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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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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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Singular integrals and differentiability properties of functions. The Littlewood-Paley

Hi. I can not understand some things in this book. $|\frac{\partial u} {\partial y}|^2=<\frac{\partial u} {\partial y},\frac{\partial u} {\partial y}>?$ What internal product is it? Why $\int_{...
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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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Motivation and definition of square functions in harmonic analysis

Let $f \in L^1_{loc}([0,1))$. Let $h_I$ denote the Haar function supported on the dyadic interval $I$. The dyadic Littlewood-Paley square function is defined by $$Sf(x) := \left(\sum_I \frac{|\...
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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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Littlewood-Paley decomosition

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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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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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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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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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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$W^{s,p}(\mathbb{R}^{n})$ 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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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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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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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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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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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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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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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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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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$L^{p}$ 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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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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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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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 ...