# Welcome.

## February 14, 2011

### Harmonic Analysis Lecture Notes 14

Filed under: 2011 Harmonic Analysis Course — xuhmath @ 6:52 pm

Definition (Construction of the dyadic partition of unity) Fix a ${C^\infty}$ mapping ${\phi : [0,\infty) \rightarrow [0,1]}$ with

$\displaystyle \phi(r) = \left\{\begin{matrix} 1 & r < 1.40 \\ \\ 0 & r > 1.42 \end{matrix}\right. \hbox{\hskip 28pt (note } 1.40 < \sqrt{2} < 1.42)$

which we also regard as a ${C^\infty}$ map ${\phi : {\mathbb R}^d \rightarrow [0,1]}$ via ${\phi(\xi) = \phi(|\xi|)}$. We further define ${\psi: {\mathbb R}^d \rightarrow [0,1]}$ by

$\displaystyle \psi(\xi) := \phi(\xi) - \phi(2\xi)$

Note that

$\displaystyle 1 = \phi(\xi) + \sum_{N\in 2^{\mathbb Z}, N \geq 2} \psi( \frac{\xi}{N}) \quad\underbrace{=}_{a.e}\quad \sum_{N \in 2^{\mathbb Z}} \psi(\xi/N).$

Now, from previous time for this time:

Theorem (Mikhlin multiplier theorem, or possibly Marcinkiewicz, or Hörmander) If ${m : {\mathbb R}^d \setminus\{0\} \rightarrow {\mathbb C}}$ obeys

$\displaystyle \Big|\frac{\partial^\alpha}{\partial \xi^\alpha} m(\xi) \Big| \lesssim |\xi|^{-|\alpha|} \hbox{ for } 0 \leq |\alpha| \leq \left\lceil \frac{d+1}{2} \right\rceil$

Then

$\displaystyle f \mapsto (m \hat f)\check{\:} = \check m * f$

maps ${L^p \rightarrow L^p}$ (in a bounded manner) for ${1 < p < \infty}$.

Proof: We will just prove the theorem for ${m}$ obeying “symbol estimates” for multi-indices ${0 \leq \alpha \leq d+2}$. (For the full case, see [Stein, Harmonic Analysis, VI.4.4]) Clearly ${m}$ is bounded on ${L^2}$ (just requires ${m \in L^\infty(d\xi)}$, i.e., ${\alpha = 0}$). To prove it is bounded on ${L^p}$ we just need to check that ${\check m}$ obeys the Calderón–Zygmund cancellation condition:

$\displaystyle \int_{|x| \geq 2|y|} |\check m (x-y) - \check m(x) | \; dx \lesssim 1.$

This is implied by

$\displaystyle | \nabla \check m (x)| \lesssim |x|^{-d-1}$

which is what we will show. Decompose ${m(\xi) = \sum_{N \in 2^{\mathbb Z}} m_N(\xi)}$, where ${m_N(\xi) = \psi(\frac{\xi}{N}) m(\xi)}$. Now

$\displaystyle \begin{array}{rcl} \displaystyle \frac{\partial^\gamma}{\partial \xi^\gamma} [\xi m_N(\xi)] &=& \displaystyle \frac{\partial^\gamma}{\partial \xi^\gamma} [ \psi(\frac{\xi}{N}) \xi m(\xi)] \\ \\ &=& \sum_{\alpha + \beta = \gamma} c^{\alpha\beta}_{\gamma} N^{-|\beta|} (\partial^\beta \psi)(\frac{\psi}{N}) [\frac{\partial^\alpha}{\partial \xi^\alpha} (\xi m(\xi))] \end{array}$

where ${c^{\alpha\beta}_\gamma}$ are some combinatorial coefficients. So,

$\displaystyle \begin{array}{rcl} \displaystyle \left\| \frac{\partial^\gamma}{\partial \xi^\gamma} [\xi m_N(\xi)] \right\|_{L^1(d\xi)} &\lesssim_\gamma& \displaystyle \sum_{\alpha + \beta = \gamma} N^{-|\beta|} \int_{|\xi| \sim N} |\xi|^{1-|\alpha|} d\xi \\ \\ &\lesssim_\gamma& N^{d+1-\gamma} \end{array}$

Hence, because F.T${:L^1 \rightarrow L^\infty}$,

$\displaystyle |x^\gamma (\nabla \check m_N)(x) | \lesssim_\gamma N^{d+1 -\gamma}$

and so, choosing various ${\gamma}$‘s,

$\displaystyle |(\nabla \check m_N)(x) | \lesssim \min( \;\underbrace{N^{d+1}}_{\gamma = 0} \; ,\; \underbrace{N^{-1}|x|^{-(d+2)}}_{|\gamma| = d+2} \;).$

Now we sum

$\displaystyle |\nabla \check m (x) | \lesssim \displaystyle \sum_{N \leq |x|^{-1}} N^{d+1} + \sum_{N \geq |x|^{-1}} N^{-1}|x|^{-(d+2)} \lesssim |x|^{-d-1}.$

$\Box$

Definition (Littlewood–Paley projections) Let ${\psi}$ be as above and ${N \in 2^{\mathbb Z}}$, then

$\displaystyle f_N := P_N f := ( \psi(\frac{\xi}{N} \hat f ))\check\; = \int N^d \check\psi(Ny) f (x - y) \; dy$

and similarly

$\displaystyle f_{\leq N} := P_{\leq N} f := (\phi(\frac{\xi}{N}) \hat f(\xi))\check\; = \int N^d \check \phi(Ny) f(x-y) \; dy.$

These are not honest projections ${P_N^2 \neq P_N}$ (namely, ${\phi^2 \neq \phi}$).

Proposition

1. If ${f \in L^p}$, ${1 \leq p < \infty}$, then

$\displaystyle \sum_{N \in 2^{\mathbb Z}} P_Nf = P_{\leq 1} f + \sum_{N \in 2^{\mathbb Z}, N \geq 2} P_N f = f$

both a.e. and in ${L^p}$.

2. We have

$\displaystyle ||P_N f||_{L^p} + ||P_{\leq N} f||_{L^p} \lesssim ||f||_{L^p}$

for all ${1 \leq p \leq \infty}$.

3. ${\big| [P_N f](x) \big| + \big| [P_{\leq N} f] (x) \big| \lesssim [Mf](x)}$.
4. For ${1 \leq p \leq q \leq \infty}$,

$\displaystyle ||P_N f||_{L^q} + ||P_{\leq N} f||_{L^q} \lesssim N^{d \left( \frac{1}{p} - \frac{1}{q}\right)} ||f||_{L^p} \hbox{\hskip 28pt (S. Bernstein's inequalities)}$

5. ${\| |\nabla|^s f_N\|_{L^p} \approx N^s \| f_N \|_{L^p}}$ for all ${1 \leq p \leq \infty}$ and for all ${s \in {\mathbb R}}$.
6. ${\| |\nabla|^s f_{\leq N} \|_{L^p} \lesssim N^s \| f \|_{L^p}}$ for all ${1 \leq p \leq \infty}$ and for all ${s > 0}$.

Remark

1. ${L^\infty}$ norm convergence fails because smooth functions are not dense in ${L^\infty}$. A.e. convergence holds for the second decomposition in ${L^\infty}$ but not for the first, consider ${f \equiv 1}$.
2. Note that ${P_{\leq N} = P_{\leq 2N} P_{\leq N}}$ and so ${\|P_{\leq N} f\|_{L^q} \lesssim N^{\frac{d}{p} - \frac{d}{q}} \| P_{\leq N} f\|}$. Similarly for ${P_N = (P_{N/2} + P_{N} + P_{2N}) P_N}$. Bernstein’s name is also attached to inequalities of the form ${\| P'\|_{L^\infty([0,1])} \lesssim (\text{deg of }P) \| P\|_{L^\infty([0,1])}}$ for polynomials ${P}$.

Proof: For assertion 2, this is Minkowski’s inequality:

$\displaystyle \| P_{\leq N} f\|_{L^p} = \| N^d \check \phi (N \cdot) * f\|_{L^p} \leq \underbrace{\| N^d \check \phi(Nx)\|_{L^1}}_{= \|\check \phi\|_{L^1} < \infty, \text{ because } \phi \in \mathcal{S}} \| f \|_{L^p}$

and the reasoning for ${\|P_N\|}$ is similar.

For assertion 3,

$\displaystyle \big| [P_{\leq N} f[ (x) \big| \leq \int | N^d \check \phi(Ny)| \; |f(x-y)| \; dy \leq \int \frac{N^d}{(1 + |Ny|)^{100(d+1)}} |f(x-y)| \; dy \leq [Mf](x).$

To realize the final inequality, observe

$\displaystyle g(r) = \int_r^\infty - g'(\rho) \; d\rho = \int_0^\infty -\frac{\chi_{B(0,\rho)}(r)}{\rho^d} g'(\rho) \rho^d, \; d\rho$

i.e., ${g}$ is a positive linear combination of characteristic functions of balls\footnote{${|B(0,1)| \int_0^\infty \rho^d [-g'(\rho) d\rho] = \underbrace{d|B(0,1)|}_{(d-1)\hbox{-volume of }\partial B(0,1)} \int_0^\infty g(\rho) \rho^{d-1} d\rho = \|g\|_{L^1({\mathbb R}^d)}}$} and so

$\displaystyle \int g(y) |f(x-y)| \; dy = \int_0^\infty \int \frac{1}{\rho^d} \chi_{B(0,\rho)}(y) |f(x-y)| \; |g'(\rho)| \rho^d \; dy \lesssim \int_0^\infty [Mf](x) |g'(\rho)| \rho^d \; d\rho$

For assertion 1, when ${f \in \mathcal{S}({\mathbb R}^d)}$, ${\| P_N f\|}$ converges to ${f}$ when ${N \rightarrow \infty}$, to ${0}$ when ${N \rightarrow 0}$, both a.e. and in ${L^p}$. This extends to ${f \in L^p}$ via assertions 3 and 2, respectively.

For assertion 4,

$\displaystyle \begin{array}{rcl} \|P_{\leq N} f\|_{L^q} &=& \|N^d \check \phi(N \cdot) * f\|_{L^q}\\ \scriptsize \begin{bmatrix} \text{Young's} \end{bmatrix} &\leq& \displaystyle \underbrace{\| N^d \check \phi(N x) \|_{L^r}}_{\lesssim N^{d-\frac{d}{r}}} \| f \|_{L^p}, \hbox{\hskip 18pt} \frac{d}{q} + d = \frac{d}{p} + \frac{d}{r} \end{array}$

For assertion 5, let ${\theta_s(\xi) := |\xi|^s (\psi(\xi/2) + \psi(\xi) + \psi(2\xi))}$ and so ${\theta_s(\xi/N) \psi(\xi/N) = N^{-s} |\xi|^s \psi(\xi/N)}$. Thus, by Minkowski’s inequality

$\displaystyle \| N^{-s} |\nabla|^{s} P_N f\|_{L^p} = \| N^d \check \theta_s (N \cdot) * f_N \|_{L^p} \lesssim \| \check \theta_s\|_{L^1} \| f_N \|_{L^p} \lesssim_s \|f_N \|_{L^p}.$

For the converse inequality note that

$\displaystyle \theta_{-s}(\xi/N) \theta_{s} (\xi/N) \psi(\xi/N) = \psi(\xi/N)$

and so

$\displaystyle \| f_N \|_{L^p} = \| \underbrace{N^d \check \theta_{-s} (N \cdot )}_{\theta_{-s}(i \nabla)} * N^d \check \theta_s(N \cdot) * f_N \|_{L^p} \lesssim \underbrace{\|\check \theta_{-s} \|_{L^1}}_{\lesssim 1} \|N^{-s} |\nabla|^s f_N\|_{L^p}.$

For assertion 6,

$\displaystyle \begin{array}{rcl} \| |\nabla|^s f_{\leq N}\| &\leq& \displaystyle \sum_{M \leq N} \| |\nabla|^s f_M \|_{L^p} \\ \\ &\lesssim& \sum_{M \leq N} M^s \| f_M \|_{L^p} \\ \\ &\leq& \underbrace{(\sum_{M \leq N} M^s)}_{\lesssim N^s} \|f \|_{L^p} \end{array}$

$\Box$

Theorem (Littlewood–Paley Square function estimate) Given ${f \in L^p}$ when ${1 < p < \infty}$, define

$\displaystyle S(f)(x) = \sqrt{\sum_{N \in 2^{\mathbb Z}} |f_N(x)|^2}$

Then

$\displaystyle \| S(f) \|_{L^p} \approx_p \| f \|_{L^p}.$

Remark

1. We can view ${S}$ as a sub-linear operator, or as a vestige of the linear operator ${f \mapsto (f_N) \in \ell^2}$ and then the square function estimate is about its mapping properties from ${L^p \rightarrow L^p({\mathbb R}^d \rightarrow\ell^2)}$
2. The Littlewood–Paley operators decompose ${f}$ into its constituent frequencies/wave-lengths/length-scales. The square function says that we do not need to fear subtle cancellations between these different components—note the absolute value signs.
3. Compare ${\|f\|_{L^p} \leq \sum \|f_N\|_{L^p}}$, the triangle inequality, but this is not reversible.

Lemma (Khinchin) Let ${X_n}$ be (statistically) independent identically distributed random variables, Bernoulli ${X_n = \pm 1}$ with equal probability. For complex numbers ${c_n}$,

$\displaystyle \mathop{\mathbb E}( |\sum c_n x_n|^p )^{1/p} \approx_p \left( \sum |c_n|^2\right)^{1/2}$

for each ${0 < p < \infty}$.

Proof: We will just treat ${c_n}$ as real numbers. Note that for all ${t > 0}$

$\displaystyle \mathbb{P}( \sum c_n X_n > \lambda ) \underbrace{\leq}_{\text{Markov}} e^{-\lambda t} \mathop{\mathbb E} \{ e^{t\sum c_n X_n}\} \leq e^{-\lambda t} \prod_n \cosh(t c_n) \leq^* e^{-\lambda t} \prod_n e^{t^2 c_n^2/2} \leq e^{-\lambda t} \exp\{ \frac{t^2}{2} \sum |c_n|^2\}$

and optimizing in ${t}$,

$\displaystyle \mathbb{P} (\sum c_n X_n > \lambda)\leq \exp \left\{ -\frac{1}{2} \frac{\lambda^2}{\sum |c_n|^2} \right\}$

where the ${*}$‘ed inequality follows from noting ${\cosh(x) = \sum_{k=0}^\infty \frac{x^{2k}}{(2k)!}\leq \exp(x^2/2) = \sum_{k=0}^\infty \frac{x^{2k}}{2^k k!}}$. We will finish this proof next time. $\Box$