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 < ∞.
30
questions
1
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0answers
22 views
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 ...
1
vote
1answer
56 views
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 $T_m$ ...
1
vote
0answers
28 views
Littlewood-Paley Inequality for dyadic intervals
I'm at the moment in reading a book about Littlewood-Paley and Multiplier theory and am mainly interested in the Littlewood-Paley inequality for dyadic and arbitrary intervals. I have some question ...
1
vote
1answer
39 views
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 ...
-1
votes
1answer
18 views
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?
$$\sum_{k\in\mathbb{Z}}\|P_{...
2
votes
1answer
61 views
$L^p$-norm estimate of Littlewood-Paley multiplier operator
My goal is to
show that $$\|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)=...
1
vote
0answers
31 views
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|&...
4
votes
0answers
82 views
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 ...
4
votes
0answers
81 views
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 ...
0
votes
1answer
140 views
$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 ...
5
votes
3answers
665 views
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 ...
3
votes
1answer
883 views
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 ...
2
votes
1answer
286 views
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)$ ...
0
votes
0answers
44 views
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 ...
4
votes
0answers
199 views
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 ...
10
votes
2answers
338 views
$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}\...
4
votes
1answer
577 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 ...
3
votes
1answer
1k views
Relation between Besov and Sobolev spaces (Littlewood-Paley-theory)
For the Sobolev-norm there holds $\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)}\sim_{n,p,s}\left\Vert\left(\sum_{k\in\mathbb{N}_0}\left\vert (1+2^k)^sP_kf(x)\right\vert^2\right)^\frac{1}{2}\right\Vert_{L^p}$ ...
4
votes
0answers
353 views
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}=\...
1
vote
1answer
797 views
Version of Littlewood-Paley inequality
For any $p\in(1,\infty)$ there holds the Littlewood-Paley inequality
$$ \Vert f\Vert_{L^p}\sim_{n,p}\left\Vert \left(\sum_{k\in\mathbb{Z}}\left\vert \dot P_kf(x)\right\vert^2\right)^\frac{1}{2}\right\...
5
votes
0answers
225 views
Continuous Littlewood-Paley Inequality
I am trying to prove Q13 from Terence Tao's Fourier Analysis notes (number 4):
For every $t>0$, let $\psi_{t}:\mathbb{R}^{d}\rightarrow\mathbb{C}$ be a function obeying the estimates
$$|\...
3
votes
1answer
155 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 ...
5
votes
0answers
143 views
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
$$\psi(\xi) = \phi(\xi) - \phi(...
1
vote
1answer
239 views
Homogeneous Littlewood-Paley decomposition
I have a question concerning Littlewood-paley-theory. Suppose we have test functions $\psi_k$ supported in annuli $\{2^{k-1}\leq\vert\xi\vert\leq2^{k+1}\}$ such that $\sum_{k\in\mathbb{Z}}\psi_k(\xi)=...
0
votes
1answer
231 views
Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation
Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L^p(\mathbb R)$.
Here, $F$ denotes the ...
4
votes
1answer
252 views
Littlewood-Paley theorem on an annulus
Suppose a smooth function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ satisfies $$\text{supp}~\hat{f}\subset \{\xi:1<|\xi|<2\}$$ and set functions $f_k$ by saying $$\hat{f_k}:=\hat{f}~\chi_{\{1+2^{-...
3
votes
1answer
521 views
$L^{p}$ 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^{...
28
votes
2answers
850 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 ...
3
votes
0answers
338 views
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 ...
1
vote
1answer
376 views
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 ...
