# Concrete examples of elliptic pseudo-differential equations

Remember that $p \in S^{m}_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$ or that $p: \mathbb{R}^n \times \mathbb{R}^n \longrightarrow \mathbb{C}$ is a simbol if it is a smooth function such that \begin{align*} \displaystyle |D^\alpha_{\xi} D^\beta_{x} p(x,\xi)| \leq C_{\alpha , \beta}(1+|\xi|)^{m-|\alpha|} \end{align*} then the associated psuedo-differential operator is \begin{align*} \displaystyle p(x,D_x)f=\mathrm{OP}(p)f=\int_{\mathbb{R}^n} e^{i x \cdot \xi}p(x,\xi) \widehat{f}(\xi) d\xi \end{align*} where $f \in \mathcal{S}(\mathbb{R}^n)$. For example if $c_\alpha \in C^\infty(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n):=C^{\infty}_b$ and we consider \begin{align*} \displaystyle p(x,\xi)=\sum_{|\alpha| \leq m} c_{\alpha}(x) \xi^\alpha \end{align*} then $p \in S^{m}_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$ and using Fourier transform we have that \begin{align*} \displaystyle p(x, D_x)f = \sum_{|\alpha| \leq m} c_{\alpha}(x) D^{\alpha}_x f \end{align*} for each $f \in \mathcal{S}(\mathbb{R}^n)$, and any differential operator with limited regular coefficients is a pseudo-differential operator. We say that the simbol $p \in S^{m}_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$ is elliptic if $|p(x,\xi)| \geq C|\xi|^m$, and consequently the pseudo-differential operator is said elliptic. Therefore we can consider the elliptic pseudo-elliptic differential equation: \begin{align*} \displaystyle p(x, D_x)u = f \end{align*} where for example we assume that $f \in C^\infty(\mathbb{R}^n)$. Clearly the Laplace operator is also an elliptic pseudo-differential operator.

My question is: which other concrete examples and "famous" (as Laplace operator) of elliptic pseudo-differential operators are there? thank you.

Perhaps, the most famous one is the fractional Laplacian $$(-\Delta)^{s/2}u(x)=\frac{1}{(2\pi)^n}\int_{\mathbb R^n}\mathrm{e}^{ix\xi}|\xi|^s\hat u(\xi)\,d\xi.$$ A version for periodic functions is $$|\partial_x|^a u(x)=\sum_{k\in\mathbb Z}|k|^a\mathrm{e}^{ikx}\hat u_k$$ whenever $\,u(x)=\sum_{k\in\mathbb Z}\mathrm{e}^{ikx}\hat u_k$.