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For $f, g \in \mathcal{S}$, define $f_k = P_k f$, $g^k = \sum_{j=-\infty}^{k-3}P_j g$, where $P_j$ is the Littlewood-Paley projection. Is it true that

$$ P(f, g) = \sum_{j=-\infty}^\infty f_j g^j $$

converges in $L^p$ and satisfies the Holder type inequality

$$ \|P(f, g)\|_{L^p} \le C \|f\|_{L^q}\|g\|_{L^r} $$

for $\dfrac{1}{p} = \dfrac{1}{q} + \dfrac{1}{r}$ ? Why?

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    $\begingroup$ I think this is true as long as $1 < p < \infty$. The rough idea is that $\| P(f,g) \|_{L^p}$ can be controlled by the $L^p$ norm of the square function $( \sum_j | f_j g^j |^2 )^{1/2}$, which can in turn be controlled pointwise by the product of the Hardy-Littlewood maximal function of $g$ times the Littlewood-Paley square function of $f$. See page 9 of notes here: math.ucla.edu/~tao/247b.1.07w/notes6.pdf $\endgroup$
    – Jason
    Nov 18, 2020 at 3:42

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