Monday, August 06, 2012

work-in-progress, 3 of 4: a suspicion or two, but generally proud.

this post is a little more cheerful, since by this point i start to see light at the end of the tunnel. in fact, at this point i thought i had fully patched the paper .. only to find, the next day, another gap in another part of the paper.

it never ends, does it?

no matter: the ending (that you'll later read) is near and this post gave signs of things to come. it is also a bit more technical, as you can tell right away. (to explain, all of my colleagues were away on holiday, so there was really nobody to share the details with.)

// initially written: tues, 24 july 2012 //

i'm a little suspicious now.

i saved one lemma with a new proof, and i think i may have patched the full one. for the latter case, i've narrowed down the technical issues to measure theory, so it should be a matter of checking standard facts ..

.. but i'm still suspicious:

i've never seen luzin's theorem used like this before, and i wonder if the statement is as general as i would like it to be.

added later: yes, the version of luzin's theorem
that i need is, in fact, true.
..
added even later: there's a more elegant approach,
so i don't need a luzin-type theorem after all!

part of me is seething, upset with myself for having been so stupid again.

how careless could i have been, to miss that error?
maybe i should collaborate more in this metric geometry stuff:
having another pair of eyes around has got to help, right?

something good .. or at least, something useful has come out of this, though. i know my proof much better than i did before, and i'm more aware of the subtleties involved.

moreover, i feel like i've become stronger with these techniques now.
to compare with rock climbing, it's like realising that, sometimes, you don't need cracks or holes in the rock to have a good grip.

if the rock is granite-based, then the surface is already rough with friction, i.e. a slab climb: you just have to be careful and not lose all the layers of skin on your fingertips, while doing so.
so i feel stronger and more able, and now i want to climb harder things ..!

in particular, part of me is proud. i honestly think that i invented (or at least, re-invented) a new construction ..

in geometric measure theory, there is a notion of tangent measure that was introduced by marstraηd and then weaponised by preιss [1].
roughly speaking, one takes rescalings of Euclidean space at a point, restricts the measure to that scale, and takes a limit measure. (the process is geometrically intuitive, but for rigor one has to appeal to the right weak-star topology.)

some version of this also makes sense on metric spaces: rescalings of space are well-defined as weak tangents (a specific case of gromov-hausdorff limits of sequences of metric spaces) and sometimes there is a way to bring the measure with you; in the case of doubling measures, then the limit is well-defined.
so what i've done is study tangent derivations.

roughly speaking (again), a derivation is a generalisation of a partial differential operator, but in the context of arbitrary metric spaces (with a fixed Borel measure). so a tangent derivation is just a derivation with respect to the tangent measure on a weak tangent.
fine and dandy, you might say ..
.. but how do you even know they exist?

i'm saying that they do exist, when you have derivations downstairs, that is: on the original metric space.

(this shouldn't be too surprising: in the case of spaces with doubling measures, it's already well-known that poincaré inequalities persist under weak tangents. what i'm saying is that the hypothesis can be made weaker.)
there has always been this technical problem of mine, and that concerns iterated limits (for doubly-index sequences). this always comes up when trying to build derivations and while simultaneously passing to different measures. you see, derivations have continuity properties, but they are with respect to another weak topology that is specific to the underlying measure.

it is always a "risk" to take weak limits of weak limits: there's no guarantee of what comes out. (if something does, then it usually has to do with some extra information about the situation.) i'm saying, however, that you can now avoid that ..

.. and that's what i'm proud of.

[1] in that he used them to prove the following theorem: given a Radon measure $\mu$ on $\mathbb{R}^n$, if there exists $s \in [1,n]$ that satisfies the density condition $$0 \;<\; \lim_{r \to 0} \frac{\mu(B(x,r))}{r^s} \;<\; \infty$$ $\mu$-almost everywhere --- that is, the limit exists and is positive and finite --- then $s$ is an integer and the support of $\mu$ is a countable union of $s$-dimensional $C^1$-smooth submanifolds in $\mathbb{R}^n$ (up to a set of $\mu$-measure zero).

amazing: from a purely measure-theoretic criterion, one gets a vastly improved regularity. (FYI: the fact that $s$ must be an integer is due to marstraηd.)