## The Alabern-Mateu-Verdera characterization of Sobolev spaces

I’ve been trying to understand how the Alabern-Mateu-Verdera characterization of Sobolev spaces works. Consider a function $f\colon \mathbb R\to\mathbb R$. Let $f_t$ be the average of $f$ on scale $t>0$; that is, $\displaystyle f_t(x)=\frac{1}{2t}\int_{x-t}^{x+t}f(y)\,dy$. The difference $|f_t-f|$ measures the deviation of $f$ from a line. AMV define the square function $S_f$ as a weighted average of the squares of such deviations:

$\displaystyle S_f^2(x)=\int_0^{\infty} \left|\frac{f_t(x)-f(x)}{t}\right|^2 \frac{dt}{t}$

Since I’m mostly interested in the local matters, I’ll use the truncated square function $S_f^2(x)=\int_0^{1}\dots$ which avoids the large-scale considerations. If $f$ is very nice (say, twice continuously differentiable), then $|f_t-f|$ is of order $t^2$, and the integral converges with room to spare. For example, here is the Gaussian ($f$ is in blue, $S_f$ in red):

Gaussian

This looks suspicious. Clearly, $S_f$ measures the size of the second derivative, not of the first. Yet, one of the Alabern-Mateu-Verdera theorems is for the Sobolev space of first order $W^{1,p}$: namely, $f\in W^{1,p} \iff S_f\in L^p$ (for finite $p$). So, the degree of integrability of $|f'|$ matches that of $S_f$, even though the functions look very different.

For functions that are not very nice $S_f$ may be infinite for some values of $x$. For example, if the graph of $f$ has a corner at some point, then $|f_t-f|$ is of order $t$ there, and the integral defining $S_f$ diverges as $\displaystyle \frac{dt}{t}$. For example, take the triangle $f(x)=(1-|x|)^+$:

Triangle

The triangle is a Lipschitz, i.e., $s\in W^{1,\infty}$, but its $S_f$ is not bounded. So, the AMV characterization $f\in W^{1,p} \iff S_f\in L^p$ does not extend to $p=\infty$. However, the blow-up rate of $S_f$ in this example is merely logarithmic ($|\log{}|^{1/2}$ to be precise), which implies $S_f\in L^p$ for all $p<\infty$, in accordance with the AMV theorem. Again, we notice that $S_f$ and $|f'|$ look rather unlike each other… except that $S_f$ now resembles the absolute value of the Hilbert transform of $f'$.

Here is the semicircle $f(x)=\sqrt{1-x^2}$:

At the endpoints of the arc $|f'|$ blows up as $(1-|x|)^{-1/2}$, and therefore $f\in W^{1,p}$ only when $p<2$. And indeed, near the endpoints the nonlinearity on scale $t$ is about $\sqrt{t}$, which turns the integrand in the definition of $S_f$ into $\displaystyle \frac{dt}{t^2}$. Hence, $S_f^2(x)\sim \frac{1}{|x-1|}$ as $x\to 1$. We have $S_f\in L^p$ iff $p<2$, as needed.

The last example, $f(x)=(1-\sqrt{|x|})^+$, has both a cusp and two corners, demonstrating the different rates at which $S_f$ blows up.

Cusp and corners

My Scilab code is probably not the most efficient one for this purpose. I calculated $f_t-f$ using a multidiagonal matrix with $-1$ on the main diagonal and $1/(2k)$ on the nearest $2k$ diagonals.
 step=0.01; scale=1; x=[-3:step:3]; n=length(x); s=zeros(n) f=max(0,1-sqrt(abs(x))) for k=1:(scale/step) do avg=-diag(ones(n,1)) for j=1:k do avg=avg+diag(ones(n-j,1)/(2*k),j)+diag(ones(n-j,1)/(2*k),-j) end s=s+(avg*f').^2/(step^2*k^3) end s=s.^(1/2) clf(); plot(x,f); plot(x,s,'red') 

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