is finite. In this case the function $W$ is called the *weight function* for $\mathcal{L}\_W^p$. Furthermore, it is well known that the integral above constitutes a norm on $\mathcal{L}^p\_W(X)$. Under which conditions on the weight functions $W\_1$ and $W\_2$ are $\mathcal{L}\_{W\_1}^p(X)$ and $\mathcal{L}^p\_{W\_2}(X)$ topologically isomorphic? (i.e. when is there a Banach space isomorphism which is also a homeomorphism with respect to the norm-topologies due to the two weight functions?)



When are Weighted $\mathcal{L}^p$-Spaces Topologically Isomorphic?
The space $L^p\_W(X)$ is nothing else than the $L^p$ with respect to the finite measure $\mu\_W$ whose density with respect to $\mu$ is $W$. Now you can use the fact that the measure spaces determined by purely non-atomic Borel probability measures on "nice" spaces $X$ (separable, metrizable, complete) are all isomorphic to the unit interval endowed with the Lebesgue measure, which implies that for any fixed $p$ the corresponding $L^p$ spaces are all isometrically isomorphic.


