# Littlewood-Paley characterisation of Hölder regularity

I am going through Terence Tao's "Nonlinear Dispersive Equations (Local & Global Analysis)" and trying to work through some of his exercises. However, I find myself being stumped by Exercise A.21. The question goes as follows:

Let $$0<\alpha<1$$ and $$1\le p\le\infty$$. If $$f\in S_x(\mathbb{R}^d)$$, we define the Hölder norm $$\|f\|_{\Lambda_\alpha^p(\mathbb{R}^d)}$$by the formula $$\|f\|_{\Lambda_\alpha^p(\mathbb{R}^d)}:=\|f\|_{L^p_x(\mathbb{R}^d)} + \sup_{h\in\mathbb{R}^d; 0<|h|\le1}\frac{\|f^h-f\|_{L^p_x(\mathbb{R}^d)}}{|h|^\alpha}$$ where $$f^h(x)=f(x+h)$$ is the translate of $$f$$ by $$h$$. Show that $$\|f\|_{\Lambda_\alpha^p(\mathbb{R}^d)} \sim_{p,\alpha,d} \|f\|_{L^p_x(\mathbb{R}^d)} + \sup_{N\ge 1}N^\alpha\|P_{N}f\|_{L^p_x(\mathbb{R}^d)}$$ (Hint: to control the former by the latter, obtain two bounds for the $$L^p_x(\mathbb{R}^d)$$ norm of $$P_Nf^h-P_Nf$$, using triangle inequality for the high frequency case $$N\gtrsim |h|^{-1}$$ and the fundamental theorem of calculus in the low frequency case $$N\lesssim |h|^{-1}$$.)

So when attempting to control the former with the latter in the high frequency case, I took his hint and applied triangle inequality, which landed me at $$\|P_Nf^h-P_Nf\|_{L^p_x(\mathbb{R}^d)} \le 2\|P_Nf\|_{L^p_x(\mathbb{R}^d)}$$

Thereafter, $$\frac{\|P_Nf^h-P_Nf\|_{L^p_x(\mathbb{R}^d)}}{|h|^\alpha} \lesssim N^\alpha\|P_Nf\|_{L^p_x(\mathbb{R}^d)}$$

From here, I found myself in a rather unpleasant dead-end since moving forward will require me to take the infinite sum over $$N$$'s, which in this present form will not converge.

May I know how should I side-step this problem? Or am I misunderstanding his hint? Thanks in advance!

You need to remember that $$N$$ is a dyadic number, so that $$\sum_{N \geq c} N^{-\alpha}$$ converges for positive $$\alpha$$.
$$\sum_{N \geq |h|^{-1}} \frac{\|P_Nf^h-P_Nf\|_{L^p_x(\mathbb{R}^d)}}{|h|^\alpha} = \sum_{N \geq |h|^{-1}} N^\alpha\|P_Nf^h-P_Nf\|_{L^p_x(\mathbb{R}^d)}\cdot \frac{1}{N^\alpha|h|^\alpha}$$
Split the sum using Holder and put the first factor in $$\ell^\infty$$ and sum the second factor to get a constant (which is $$\approx 1$$, independently of $$h$$.)