Questions tagged [littlewood-paley-theory]

For questions about the littlewood-paley theory, a theoretical framework used to extend certain results about L2 functions to Lp functions for 1 < p < ∞.

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Dual of homogeneous Besov space

I am trying to prove a trace theorem from the homogeneous Besov space $\overset{\cdot}{B}_{2,1}^{1/2}(\mathbb{R}^n)$ to $L^2(K)$, where $K$ is a compact euclidean hypersurface. The homogeneous Besov ...
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How to show that Littlewood-Paley operator is almost projection?

The Littlewood-Paley operator is sometimes also improperly known as a projection since $$||P_kP_kf||_{L^p}\approx ||P_kf||_{L^p}$$ I wish to prove this fact so this would mean I'd need to show that ...
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Books on Littlewood-Paley Theory

What are some good references for a beginner to learn about Littlewood-Paley theory? Most books I found are all not too beginner-friendly and I find myself spending more time doing background reading ...
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$\|\nabla_x(-\Delta_x)^{-1} P_{\leq 1} f\|_{L^p}\lesssim_p\|\langle x\rangle f\|_{L^1}$ for all mean-zero $f$ and all $p\in(1,\infty]$.

Let $X=\left\{f\in L^1(\mathbb R^3): \int_{\mathbb R^3} f\,dx=0, \langle x\rangle f\in L^1\right\}$, where $\langle x\rangle :=(1+|x|^2)^\frac12$ is the Japanese bracket. Consider the operator $Tf=\...
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Show that $|||\nabla^a f||\nabla^b g|||_{L^2}\leq_{a,b,d} ||f||_{L^{\infty}}||g||_{H^{a+b}}+||f||_{H^{a+b}}||g||_{L^{\infty}}$

I am preparing talk on Beale-Kato-Majda blow condition for Euler/NS equation. I came across the following exercise (Exercise 2) in Tao's blog: https://terrytao.wordpress.com/2018/10/09/254a-notes-3-...
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Paley-Littlewood blocks - $| u^ j|_{B^{-1}_2} \leq C 2^{-j\sigma} | u^j |_{B^{\sigma-1}_2} $ if $\hat{u^j} \subset 2^j B$

As in the title, I'd like to know if it's true that: $$| u^j|_{B^{-1}_{2,\infty}} \leq C 2^{-j\sigma} | u^j |_{B^{\sigma-1}_{2,\infty}} $$ if $\hat{u^j} \subset 2^j B$ ($B$ being a ball, $| \cdot |_{B^...
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Inequality regarding Littlewood-Paley decomposition

I have been reading a paper written by J.Bourgain and I am stuck at one inequality. So the author is trying to estimate the $L^2$ norm of $$F(t)=\sum_{m\in \mathbb{Z}} e^{2im\Delta t}(f_m \overline{f}...
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Bound on fourier coefficient series

Im trying to work on the following exercise, Let $\phi$ be a function s.t. $\|\widehat{\phi}(\xi)\| \leq B \min (\| \xi\|^\delta, \| \xi\|^{-\epsilon}) $, where $\epsilon,\delta >0$. Then $$ \sum_{...
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Orthogonality in $L^2$ in Littlewood-Paley theory

I recently began studying Litllewood-Paley Theory trough Grafakos book on fourier analysis, and yet in the beginning of the chapter the author states a motiovational fact about the $L^2$ spaces, being ...
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Reference request: Commutator estimate

I am wondering if the following commutator estimate is true, and in such case, where can I find a proof for it. Let $N\in 2^{\mathbb{Z}}$ dyadic, and let's denote by $P_N$ the standard Littlewood-...
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Arbitrarily dilated Triebel-Lizorkin space have equivalence with original Triebel-Lizorkin space?

The original Triebel-Lizorkin space is defined as follows: For $\alpha\in\mathbb{R}$, $0<p<\infty$, $0<q<\infty$ and $\phi\in \mathcal{S}(\mathbb{R})$ whose Fourier transform is supported ...
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Why is the space of $L^p$ (Fourier) multipliers a Banach space? (Stein)

I am reading Stein's book 'Singular Integrals and Differentiability Properties of Functions', and he introduces the class $\mathcal{M}_p$ of $L^{\infty}$ multipliers $m$ such that the operator $T_m$ ...
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Couterexample to Littlewood-Paley theorem

Let $d\geq 2$ and let $P_j$ be the Fourier multiplier defined on $L^2(\mathbb{R}^d)$ by $\widehat{P_kf}(\xi)=\mathbf{1}_{2^k<|\xi|\leq2^{k+1}} \hat f(\xi)$, for any $k \ge 0$. It has been proven by ...
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A Littlewood-Paley estimate

For $f\in L^1(\mathbb{R}^n)$, let $P_{2^k}f,k\in\mathbb{Z}$ be the inverse Fourier transform of $1_{2^k\le|\xi|<2^{k+1}}\hat{f}(\xi)$. Then, do we have the following? $$\sum_{k\in\mathbb{Z}}\|P_{...
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$L^p$-norm estimate of Littlewood-Paley multiplier operator

My goal is to show that $$\|P_j f\|_p \lesssim 2^{-js}\||\nabla |^s P_{\geq j}f\|_p$$ where $$\widehat{P_{\geq j}f}(\xi)=(1-\phi(2^{-j}\xi))\widehat{f}(\xi)$$ $$\widehat{P_jf}(\xi)=...
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A Besov norm estimate of analytic function based on Paley-Wiener theorem

Assume that $m$ is a bounded and analytic function over a sector $\Sigma_\omega$ and $m_e=m\circ \exp$. Hence $m_e$ is a bounded and analytic function over the strip $$ \{z\in\mathbb{C}\mid |\Im z|&...
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What is Littlewood–Paley theory?

Recently I have been reading Singular Integrals and Differentiability Properties of Functions by E. M. Stein, where Chapter IV is devoted to Littlewood–Paley theory. Whereas Stein explains clearly the ...
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Littlewood-Paley decomposition with arbitrary dilation factor

In all harmonic analysis literature I've seen, the dilation factor people always use in the Littlewood-Paley decomposition is $2$, i.e. decompose the function dyadically. To be specific here's what ...
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1 vote
1 answer
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$L^p$ norm of Littlewood-Paley block of a smooth function

I'm working on a thesis concerning the Littlewood-Paley decomposition of distributions and the use of paraproducts. I'm referring to the book of Bahouri, Chemin, Danchin and during a proof (page 88 ...
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6 votes
3 answers
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Littlewood-Paley decomposition

Littlewood-Paley decomposition is a particular way of decomposing the phase space which takes a single function and writes it as a superposition of a countably infinite family of functions of varying ...
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  • 1,070
3 votes
1 answer
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Why using dyadic decomposition

In the context of Paley-Littlewood theory, based on divide and conquer strategy in order to separate high frequencies from medium and low ones authors use the dyadic decomposition that is they take ...
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  • 1,070
2 votes
1 answer
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Holder spaces via Littlewood-Paley decomposition

Consider the Littlewood-Paley decomposition of a tempered distribution $f$: \begin{equation} f = \tilde{\theta}(D)f + \sum_{k \geq 1} \theta(2^{-k}D)f. \end{equation} Here $(\tilde{\theta}, \theta)$ ...
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  • 607
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Search for a reference on characterising Lebesgue spaces in terms of (inhomogeneous) Besov spaces.

I am trying to find a proof that $||f||_{L^{p}}\sim{}||f||_{B^0_{p,2}}$ where $||f||_{B^s_{p,r}}=||(2^{js}||f_j||_{L^p})_{j\in\mathrm{Z}_{\geq{}-1}}||_{\ell^r}$ is the norm characterising the ...
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  • 905
4 votes
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Proving Div-Curl Lemma through Paraproducts

I want to to prove the classical div-curl lemma of Coifman-Lions-Meyer-Semmes in the following form: Div-Curl Lemma. Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a Schwartz function such that ...
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10 votes
2 answers
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$W^{s,p}(\mathbb{R}^{n})$ Is Not Closed Under Multiplication when $s\leq n/p$

For $s\in\mathbb{R}$, $1<p<\infty$, and $n\geq 1$, define the Sobolev space $W^{s,p}(\mathbb{R}^{n})$ by $$W^{s,p}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}(\mathbb{R}^{n}) : \|(\langle{\xi}\...
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6 votes
1 answer
848 views

Critical Homogeneous Sobolev Embedding

For $s\in\mathbb{R}$, $1<p<\infty$, define the homogeneous Sobolev space $\dot{W}^{s,p}(\mathbb{R}^{n})$ as follows: For $f\in\mathcal{S}_{0}(\mathbb{R}^{n})$ (Schwartz functions with Fourier ...
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4 votes
1 answer
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Relation between Besov and Sobolev spaces (Littlewood-Paley-theory)

For the Sobolev-norm there holds $\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)}\sim_{n,p,s}\left\Vert\left(\sum_{k\in\mathbb{N}_0}\left\vert (1+2^k)^sP_kf(x)\right\vert^2\right)^\frac{1}{2}\right\Vert_{L^p}$ ...
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Fractional Sobolev Space Trace Inequality

Let $\phi\in\mathcal{S}(\mathbb{R}^{n})$ be a Schwartz function, such that ${\phi}\equiv 1$ on the unit ball $|\xi|\leq 1$ and $\text{supp}({\phi})\subset B_{2}(0)$. Set $\phi_{0}=\phi$ and $\phi_{j}=\...
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1 vote
1 answer
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Version of Littlewood-Paley inequality

For any $p\in(1,\infty)$ there holds the Littlewood-Paley inequality $$ \Vert f\Vert_{L^p}\sim_{n,p}\left\Vert \left(\sum_{k\in\mathbb{Z}}\left\vert \dot P_kf(x)\right\vert^2\right)^\frac{1}{2}\right\...
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  • 439
5 votes
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Continuous Littlewood-Paley Inequality

I am trying to prove Q13 from Terence Tao's Fourier Analysis notes (number 4): For every $t>0$, let $\psi_{t}:\mathbb{R}^{d}\rightarrow\mathbb{C}$ be a function obeying the estimates $$|\...
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4 votes
1 answer
287 views

where to find a proof of Littlewood-Paley theorem by Khintchine's inequality

I heard that one application of Khintchine's inequality is the proof of Littlewood-Paley theorem. I only know a proof of Littlewood-Paley theorem using vector version of Calderon-Zygmund theorem. Can ...
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7 votes
0 answers
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Littlewood-Paley theorem at endpoints

Given $\phi$ a smooth real radial function supported on the closed ball supported on $\{0\leq \xi \leq 2\}$ and is identically $1$ on $\{0\leq \xi \leq 1\}$, we define $$\psi(\xi) = \phi(\xi) - \phi(...
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  • 1,946
3 votes
2 answers
330 views

Homogeneous Littlewood-Paley decomposition

I have a question concerning Littlewood-paley-theory. Suppose we have test functions $\psi_k$ supported in annuli $\{2^{k-1}\leq\vert\xi\vert\leq2^{k+1}\}$ such that $\sum_{k\in\mathbb{Z}}\psi_k(\xi)=...
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  • 439
0 votes
1 answer
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Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L^p(\mathbb R)$. Here, $F$ denotes the ...
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  • 779
4 votes
1 answer
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Littlewood-Paley theorem on an annulus

Suppose a smooth function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ satisfies $$\text{supp}~\hat{f}\subset \{\xi:1<|\xi|<2\}$$ and set functions $f_k$ by saying $$\hat{f_k}:=\hat{f}~\chi_{\{1+2^{-...
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4 votes
1 answer
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$L^{p}$ Boundedness of Fourier Multiplier without Littlewood-Paley

Suppose $\zeta\in\mathcal{S}(\mathbb{R}^{n})$ is a Schwartz function such that $\widehat{\zeta}$ is compactly supported away from the origin (say in the annulus $2^{-l_{0}}<\left|\xi\right|<2^{...
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28 votes
2 answers
960 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
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3 votes
0 answers
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Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
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  • 269
1 vote
1 answer
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Littlewood Paley characterization of BMO spaces

I know that there is a Littlewood-Paley characterization of Hardy spaces (for instance, this is found in Grafakos, Modern Fourier Analysis, section 6.4.6). I'd like to know if a similar ...
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