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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

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