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 $$\psi(\xi) = \phi(\xi) - \phi(2\xi)$$ note that we have $\sum_{k\in\mathbb{Z}} \psi(\xi/2^k) = 1$. Given $f\in L^p$, we define the projection $$P_k f = \mathcal F^{-1}\Big( \psi(\xi/2^k)\mathcal{F}(f)\Big)$$ and the square function is define by $$Sf = \left( \sum_{k\in \mathbb{Z}} (P_k f)^2 \right)^{1/2}.$$
Littlewood-Paley theorem says that the $L^p$ norm of the square function associated with $f$ is equivalent to the $L^p$ of $f$ when $p\in (1,\infty)$. I'm interested in the endpoint case.
- Why it fails for $p=1$ and $p=\infty$?
- Any positive results for $p=1$ or $p=\infty$? e.g. Does one-side inequality holds? i.e. Can we bound the $L^{\infty}$ norm of the square function by $\|f\|_{\infty}$?