## Mat-1.C – Harmonic Analysis

For those of you who followed my Harmonic Analysis notes a couple of years back, here is an updated (and slightly more polished) version in pdf  form, from an advanced course I gave at Aalto university:

## Circle discrepancy for checkerboard measures

This week I am giving a talk at the Department of Mathematics and Systems Analysis of Aalto University where I will discuss results from a recent paper with Mihalis Kolountzakis. I will give a short introduction to different notions of discrepancy together with a description of the main set-up and results in the checkerboard setting. I will also describe our main questions and results and some further problems related to checkerboard colorings. Our work with Mihalis Kolountzakis is a natural continuation of the previous papers [K], [IK] where similar questions have been considered. In fact our current papers answers some of the questions posed by Kolountzakis and Iosevich in [IK].

## DMat0101, Notes 8: Notes Littlewood-Paley inequalities and multipliers

In this final set of notes we will study the Littlewood-Paley decomposition and the Littlewood-Paley inequalities. These consist of very basic tools in analysis which allow us to decompose a function, on the frequency side, to pieces that have almost disjoint frequency supports. These pieces, the Littlewood-Paley pieces of the function, are almost orthogonal to each other, each piece oscillating at a different frequency.

1. The Littlewood-Paley decomposition

We start our analysis with forming a smooth Littlewood-Paley decomposition as follows. Let ${\phi}$ be a smooth real radial function supported on the closed ball ${\{\xi\in{\mathbb R}^n:0<|\xi|\leq 2\}}$ of the frequency plane, which is identically equal to ${1}$ on ${\{\xi\in{\mathbb R}^n:0\leq|\xi|\leq 1\}}$. We then form the function ${\psi}$ as

$\displaystyle \psi(\xi):=\phi(\xi)-\phi(2\xi),\quad \xi \in {\mathbb R}^n.$

Observing that ${\phi(2\xi)=\phi(\xi)=1}$ if ${|\xi|<1/2}$ and also that ${\phi(\xi)=\phi(2\xi)=0}$ if ${|\xi|>2}$ we see that ${\psi}$ is supported on the annulus ${\{\xi\in {\mathbb R}^n:1/2 \leq |\xi|\leq 2\}}$.

Now the sequence of functions ${\{\psi(\xi/2^k)\}_{k\in {\mathbb Z}}}$ forms a partition of unity:

$\displaystyle \sum_{k\in {\mathbb Z}}\psi(\xi/2^k)=1,\quad \xi \in {\mathbb R}^n\setminus\{0\}.$

To see this first observe that each function ${\psi(\xi/2^k)}$ is supported on the annulus ${\{2^{k-1}\leq |\xi|\leq 2^{k+1}\}}$. Thus for each given ${\xi \in {\mathbb R}^n}$ there are only finite terms in the previous sum. In particular if ${2^\ell<|\xi_o|\leq 2^{\ell+1}}$, then

$\displaystyle \sum_{k\in{\mathbb Z}}\psi(\xi_o/2^k)=\psi(\xi_o/2^{\ell})+\psi(\xi_o/2^{\ell+1})=\phi(\xi_o/2^{\ell+1})-\phi(\xi_o/2^{\ell-1})=1.$

Note that we miss the origin in our decomposition of the frequency space as each piece ${\psi(\xi/2^k)}$ is supported away from ${0}$. Some attention is needed concerning this point but usually it creates no real difficulty.

Thus we partition the unity in the form ${1=\sum \psi_k}$ and each ${\psi_k}$ is smooth and has frequency support on an annulus of the form ${|\xi|\simeq 2^k}$. Now for ${k\in {\mathbb Z}}$ let us define the multiplier operators

$\displaystyle \widehat {\Delta_k(f)}(\xi)=\psi(\xi/2^k)\hat f(\xi),$

and

$\displaystyle \widehat {S_k(f)}(\xi)=\sum_{\ell\leq k} \widehat{\Delta_\ell (f)}(\xi)=\phi(\xi/2^k)\hat f(\xi),$

initially defined for ${f\in L^2({\mathbb R}^n)}$ or ${f\in {\mathcal S(\mathbb R^n)}}$. The operator frequency cut-off operator ${\Delta_k}$ is almost a projection to the corresponding frequency annulus ${|\xi|\simeq 2^k}$. It is not exactly a projection since the function ${\psi(\xi/2^k)}$ is a smooth approximation of the indicator function ${\chi_{\{\xi\in{\mathbb R}^n:2^{k-1}\leq |\xi|\leq 2^k\}}}$, introducing a small tail in the region ${\{\xi\in\mathbb R^n:2^k<|\xi|\leq 2^{k+1}}$ which is mostly harmless. Similarly, the operator ${S_k}$ is almost a projection on the ball ${|\xi|\lesssim 2^k}$. Continue reading

## DMat0101, Notes 7: General Calderón-Zygmund Operators

After having studied the Hilbert transform in detail we now move to the study of general Calderón-Zygmund operators, that is operators given formally as

$\displaystyle T(f)(x)=\int K(x,y)f(y)dy,$

for an appropriate kernel ${K}$. Let us quickly review what we used in order to show that the Hilbert transform ${H}$ is of weak type ${(1,1)}$ and strong type ${(2,2)}$. First of all we essentially used the fact that the linear operator ${H}$ is defined on ${L^2}$ and bounded, that is, that it is of strong type ${(2,2)}$. This information was used in two different ways. First of all, the fact that ${H}$ is defined on ${L^2}$ means that it is defined on a dense subspace of ${L^p}$ for every ${1\leq p <+\infty}$. Furthermore, the boundedness of the Hilbert transform on ${L^2}$ allowed us to treat the set ${\{|H(g)|>\lambda\}}$ where ${g}$ is the good part’ in the Calderón-Zygmund decomposition of a function ${f}$. Secondly, we used the fact that there is a specific representation of the operator ${H}$ of the form

$\displaystyle H(f)(x)=\int K(x,y)f(y)dy,$

whenever ${f\in L^2}$ and has compact support and ${x\notin {\mathrm{supp}}(f)}$. For the Hilbert transform we had that the kernel ${K}$ is given as

$\displaystyle K(x,y)=\frac{1}{x-y}.$

We used the previous representation and the formula of ${K}$ to prove a sort of restricted ${L^1}$ boundedness of ${H}$ on functions which are localized and have mean zero, which is the content of Lemma 7 of Notes 6. This, in turn, allowed us to treat the bad part’ of the Calderón-Zygmund decomposition of ${f}$. From the proof of that Lemma it is obvious that what we really need for ${K}$ is a Hölder type condition. Note as well that for the Hilbert transform we first proved the ${L^p}$ bounds for ${1 and then the corresponding boundedness for ${2 followed by the fact that ${H}$ is essentially self-adjoint. Continue reading

## DMat0101, Notes 6: Introduction to singular integral operators; the Hilbert transform

This week we come to the study of singular integral operators, that is operators of the form

$\displaystyle T(f)(x)=\int K(x,y)f(y)dy, \quad x\in {\mathbb R}^n, \ \ \ \ \ (1)$

defined initially for nice’ functions ${f\in\mathcal S({\mathbb R}^n)}$. Here we typically want to include the case where ${K}$ has a singularity close to the diagonal

$\displaystyle \Delta=\{(x,x):x\in{\mathbb R}^n\}\subset {\mathbb R}^{2n},$

which is not locally integrable. Typical examples are

$\displaystyle K(x,y)=\frac{1}{|x-y|^n},\quad x,y\in{\mathbb R}^n,$

$\displaystyle K(x,y)=\frac{x_j-y_j}{|x-y|^{n+1}},\quad x,y\in {\mathbb R}^n$

and in one dimension

$\displaystyle K(x,y)=\frac{1}{x-y},\quad x,y\in {\mathbb R},$

and so on. Observe that these kernels have a non integrable singularity both at infinity as well as on the diagonal ${\Delta}$. It is however the local singularity close to the diagonal that is important and will lead us to characterize a kernel as a singular kernel. For example, the kernel

$\displaystyle K(x-y)=\frac{1}{|x-y|^{n-\epsilon}},\quad \epsilon>0$

is not a singular kernel since its singularity is locally integrable. Observe that for Schwartz functions ${f\in{\mathcal S(\mathbb R^n)}({\mathbb R}^n)}$ it makes perfect sense to define

$\displaystyle T(f)(x)=\int_{{\mathbb R}^n}\frac{f(y)}{|x-y|^{n-\epsilon}}dy,$

and in fact the previous integral operator was already considered in the Hardy-Littlewood-Sobolev inequality of Exercise 12 in Notes 5 and can be treated via the standard tools we have seen so far.

Thus, if one insists on writing the representation formula (1) throughout ${{\mathbb R}^n}$ then ${K}$ will not be a function in general. Indeed, the discussion in Notes 4 reveals that if the operator ${T}$ is translation invariant then the kernel ${K}$ must necessarily be of the form ${K(x-y)}$ for an appropriate tempered distribution ${K\in {\mathcal S'(\mathbb R^n)}}$:

$\displaystyle T(f)=K*f.$

Bearing in mind that there are tempered distributions which do not arise from functions or measures we see that (1) does not make sense in general and it should be understood in a different way. To give a more concrete example, think of the principal value distribution ${K=\textnormal{p.v.}\frac{1}{y}\in \mathcal S'({\mathbb R})}$ and write

$\displaystyle T(f)=(f*\textnormal{p.v.}\frac{1}{y})(x).$

Here we would like to rewrite this in the form

$\displaystyle T(f)=\int_{\mathbb R} \frac{f(y)}{x-y}dy,$

but this does not make sense even for ${f\in \mathcal S ( {\mathbb R} )}$ since the function ${\frac{1}{x-y}}$ is not locally integrable on the diagonal ${x=y}$.

In fact, the representation (1) of the operator will not be true in general but we will satisfy ourselves with its validity for functions ${f\in L^2({\mathbb R}^n)}$, of compact support, and whenever ${x}$ does not lie in the support of ${f}$. Indeed, if ${f}$ has compact support and ${x\notin{\mathrm{supp}}(f)}$ then ${|y-x|>\epsilon}$ in (1) and thus we are away from the diagonal. Indeed, returning to the principal value example, observe that the integral

$\displaystyle \int_{{\mathbb R}}\frac{f(y)}{x-y}dy,$

makes perfect sense when ${f}$ has compact support and ${x\notin {\mathrm{supp}}(f)}$. Continue reading

## DMat0101, Notes 5: The Hardy-Littlewood maximal function

1. Averages and maximal operators

This week we will be discussing the Hardy-Littlewood maximal function and some closely related maximal type operators. In order to have something concrete let us first of all define the averages of a locally integrable function ${f\in L^1 _{\textnormal{loc}}({\mathbb R}^n)}$ around the point ${x\in{\mathbb R}^n}$:

$\displaystyle A_r(f)(x)=\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y)dy,$

where ${B(x,r)}$ is the Euclidean ball with center ${x\in{\mathbb R}^n}$ and radius ${r>0}$ and ${|B(x,r)|}$ denotes its Lebesgue measure. Note that since Lebesgue measure is translation invariant we have

$\displaystyle |B(x,r)|=|B(0,r)|=r^n |B(0,1)|=\Omega_n r^n,$

where ${\Omega_n}$ denotes the Lebesgue measure (or volume in this case) of the ${n}$-dimensional unit ball ${B(0,1)\subset{\mathbb R}^n}$. Denoting by ${\chi}$ the indicator function of the normalized unit ball

$\displaystyle \chi(x)=\frac{1}{|B(0,1)|}\chi_{B(0,1)}(x),$

and noting that the balls centered at zero are ${0}$-symmetric, we can write

$\displaystyle \begin{array}{rcl} A_r(f)(x)&=&\frac{1}{|B(0,1)|r^n }\int_{B(0,r)}f(x-y)dy\\ \\ &=&\int_{{\mathbb R}^n} f(x-y)\frac{1}{|B(0,1)|r^n}\chi_{B(0,1)}({y}/{r})dy \\ \\ &=& (f*\chi_r)(x). \end{array}$ Continue reading

## DMat0101, Notes 4: The Fourier transform of the Schwartz class and tempered distributions

In this section we go back to the space of Schwartz functions ${\mathcal S({\mathbb R}^n)}$ and we define the Fourier transform in this set up. This will turn out to be extremely useful and flexible. The reason for this is the fact that Schwartz functions are much nicer’ than functions that are just integrable. On the other hand, Schwartz functions are dense in all ${L^p}$ spaces, ${p<\infty}$, so many statements established initially for Schwartz functions go through in the more general setup of ${L^p}$ spaces. A third reason is the dual of the space ${\mathcal S({\mathbb R}^n)}$, the space of tempered distributions, is rich enough to allow us to define the Fourier transform of much rougher objects than integrable functions

1. The space of Schwartz functions as a Fréchet space

We recall that the space of Schwartz functions ${\mathcal S({\mathbb R}^n)}$ consists of all smooth (i.e. infinitely differentiable) functions ${f:{\mathbb R}^n\rightarrow {\mathbb C}}$ such that the function itself together with all its derivatives decay faster than any polynomial at infinity. To make this more precise it is useful to introduce the seminorms ${p_N}$ defined for any non-negative integer ${N}$ as

$\displaystyle p_N(f)=\sup_{|\alpha|\leq N,|\beta|\leq N}\sup_{x\in{\mathbb R}^n}|x^\alpha \partial^\beta f(x)|,$

where ${\alpha,\beta\in\mathbb N^n _o}$ are multi-indices and as usual we write ${|\alpha|=\alpha_1+\cdots+\alpha_n}$. Thus ${f\in \mathcal S({\mathbb R}^n)}$ if and only if ${f\in C^\infty({\mathbb R}^n)}$ and ${p_N(f)<+\infty}$ for ${N\in{\mathbb N}_o}$.

It is clear that ${\mathcal S({\mathbb R}^n)}$ is a vector space. We have already seen that a basic example of a function in ${\mathcal S({\mathbb R}^n)}$ is the Gaussian ${f(x)=e^{-\pi|x|^2}}$ and it is not hard to check that the more general Gaussian function ${f(x)=e^{-\langle Ax,x\rangle}}$, where ${A}$ is a positive definite real matrix, is also in ${\mathcal S({\mathbb R}^n)}$. Furthermore, the product of two Schwartz functions is again a Schwartz function and the space ${\mathcal S({\mathbb R}^n)}$ is closed under taking partial derivatives or multiplying by complex polynomials of any degree. As we have already seen (and it’s obvious by the definitions) the space of infinitely differentiable functions with compact support is contained in ${\mathcal S({\mathbb R}^n)}$, ${\mathcal D({\mathbb R}^n)=C^\infty _c({\mathbb R}^n)\subset \mathcal S({\mathbb R}^n)}$, and each one of these spaces is a dense subspace of ${L^p({\mathbb R}^n)}$ for any ${1\leq p<\infty}$ and also in ${C_o({\mathbb R}^n)}$, in the corresponding topologies.

The seminorms defined above define a topology in ${\mathcal S({\mathbb R}^n)}$. In order to study this topology we need the following definition:

Definition 1 A Fréchet space is a locally convex topological vector space which is induced by a complete invariant metric. Continue reading