Semicircle

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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