There are still some things left, I wanted to add about the issue of weak-* convergence in {L^\infty}, non-linear distortions and Young measures. The first is, that Young measures are not able to describe all effects of weak-* convergence, namely, the notion does not handle contractions properly. The second thing is, that there is an alternative approach based on the {\chi}-function which I also find graphically appealing.

1. Concentrations and Young measures

One can distinguish several “modes” that a sequence of functions can obey: In this blog entry of Terry Tao he introduces four more modes apart from oscillation:

  1. escape to horizontal infinity
  2. escape to width infinity
  3. escape to vertical infinity
  4. typewriter sequence

1.1. Escape to horizontal infinity

This mode is most easily described by the sequence {f_n = \chi_{[n,n+1]}}, i.e. the characteristic functions on an interval of unit length which escapes to infinity. Obviously, this sequence does not convergence in any {L^p({\mathbb R})} norm and its weak convergence depends on {p}:

For {p>1} the sequence does converge weakly (weakly-* for {p=\infty}) to zero. This can be seen as follows: Assume that for some non-negative {g\in L^q} (with {1/p + 1/q = 1}) we have {\epsilon \leq \int g f_n = \int_n^{n+1} g}. The we get with Hölders inequality that {\epsilon^q \leq \int_n^{n+1} |g|^q}. But this contradicts the fact that {g\in L^q}.

For {p=1} the sequence does not convergence weakly to zero as can be seen by testing it with the function {g \equiv 1} and also does not converge weakly at all (test with {g = \sum_n (-1)^n\chi_{[n,n+1]}} and observe that the dual pairings do not converge).

However, this type of convergence does not occur in bounded domains, and hence, can not be treated with Young measures as they have been introduced the in my previous entry.

1.2. Escape to width infinity

The paradigm for this mode of convergence is {f_n = \frac{1}{n}\chi_{[0,n]}}. This sequence even convergence strongly in {L^p} for {p>1} but not in strongly in {L^1}. However, it converges weakly to 0 in {L^1}. This mode needs, similar to the previous mode, an unbounded domain.

1.3. Escape to vertical infinity

The prime example, normalized in {L^2(]-1,1[)}, for this mode is {f_n = \sqrt{n}\chi_{[-1/n,1/n]}}. By testing with continuous functions (which is enough by density) one sees that the weak limit is zero.

If one wants to assign some limit to this sequence {f_n} one can say that the measure {f_n^2\mathfrak{L}} does converge weakly in the sense of measures to {2\delta_0}, i.e. twice the point-mass in zero.

Now, what does the Young measure say here?

We check narrow convergence of the Young measures {\mu^{f_n}} by testing with a function of type {\psi(x,y) = \chi_B(x)\phi(y)} for a Borel set {B} and bounded continuous function {\phi}. Then we get for {n\rightarrow \infty}

\displaystyle  |\int_{\Omega\times{\mathbb R}} \psi(x,y){\mathrm d}\mu^{f_n}(x,y)| \leq \int_{-1/n}^{1/n}|\psi(\sqrt{n})|{\mathrm d}\mathfrak{L}(x)\rightarrow 0

Hence,

\displaystyle  \mu^{f_n}\rightharpoonup 0.

We conclude that this mode of convergence can not be seen by Young measures. As Attouch et. al say in Section 4.3.7: “Young measures do not capture concentrations”.

1.4. Typewriter sequence

A typewriter sequence on an interval (as described in Example 4 of this blog entry of Terry Tao) is a sequence of functions which are mostly zero and the non-zero places revisit every place of the interval again, however, with smaller support and integral. This is an example of a sequence which converges in the {L^1}-norm but not pointwise at any point. However, this mode of convergence is not very interesting with respect to Young measures. It basically behaves like “Escape to vertical infinity” above.

2. Weak convergence via the {\chi}-function

While Young measures put a uniformly distributed measure on the graph of the function, and thus, are a more “graphical” representation of the function, the approach described now uses the area between the graph and the {x}-axis.

We consider an open and bounded domain {\Omega\subset {\mathbb R}^d}. Now we define the {\chi}-function as {\chi:{\mathbb R}\times{\mathbb R}\rightarrow {\mathbb R}}

\displaystyle  \chi(\xi,u) = \begin{cases} 1, & 0<\xi<u\\ -1, & u<\xi<0\\ 0, &\text{else}. \end{cases}

The function looks like this:

We then associate to a given function {u:\Omega\rightarrow{\mathbb R}} the function {U:(x,\xi) \mapsto \chi(\xi,u(x))}. Graphically, this function has the value {1}, if the value {\xi} is positive and between zero and {u(x)} and it is {-1}, if {\xi} is negative and again between zero and {u(x)}. In other words: the function {(x,\xi)\mapsto \chi(\xi,u(x))} is piecewise constant of the area between zero and the graph of {u} encoding the sign of {u}. For the functions {f_n} from this Example 1 in the previous post this looks like this:

Similar to the approach via Young measure, we now consider the sequence of the new objects, i.e. the sequence of {(x,\xi)\mapsto \chi(x,f_n(x))} and use a weak form of convergence here. For Young measures we used narrow convergence and here we use simple weak-* convergence.

On can show the following lemma:

Lemma 1 Assume that {f_n} converges weakly-* in {L^\infty({\mathbb R})}. Then, for the weak-* limit of the mappings {(x,\xi)\mapsto\chi(\xi,f_n(x))}, denoted by {F} there exists a probability measure {\nu_x} such that

\displaystyle  \partial_\xi F(x,\cdot) = \delta_0 - \nu_x.

The proof (in a slightly different situation) can be found in Kinetic Formulation of Conservation Laws, Lemma 2.3.1.

Example 1 We again consider Example 1 from my previous post:

\displaystyle  f_n(x) = \begin{cases} a & \text{for }\ \tfrac{2k}{n} \leq x < \tfrac{2k+1}{n},\ k\in{\mathbb Z}\\ b & \text{else.} \end{cases}

The graph of some {f_n} and the corresponding function {F_n:(x,\xi) \mapsto \chi(x,f_n(x))} was shown above. Obviously the weak-* limit of these {\chi}-functions {F_n} is (in the case {b<0<a}) given by

\displaystyle  F(x,\xi) = \begin{cases} \tfrac12, & 0 < \xi < a\\ -\tfrac12, & b< \xi < 0. \end{cases}

This can be illustrated as

Now take the weak derivative with respect to {\xi} (which is, as the function {F} itself, independent of {x}) to get

\displaystyle  \partial_\xi F(\cdot,x) = \delta_0 - \tfrac12 (\delta_a + \delta_b)

and, comparing with Lemma~1, we see

\displaystyle  \nu_x = \tfrac12 (\delta_a + \delta_b).

Cool: That is precisely the same limit as obtained by the Young measure!

Well, the observation in this example is not an accident and indeed this approach is closely related to Young measures. Namely, it holds that {\mu^{f_n}_x\rightharpoonup \nu_x}.

Maybe, I’ll come back to the proof of this fact later (which seemed not too hard, but used a different definitions of a Young measure I used here).

To conclude: Both the approach via Young measures and the approach via the {\chi}-function lead to the same new understanding of weak-* limits in {L^\infty}. This new understanding is a little bit deeper than the usual one as it allows to goes well with non-linear distortions of functions. And finally: Both approaches use a geometric approach: Young measures put a equidistributed measure on the graph of the function and the {\chi}-function puts {\pm1} between the graph and the {x}-axis.

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