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

Posted on May 28, 2011 by

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 →

Posted in Dmat0101 - Harmonic Analysis, math.CA, Teaching | Tagged , , , , , , , , | 1 Comment

DMat0101, Notes 7: General Calderón-Zygmund Operators

Posted on May 15, 2011 by

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<p<2} and then the corresponding boundedness for {2<p<\infty} followed by the fact that {H} is essentially self-adjoint. Continue reading →

Posted in Dmat0101 - Harmonic Analysis, math.CA, Mathematics, Teaching | Tagged , , , , , , , | Leave a comment

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

Posted on April 10, 2011 by

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 →

Posted in Dmat0101 - Harmonic Analysis, math.CA, Mathematics, Teaching | Tagged , , , , , , , | 1 Comment

DMat0101, Notes 5: The Hardy-Littlewood maximal function

Posted on March 29, 2011 by

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 →

Posted in Dmat0101 - Harmonic Analysis, math.CA, Teaching | Tagged , , , , , , , | Leave a comment

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

Posted on March 20, 2011 by

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 →

Posted in Dmat0101 - Harmonic Analysis, math.CA, Teaching | Tagged , , , , , , , , , , | Leave a comment

DMat0101, Notes 3: The Fourier transform on L^1

Posted on March 10, 2011 by

1. Definition and main properties.

For {f\in L^1({\mathbb R}^n)}, the Fourier transform of {f} is the function

\displaystyle \mathcal{F}(f)(\xi)=\hat{f}(\xi)=\int_{{\mathbb R}^n}f(x)e^{-2\pi i x\cdot \xi}dx,\quad \xi\in{\mathbb R}^n.

Here {x\cdot y} denotes the inner product of {x=(x_1,\ldots,x_n)} and {y=(y_1,\ldots, y_n)}:

\displaystyle x\cdot y=\langle x,y\rangle=x_1y_1+\cdots x_n y_n.

Observe that this inner product in {{\mathbb R}^n} is compatible with the Euclidean norm since {x\cdot x=|x|^2}. It is easy to see that the integral above converges for every {\xi\in{\mathbb R}^n} and that the Fourier transform of an {L^1} function is a uniformly continuous function.

Theorem 1 Let {f,g\in L^1({\mathbb R}^n)}. We have the following properties.

(i) The Fourier transform is linear {\widehat{f+g}=\hat f + \hat g} and {\widehat{cf}=c \hat f} for any {c\in{\mathbb C}}.

(ii) The function {\hat f(\xi)} is uniformly continuous.

(iii) The operator {\mathcal F} is bounded operator from {L^1({\mathbb R}^n)} to {L^\infty({\mathbb R}^n)} and

\displaystyle \|\hat f \|_{L^{\infty}({\mathbb R}^n)}\leq \|f\|_{L^1({\mathbb R}^n)}.

(iv) (Riemann-Lebesgue) We have that

\displaystyle \lim _{|\xi|\rightarrow +\infty} \hat f(\xi)=0. Continue reading →

Posted in Dmat0101 - Harmonic Analysis, math.CA, Teaching | Tagged , , , , , , , , , , , | Leave a comment

DMat0101, Notes 2: Convolution, Dense subspaces and interpolation of operators

Posted on February 23, 2011 by

1. Convolutions and approximations to the identity

We restrict our attention to the Euclidean case {({\mathbb R}^n,\mathcal L,dx)}. As we have seen the space {L^1({\mathbb R}^n)} is a vector space; linear combinations of functions in {L^1({\mathbb R}^n)} remain in the space. There is however a `product’ defined between elements of {L^1({\mathbb R}^n)} that turns {L^1} into a Banach algebra. For {f,g\in L^1({\mathbb R}^n)} we define the convolution of {f*g} to be the function

\displaystyle (f*g)(x)=\int_{{\mathbb R}^n} f(y)g(x-y)dy = \int_{{\mathbb R}^n} g(y)f(x-y)dy.

Furthermore, using Fubini’s theorem to change the order of integration we can easily see that

\displaystyle \begin{array}{rcl} \|f*g\|_{L^1({\mathbb R}^n)}\leq \|f\|_{L^1({\mathbb R}^n)}\|g\|_{L^1({\mathbb R}^n)}. \end{array}

Thus for {f,g\in L^1({\mathbb R}^n)} we have that their convolution {f*g} is again an element of {L^1({\mathbb R}^n)}. Note that the previous estimate is the main difficulty in showing that {(L^1({\mathbb R}^n),*)} is a Banach algebra.

More generally, the convolution of {f\in L^p({\mathbb R}^n)}, {1\leq p \leq +\infty}, and {g\in L^1({\mathbb R}^n)}, is a well defined element of {L^p({\mathbb R}^n)} and we have that

\displaystyle \|f*g\|_{L^p({\mathbb R}^n)}\leq \|f\|_{L^p({\mathbb R}^n)}\|g\|_{L^1({\mathbb R}^n)}. \ \ \ \ \ (1)

Continue reading →

Posted in Dmat0101 - Harmonic Analysis, math.CA, Teaching | Tagged , , , , , , , | Leave a comment

DMat0101, Notes 1: Quick review of measure theory

Posted on February 17, 2011 by

0. About these notes

The notes that will follow are meant to be a companion to the Harmonic Analysis course that I’m giving this semester at IST. These notes are inspired, influenced and sometimes shamelessly copied from books, lecture notes of other people, research papers and online material. The whole idea and structure of the course and, in particular, the use of the blog as a general platform of communication and interaction in relevance to the course is highly influenced by similar efforts of the other people and, especially, from Terence Tao’s blog as well as his lecture notes. Be sure to check the originals!

1. Introduction and notations

As mentioned in the overview of the course, we will be mainly concerned with operators acting on certain function spaces, or even spaces of more rough objects such as measures or distributions. Typically we will want to study the mapping properties of such an operator, that is whether it maps one function space to another and so on. A typical estimate in this context is of the form

\displaystyle \|Tf\|_Y\leq C \|f\|_X, Continue reading →

Posted in Dmat0101 - Harmonic Analysis, math.CA, Mathematics, seminar notes, Teaching | Tagged , , , , , , , | 1 Comment

Course Announcement DMat0101: Harmonic Analysis, PhD course at IST

Posted on January 19, 2011 by

This semester, starting the week 14-21 of February, I’m giving a course in Harmonic Analysis on the PhD level as part of the Doctoral Program in Mathematics at the Department of Mathematics at IST. I will use this blog in order to coordinate the course, post comments and notes, as well as a means of communicating with whoever is interested in following the course. For this I expect the comment function of the blog to play a central role and give everyone an easy way to comment on the content of a lecture or ask questions related to the course in general.

Continue reading →

Posted in Dmat0101 - Harmonic Analysis, math.CA, Mathematics, Teaching | Tagged , , , , , , , | 2 Comments

The maximal function along a polynomial curve; effective dimension bounds.

Posted on November 12, 2010 by

In this post I will try to give a description of an older result of mine that studies the {L^2\rightarrow L^2} operator norm of the maximal function along a polynomial curve. The relevant paper can be found here. The main object of study in this paper is the maximal operator

\displaystyle \mathcal{M}_P(f)(x):=\sup_{\epsilon>0}\frac{1}{2\epsilon} \int_{|t|\leq \epsilon} |f(x_1-t,x_2-t^2,\ldots,x_d-t^d)|dt.

It was known since the seventies that this operator is bounded on {L^p} for {1<p<\infty}. I was however interested in getting some effective bounds for the operator norm, at least on {L^2}. In fact it is possible to do that:

Theorem 1 (Parissis, 2010) There is a numerical constant {c>0} such that

\displaystyle \|\mathcal{M}_P(f)\|_{L^2({\mathbb R}^d)} \leq c \log d\ \| f\|_{L^2({\mathbb R}^d)}.

In this post we will content ourselves to proving a slightly weaker estimate with linear (instead of logarithmic) growth in {d}. This will serve presenting the main ideas and techniques involved in the proof while keeping things as simple as possible. I will however give some clues on how to move from the linear dependence to the logarithmic one without presenting too many details. Of course the reader can always consult the original paper where all the details are presented.

The methods and ideas in this paper are somehow a mix originating in two independent investigations. The first is concerned with the dimension dependence of the operator norm of the maximal function. It was first Stein that observed that the Hardy-Littlewood maximal function associated with the Euclidean ball function is bounded on {L^p} with norm bounds that do not depend on the dimension. The second area of research has to do with the boundedness properties of maximal functions (and singular integrals) along lower dimensional varieties. The operator under study is such an example. However since here I am interested in good dimensional constants for the the norm of such an operator, tools from the first area of research will be used. I will try to give a short overview of these two areas. I will then try to describe how Bourgain’s ideas for the study of the standard maximal function can be used together with some new ones in order to get a good operator bound for the maximal function along a polynomial curve. Continue reading →

Posted in math.CA, Mathematics, open problem, paper, seminar notes | Tagged , , , , , , | Leave a comment