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Let $B = B(0,4/3)$, $C = \{x \in \mathbb R^d : 3/4 \leq \|x\|_2 \leq 8/3\}$ and $\tilde{C} = \{x \in \mathbb R^d : 1/12 \leq \|x\|_2 \leq 10/3\}$. For a fixed Littlewood-Paley decomposition $\chi \in \...

I have a function $\phi(x): \mathbb{R} \to [0, 2 \pi)$, which describes phase of another function $$f = e^{i \phi(x)}. $$ I am interested in the following problem. If I know the function/distribution $...