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 $\|K_jf^2\|_2\leq C_3e^{-c_4j}$