# Inequality involving the fractional Laplacian

Let $$s \in \mathbb{R}$$ such that $$0. Consider the fractional Laplacian $$(-\Delta)^s$$ in the real line defined via Fourier series as follows: if $$f:[-\pi,\pi] \subset \mathbb{R} \longrightarrow \mathbb{C}$$ is a periodic function and is written as $$f(x)=\sum_{n \in \mathbb{Z}} f_n e^{inx}$$ then $$(-\Delta)^{s/2}f(x)=\sum_{n \in \mathbb{Z}} |n|^{s} f_n e^{inx}.$$

Question. If we define $$g: \mathbb{R} \longrightarrow \mathbb{R}$$ by $$g(x):= |f(x)|,\; \forall \; x \in \mathbb{R}$$ then is true that $$|(-\Delta)^{s/2}g(x)| \leq |(-\Delta)^{s/2}f(x)|? \tag{1}$$

I didn't make any progress as I couldn't get any relation between the Fourier coefficients of $$f$$ and $$g$$ (it would be ideal if we had $$g_n=|f_n|$$ for each $$n \in \mathbb{Z}$$). There is some relation? The inequality in $$(1)$$ makes sense?

If true, this would follow from the integral expression for the fractional Laplacian: $$(-\Delta)^{s/2} f(x) = \int_{-\pi}^\pi (f(x) - f(y)) \nu(x - y) dy$$ for an appropriate kernel $$\nu$$. But, unfortunately, the claimed inequality is false: if, for example, $$f$$ is a non-zero odd function, then $$(-\Delta)^{s/2} f(0) = 0$$ (by symmetry), while $$(-\Delta)^{s/2} |f|(0) < 0$$ (by the integral expression given above, or by a version of the maximum principle).
On the positive side, we have the following inequality ("Markov property") for the corresponding quadratic (Dirichlet) forms: $$\langle |f|, (-\Delta)^{s/2} |f| \rangle \leqslant \langle f, (-\Delta)^{s/2} f \rangle .$$
• Nice answer! The $\langle \cdot, \cdot \rangle$ is the $L^2$-inner product? Do you have some reference of the 'Markov Property'? Sep 22 at 11:44
• Well, the Markov property follows directly from $$\langle f, (-\Delta)^{s/2} f\rangle = \frac12 \int_{-\pi}^\pi \int_{-\pi}^\pi (f(y) - f(x)) \nu(y - x) dy dx.$$ And this is essentially one of the properties of a bilinear form required to call it a Dirichlet form. And yes, $\langle\cdot,\cdot\rangle$ is the $L^2$ inner product. Sep 22 at 11:52