# 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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### 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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### 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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### 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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