## Reference for compact embedding between (weighted) Holder space on $\mathbb{R}^n$

Suppose $$0<\alpha<\beta<1$$, and $$\Omega$$ is a bounded subset of $$\mathbb{R}^n$$. Then the Holder space $$C^{\beta}(\Omega)$$ is compactly embedded into $$C^{\alpha}(\Omega)$$. But if $$\Omega=\mathbb{R}^n$$, then the compact embedding is not true.

However, if we consider the weaker weighted Holder space $$C^{\alpha, -\delta}(\mathbb{R}^n)$$ (for any $$\delta>0$$) instead of $$C^{\alpha}(\mathbb{R}^n)$$. Then is $$C^{\beta}(\mathbb{R}^n)$$ compactly embedded to $$C^{\alpha, -\delta}(\mathbb{R}^n)$$?

Here $$\|f\|_{C^{\alpha, -\delta}}=\|(1+|\cdot|^2)^{-\frac{\delta}{2}}f\|_{C^{\alpha}}.$$

I could not find a precise reference from some books on functional analysis. Any comment is welcome.

## square weighted l^2

I am looking the sequence spaces $$l^1$$ and $$\{(x_k)_k: \|x\|_{sq}^2 := \sum_{k=1}^\infty k^2\cdot x_k^2 < \infty\}.$$

I am particularly interested in relations between their respective norms: It is fairly easy to show that $$\|\cdot \|_{sq}$$ is not bounded above by $$\|\cdot\|_1$$, for example $$x_k^{(n)} := k^{-1/2}\cdot \delta_{k,n}$$, i.e. $$x^{(n)} = (0,0,\ldots,0,n^{-1/2},0,\ldots)$$. Then $$\|x^{(n)}\|_1\to 0$$ but $$\|x^{(n)}\|_{sq} \to \infty$$.

I am struggling with the other direction. I can find neither a proof that $$\|\cdot\|_1$$ is bounded above by $$\|\cdot \|_{sq}$$, neither can I construct a counterexample. I believe that the former is actually true.

## ‘Weighted’ distance transform

A distance transform (as done by DistanceTransform[]) on a scattering of background pixels effectively creates a landscape of basins with conical morphology, each one starting from the same depth (zero) and with the same slope (one). The ridges between the basins are therefore equidistant between the minima, which is why a watershed transform after a distance transform approximates a Voroni diagram.

What I want to do is create a ‘basin landscape’ with the same slope in each basin but from minima of different depth, so that a ridge between two minima would be closer to the shallower minimum.

Does anyone now if there is an established, fast algorithm to do something like this? An obvious but slow approach would be to do a separate distance transform for each background pixel to create independent basins, translate those up and down the z axis as appropriate, and then find the global minimum of each pixel over the set of basins. This would scale linearly with the number of background pixels though, which in my application at least would be a problem.

Here is some code that make an example dataset and does the basic distance transform, ignoring the depths.

locations = {{10, 157}, {26, 129}, {37, 22}, {42, 190}, {44,      158}, {58, 118}, {91, 169}, {99, 152}}; depths = {0, 40, 20, 30, 10, 25, 15, 5}; image = Image[ReplacePart[Table[1, {100}, {200}], locations -> 0]]; distances = ImageData[DistanceTransform[image]]; ListPlot3D[Reverse[distances], BoxRatios -> Full, Boxed -> False] Image[WatershedComponents[Image[distances]]] 

And here is my method:

separateBasins = Table[    image =      Image[ReplacePart[       Table[1, {100}, {200}], {locations[[l]]} -> 0]];    distances = ImageData[DistanceTransform[image]] + depths[[l]], {l,      1, Length[locations]}]; newBasins = Map[Min, Transpose[separateBasins, {3, 1, 2}], {2}]; ListPlot3D[Reverse[newBasins], BoxRatios -> Full, Boxed -> False] Image[WatershedComponents[Image[newBasins]]] 

which works but unsurprisingly is almost 10 times as long (eight distance transforms plus the finding of the minima). Anyone know if there is something more efficient, before I dig in to the various low-level distance transform algorithms?