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Let $f$ be a continuous function on $S^2$. Consider $g\in C^{\infty}(R)$, such that $g(x)=1$ for $|x|\leq 1$ and for $|x|\geq 2$. Let $h(x)=g(x)-g(2x)$. The notation $proj_k$ denotes the orthogonal projection operator onto the space of spherical harmonics of degree $k$.

Define $$K_0f(x)=\sum_{j=0}^1g(j)\, \operatorname{proj_j}f(x)$$ and $$ K_jf(x)=\sum_{k=2^{j-1}}^{2^{j+1}}h(2^{-j}k)\,\operatorname{proj_k}f(x), \quad j=1,2, \ldots. $$

Show that $\|K_jf\|_2\leq C_1e^{-c_2j}$ and $\|K_jf^2\|_2\leq C_3e^{-c_4j}$

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  • $\begingroup$ Where does this question originate? Is it an exercise? Is it part of a research article that you are trying to understand? $\endgroup$
    – Yemon Choi
    Jun 30, 2019 at 16:07
  • $\begingroup$ It is part of research article on convex geometry $\endgroup$
    – user124297
    Jul 12, 2019 at 15:38
  • $\begingroup$ Why don't you give a reference to the article so that people can look at the context? You might also have a better chance of getting responses if you indicate what you have done to try and answer the question yourself, where you are stuck, what previous background you have, and so on $\endgroup$
    – Yemon Choi
    Jul 12, 2019 at 23:17

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