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 allowed to have values in $\mathbb{R}^m$, via working with the components) Let $r=l+\alpha$ with $l \in \mathbb{N}$ and $0\le \alpha <1$. Case 1) If $\alpha >0$, $C^r_{*}(\Omega)$ is […]

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 $2^{-l_{0}}<\left|\xi\right|<2^{l_{0}}$ for some $l_{0}>0$). Let $\Delta_{j}^{\zeta}$ be the Littlewood-Paley projection defined by $$\widehat{\Delta_{j}^{\zeta}f}(\xi)=\widehat{\zeta}\left(\dfrac{\xi}{2^{j}}\right)\widehat{f}(\xi),\quad{f\in \mathcal{S}(\mathbb{R}^{n})}$$ and consider the associated Littlewood-Paley decomposition $\sum_{j}\Delta_{j}^{\zeta}f$. It is a well-known theorem, the Littlewood-Paley inequality (L. Grafakos, Classical Fourier Analysis, Theorem […]

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 or disprove that $$ \sup \left\{\frac{\int_E |f|dx}{|E|^{1/2}}: E \subset [0,1]\right\}<+\infty$$ If the claim above is false, then is it possible to prove that for any fixed […]

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