Tag Archives: Lp estimates

Fractal solutions of dispersive partial differential equations on the torus

Burak Erdoğan and I just put “Fractal solutions of dispersive partial differential equations on the torus” on the arXiv. We study cancellation in exponential sums and how this leads to bounds for the fractal dimension of solutions to certain PDE, the ultimate “square root cancellation” implying exact knowledge of the dimension.

In this post, we consider the case of Schrödinger’s equation with initial data {\chi := \chi_{[0 , \pi]}} for simplicity. Here,

\displaystyle if_t + f_{xx} = 0 , \ \ t \in \mathbb{R}, \ \ x \in \mathbb{R} / 2 \pi \mathbb{Z} \ \ , \ \ f(0,x) = \chi(x) ,

and the solution is given by

\displaystyle f(t,x) = \sum_{n \in \mathbb{Z}} \widehat{\chi}(n) e^{i t n^2 + i n x}.

Note the solution is periodic in both {x} and {t}.

For a line {\mathcal{L} \subset \left( \mathbb{R} / 2 \pi \mathbb{Z} \right)^2}, we are interested in the fractal dimension of the real and imaginary parts of the graph of {f |_{\mathcal{L}} (t,x)}. This dimension must lie in the interval {[1,2]}.

An important case to consider is {t = a/q}, which leads to so–called quantization and is related to the Talbot effect. In this case, it was shown by Berry and Klein that

\displaystyle f(t,x) = \sum_{j=0}^{q-1} c_j \chi_{[0 , 1/q] + j/q} (x), \ \ \ \ \ (1)

for some {c_j \in \mathbb{C}} that are Gauss sums. Thus {f} is a linear combination of at most {q} intervals and has fractal dimension 1. See page 4 of this paper of Chen and Olver for some pictures of this phenomenon.

The story is entirely different for irrational {t}. To see why, observe for a sequence {\frac{a_n}{q_n} \rightarrow t}, the functions {f(\frac{a_n}{q_n} , x)} as given in (1) increase in complexity as {q_n} increases.

To make this precise, we use the theory of Besov spaces and show that bounds for the {L^p} norm of

\displaystyle H_N(t,x) : = \sum_{N \leq n < 2N} e^{itn^2 + i nx}, \ \ (t,x) \in \mathcal{L},

imply fractal dimension bounds for the real and imaginary parts of {f_{\mathcal{L}}(t,x)}.

Remark: The starting point of the theory of of Besov spaces that we use can be illustrated by the following basic facts from analysis:

  • The graph of a real valued {C^{\gamma}} function has fractal dimension {\leq 2 - \gamma},
  • For {g \in C^{\gamma}}, one has {\widehat{g}(n) \ll n^{-\gamma}}.

Thus we see that both fractal dimension and Fourier decay are related to the smoothness of {g}. In our situation, we have more information than that of the conclusion of the second bullet, that is we know that

\displaystyle \sum_{N \leq n < 2N} \widehat{g}(n),

exhibits cancellation and so we are able to use more involved theory to roughly reverse the implication in the second bullet as well as provide lower bounds. ♠

Let us first consider the case of horizontal lines in space–time, that is {t} fixed. Then by orthogonality, Weyl’s inequality, and the divisor bound, one has for almost every {t},

\displaystyle ||H_N||_{L_x^2} = N^{1/2} , \ ||H_N||_{L_x^{\infty}} \ll N^{1/2 + \epsilon}, \ ||H_N||_{L_x^4} \ll N^{1/2 + \epsilon}.

By a previous blog post, this implies

\displaystyle ||H_N||_{L_x^1} \gg N^{1/2 - \epsilon}.

One can use this to show that the fractal dimension of {f_{\mathcal{L}}(t,x)} is equal to {3/2} for almost every horizontal line, recovering a theorem of Rodnianski. In one part of our paper, we adapt the above strategy to show the following.

Theorem (oblique): The fractal dimension of the real and imaginary parts of the solution to (1) restricted to almost every

\displaystyle \mathcal{L} = \{(t,x) : a t + bx = c\},

is in the interval

\displaystyle [7/4 , 19/10]. ♠

Note that the lower bound 7/4 is bigger than the 3/2 for horizontal lines. To show the upper bound in Theorem (oblique), we study exponential sums such as

\displaystyle \sup_x |\sum_{n \leq N} e^{i(c+x)n^2 + i nx}| , \ \ c \in \mathbb{R} / 2 \pi \mathbb{Z}.

In this case, square root cancellation would imply the fractal dimension is exactly {7/4}.

In our paper, we study a variety of dispersive PDE, for instance the Airy, KDV, non–linear Schrödinger, free water boundary, Boussinesq, and gravity–capillary wave equation. To handle non–linear equations, we use smoothing estimates, which roughly state that the solution to a nonlinear PDE is the linear part plus a smoother perturbation.

Hölder’s Inequality

Let {(\Omega,\mu)} be a {\sigma}–finite measure space and {f,g : \Omega \rightarrow \mathbb{C}} be measurable functions. Hölder’s inequality asserts for {p,q \in [1, \infty]}, {p^{-1} + q^{-1} = 1} one has

\displaystyle \int |fg| \leq ||f||_{p} ||g||_{q}.

One standard proof is to integrate the AM-GM inequality

\displaystyle \frac{|f(x)||g(x)|}{||f||_{p} ||g||_q} \leq \frac{|f(x)|^p}{p||f||_p^p} + \frac{|g(x)|}{q ||g||_q^q}.

Here, I provide an alternative proof. First consider the case where {g = \chi_E} is the characteristic function of a set {E} of finite measure. Let

\displaystyle f_{<M} = f(x) 1_{f(x) < M} , \ \ \ \ f_{\geq M}(x) = f(x) 1_{f(x) \geq M} .

Then

\displaystyle \int |f| \chi_E = \int |f_{<M} |\chi_E + \int |f_{\geq M} |\chi_E \leq M \mu(E) + \frac{1}{M^{p-1}} \int |f|^p.

By choosing the optimal value for {M= \frac{||f||_p}{\mu(E)^{\frac{1}{p}}}}, we obtain

\displaystyle \int |f| \chi_E \leq 2 \mu(E)^{1-\frac{1}{p}} ||f||_p.\ \ \ \ \ (1)

Remark: Note that if {|f|} is bounded away from {\lambda}, we may improve the above argument. For instance, suppose {f(x) \notin [M^{-1} \lambda , M^{\frac{1}{p-1}} \lambda]} for some {M\geq 1} and for all {x \in \Omega}. Then

\displaystyle \int |f| \chi_E \leq 2 M^{-1} \mu(E)^{1-\frac{1}{p}} ||f||_p.

We can get rid of the constant 2 in (1) by using the tensor product trick. That is, apply (1) to

\displaystyle F: \Omega^n \rightarrow \mathbb{C} , \ \ \ \ \ \ \ F(x_1 , \ldots , x_n) = f(x_1) \cdots f(x_n).

It follows that

\displaystyle \left(\int |f| \chi_E \right)^n = \int |F|\chi_{E^n} \leq \mu(E^n)^{1-\frac{1}{p}} ||F||_p = 2 \left(\mu(E)^{1-\frac{1}{p}} ||f||_p\right)^n.

Taking {n^{\rm th}} roots and letting {n \rightarrow \infty} establishes Hölder’s inequality in the case {g = \chi_E}.

Remark: In many applications, (1) is enough. Nevertheless, extending to simple functions is an easy matter and Hölder’s inequality follows by the density of simple functions in {L^p}. ♠

Getting a lower bound for an L1 norm using higher moments

I would like to discuss a principle that came up in this recent talk of Adam Harper as well as my own research with Burak Erdogan.

Let f : \Omega \to \mathbb{C} be a function on some measure space with measure \mu (for instance \Omega = [0,1] or a finite set).

Often one is interested in finding lower bounds for the L^1 norm of f, that is \int_{\Omega} |f| d\mu, but has no way to directly estimate it. As a toy example, we can consider g_N: \mathbb{R} / \mathbb{Z} \to \mathbb{C} via g_N(x) = \sum_{1 \leq n \leq N} e(x n^2). Estimating the L^1 norm directly seems hard.

But sometimes, we are able to estimate higher L^p norms of f. This is useful to our original problem, since an application of Holder’s inequality reveals that a lower bound on the L^2 norm and an upper bound on the L^4 norm gives a lower bound on the L^1. To see this, note \int |f|^2 \leq (\int |f|^4 )^{1/3} (\int |f|)^{2/3}.

We can apply this idea to our original example. Parseval’s identity gives that \int |g_N|^2 = N, while orthogonality and the divisor bound give that \int |g_N|^4 \lesssim_{\epsilon} N^{2 + \epsilon}. This gives \int |f| \gtrsim_{\epsilon} N^{1/2 - \epsilon} and is expected by the heuristic that a typical exponential sum should be about square root of the length of the sum.

My intuition is the following. Suppose the measure space is a probability space. Then \int |f| \leq (\int |f|^2)^{1/2}. We are basically trying to reverse this inequality. Equality holds when f is constant, that is f is not too concentrated. The upper bound on the L^4 norm of f implies that indeed f is not too concentrated.

We mention that there is nothing too special about the exponents 2 and 4 chosen for the above discussion (although they are convenient for the specific example I chose).

An introduction to Vinogradov’s mean value theorem

VMT-Intro: pdf file containing the post

A link to a talk I gave on the topic. The talk is much more elementary than the blog post.

 

Vinogradov’s mean value theorem (recently proved by Bourgain, Demeter and Guth) is a beautiful problem in the intersection of number theory and harmonic analysis.

As a problem in number theory, the mean value theorem requests to show most of the solutions to a certain system of Diophantine equations of the form f_j(\vec{x}) = f_j(\vec{y}) for some integral polynomials f_1 , \ldots , f_k are of the form \vec{x} = \vec{y}. As a problem in harmonic analysis, the mean value theorem requests for an upper bound for the L^p norm of a certain function. These two problems turn out to be equivalent, as explained in the pdf linked at the top of the post, thanks to Fourier analytic identities.

One goal in the above pdf is to understand the nature of the so called “critical exponent.” Interpolation reveals that Vinogradov’s mean value theorem follows from an L^{k(k+1)} bound of a certain function. While a first step to understanding this critical exponent is interpolation, consideration of the major arcs gives proper insight into why k(k+1) appears.

In the final section, I attempt to explain how the mean value theorem can be interpreted as a stronger form of orthogonality of certain complex exponentials. For a vector n \in \mathbb{Z}^k, we define e_n : \mathbb{R}^k / \mathbb{Z}^k \to \mathbb{C} via e_n(\alpha) : = e^{2 \pi i n \cdot \alpha}. Then Vinogradov’s mean value theorem can be interpreted as showing \{e_{(j , \ldots , j^k)}\}_{1 \leq j \leq N} are stronger than orthogonal (k \geq 2, not true for k =1). We make this somewhat precise in the above pdf, adopting a probabilistic perspective.

I’d like to thank Chris Gartland, a fellow graduate student, for helping me formulate the ideas in the last section. For instance, it was his idea to utilize equation 5.

I’d also like to thank Junxian Li, a fellow graduate student in number theory, for very useful discussions regarding the major arcs.

Lastly, anyone at UIUC interested in hearing more about Vinogradov’s mean value theorem (for instance Bourgain, Demeter and Guth’s recent result or classical number theoretic methods), please get in touch with me. My email can be found here, or you can visit my office in Altgeld 169.

 

Gauss Sums and Hausdorff-Young

GaussSums-Hausdorff Young

Let 1 \leq p \leq 2. The classical Hausdorff-Young inequality asserts that for any f \in L^p(\mathbb{T}), there is a A_p (equal to 1 in the present case) such that ||\hat{f}||_{p/(p-1)} \leq A_p||f||_p. There are examples that show one cannot allow p > 2. In the above pdf, we provide a construction, motivated from number theory, that shows p > 2 is impossible in the Hausdorff-Young inequality. One can think of this as a pseudo-random approach, as opposed to the random approach employed by, for instance, an application of Khintchine’s inequality.

We will first prove a discrete analog. For those a bit rusty, you are invited to check out my notes on the discrete Fourier transform.

I’d like to thank to of my fellow graduate students, Derek Jung and Xiao Li, for pointing out some mistakes in a previous version. I would also like to thank Sergei Konyagin for making me aware of the Shapiro-Rudin polynomials.

Lastly, I have not seen this example in the literature. If you have seen it, please let me know.