Knowlpedia

Definition

Principal-value singular integral

An integral operator defined by removing a neighborhood of its diagonal and taking a limit.

For a kernel singular at x=yx=y, a principal-value singular integral is an operator represented, on a stated class of inputs, by

Tf(x)=lim⁡ε↓0∫∣x−y∣>εK(x,y)f(y) dy.Tf(x)=\lim_{\varepsilon\downarrow0}\int_{|x-y|>\varepsilon}K(x,y)f(y)\,dy.

This is the radial-truncation version of a . Convergence must be specified, for example pointwise almost everywhere or in a norm.

Cancellation and scope

The limit can exist although the untruncated integral is not absolutely convergent near the diagonal. Symmetry or mean cancellation often supplies this convergence. A size estimate ∣K(x,y)∣≤C∣x−y∣−n|K(x,y)|\le C|x-y|^{-n} alone is insufficient: a positive kernel of this size diverges logarithmically on a positive constant near xx. Distributional formulas for singular operators may also include a local multiplication term in addition to the principal-value integral.