Some research in analysis at UCLA

John Garnett, Christoph Thiele, Terence Tao Abstract: We give a brief taste of some active research in harmonic and complex analysis here at UCLA.

Bounded analytic functions (John Garnett)

• We all know Louiville’s theorem: if f is bounded and analytic on all of C, then f must be constant.

• You can weaken Louiville’s theorem a bit. For instance, if f is bounded and analytic on C\{0}, then f must still be constant (since the singularity at 0 must be removable).

• In fact, there are several sets Γ with the property that bounded analytic functions on C\Γ are automatically bounded. Such sets are said to have zero analytic capacity.

• Generally, the smaller the set Γ, the more likely it is to have zero analytic capacity. Knowing which sets have zero analytic capacity tells you to the smallest sets for which you can cram all the unbounded or non-analytic behaviour of a non-constant complex function.

• If Γ has Hausdorff dimension less than 1, then Γ has zero analytic capacity. When Γ has Hausdorff dimension greater than 1, Γ has non-zero analytic capacity. (Major tool: the Cauchy integral formula).

• (All subsets of R^{2} have a Hausdorff dimension, which is a number somewhere between 0 and 2. Roughly speaking, this dimension measures how efficiently one can cover the set by small balls. The standard middlethirds Cantor set, for instance, has Hausdorff dimension log 2/ log 3).

• The intermediate case, when Γ has Hausdorff dimension 1, is not completely settled. If Γ is on the real axis, then it has zero analytic capacity if and only if it has (1-dimensional) measure zero. However, there exist sets with positive 1-dimensional measure which still have zero analytic capacity, such as the Garnett set (the product of two middle-halves Cantor sets).

• The finer resolution of this problem is ongoing research by John Garnett and his students.

Multilinear operators (Christoph Thiele, Terry Tao)

• Traditionally, real-variable harmonic analysis has concerned itself with linear operators T and how to control them in various norms. In other words, we ask questions like “If we have some sort of control on f, what does that tell us about T f?”

• More recently, though, harmonic analysis has begun to move away from the linear framework, towards genuinely non-linear problems. As an intermediate step we consider multi-linear estimates.

Example: solving an ODE

• Suppose you wanted to solve an ODE such as

u̇_{ε}(t) = εV (t)u_{ε}^{2}(t) + W (t), u(0) = 0,

where ε is a small number, and V, W are given functions. We could try to solve this by the method of power series, writing

u_{ε}(t) = u_{0}(t) + εu_{1}(t) + ε^{2}u_{2}(t) + …

• Expanding both sides and comparing co-efficients, we can solve for u_{0}, u_{1}, …:

u_{0}(t) = ∫_{0}^{t} W (s) ds
u_{1}(t) = ∫_{0}^{t} V (s)u_{0}(s)^{2} ds
u_{2}(t) = 2 ∫_{0}^{t} V (s)u_{0}(s)u_{1}(s) ds

etc. If you expand things out fully, you see that u_{0} is a linear function of W, u_{1} is a trilinear function of V and W, u_{2} is a quintilinear function of V and W, and so forth.

Example: Paraproducts

• The simplest example of a bilinear estimate is Hölder’s inequality:

∥uv∥_{L^{p}(R)} ≤ ∥u∥_{L^{q}(R)}∥v∥_{L^{r}(R)} when \frac{1}{p} = \frac{1}{q} + \frac{1}{r}.

• We can split the pointwise product uv as

u(t)v(t) = ∫_{−∞}^{t} u(s)v′(s) ds + ∫_{−∞}^{t} u′(s)v(s) ds.

(Assume some vanishing at −∞).

• Each individual integral ∫_{−∞}^{t} u(s)v′(s) ds is a bilinear function of u and v, and is an example of a paraproduct. It occurs often in the study of PDE.

• Rough theorem: Paraproducts satisfy Hölder’s inequality too, except for some endpoints.

Frequency decomposition

• The way we deal with operators like paraproducts is via frequency decomposition: breaking up a function u into low-frequency and high-frequency components u = u_{low} + u_{high}:

• Formally, this is done using the Fourier transform. There are some standard results about these types of decompositions; these results are known collectively as Littlewood-Paley theory.

• We can then split up a paraproduct into several pieces, e.g.

∫_{−∞}^{t} u_{high}(s)v′_{low}(s) ds

∫_{−∞}^{t} u_{low}(s)v′_{high}(s) ds.

The idea is that the first term is very small because v_{low} is almost constant, and the second term is almost u_{low}v_{high}, because u_{low} is almost constant. In either case one can simplify the expression to something more tractable.

• Christoph Thiele and I study much more singular operators than paraproducts, including the Bilinear Hilbert transform

B(f, g)(x) = p.v. ∫ f(x + t)g(x − t)\frac{dt}{t}.

This operator can also be analyzed by frequency decomposition, but on a level more refined than the low-high analysis above. One also needs some spatial decomposition as well. The enemy is then the Uncertainty principle, which says that you can localize frequency to a range of width dξ and space to a range of width dx only when dξdx ≥ 1.

Oscillatory integrals, Kakeya problems, PDE (Terry Tao)

• Solutions to PDE such as the free Schrödinger equation

iu_{t}(t, x) + Δu(t, x) = 0; u(0, x) = f(x)

are given by oscillatory integrals, in this case

u(t, x) = \frac{1}{(4πt)^{n/2}} ∫ e^{i|y−x|^{2}/4t}f(y) dy.

• The phase e^{i|y−x|^{2}/4t} should lead to some cancellation in this integral; it should usually be much smaller than, say, ∫ |f(y)| dy. But how to quantify this?

• One answer is via L^{p} norms. Typically one has something like L^{2} control on f, and one is interested in L^{p} spacetime control on u:

(∫ ∫ |u(t, x)|^{p} dxdt)^{1/p} ≤ C∥f∥_{2}.

This particular estimate is known to be true exactly when p = 4.

• Such estimates are very handy in the perturbation theory of the Schrödinger equation, and for instance can be used to obtain a rigorous theory for the nonlinear Schrödinger equation

iu_{t}(t, x) + Δu(t, x) = |u(t, x)|^{2}u(t, x).

In two spatial dimensions, it’s known that this solution has global L^{2} solutions if the initial data starts out in L^{2} with small norm. (The large norm problem is a big open problem). Basically one uses the method of power series mentioned earlier.

• These are linear estimates, because u depends linearly on f. But ultimately we want to study bilinear and multi-linear estimates. In the last ten years this has been an active area of research (and for many equations besides Schrödinger).

• The Schrodinger equation has some special “travelling wave” solutions, in which u is concentrated on a “tube” in space time, something of the form

{(t, x) : |t| ≤ R^{2}; |x − vt − x_{0}| ≤ R}.

By the principle of superposition, one can then construct other solutions which are supported on unions of tubes.

• One approach to understand solutions of PDE is to first understand the purely geometric problem of overlapping tubes.

• The very difficult Kakeya problem asks to what extent tubes can overlap. It states: if 0 < δ ≪ 1, and you have a collection of δ × 1 tubes in R^{3} whose directions are all δ-separated, what is the maximum median overlap of these tubes? The conjecture is O(δ^{−ε}) for all ε > 0. The best result known to date is ε > \frac{1}{2} − 10^{−10}.