Half-Laplacian

Half-Laplacian

The half-Laplacian, written $(-\Delta)^{1/2}$, is a fractional power of the Laplace operator. On the Fourier side it acts by multiplying each frequency component by a quantity proportional to the magnitude of that frequency.

Unlike an ordinary derivative, the half-Laplacian is nonlocal: its value at one point generally depends on the behaviour of the function across a wider region.

It appears in harmonic analysis, partial differential equations, probability and nonlocal models. This public glossary entry does not reproduce project-specific half-Laplacian identities from the private Riemann Console research. .