In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations.
Usually further assumptions are required to obtain results such as their boundedness on Lp(Rn).
[1] A singular integral of convolution type is an operator T defined by convolution with a kernel K that is locally integrable on Rn\{0}, in the sense that Suppose that the kernel satisfies: Then it can be shown that T is bounded on Lp(Rn) and satisfies a weak-type (1, 1) estimate.
Neither of the properties 1. or 2. is necessarily easy to verify, and a variety of sufficient conditions exist.
[2] T is said to be a singular integral operator of non-convolution type associated to the Calderón–Zygmund kernel K if whenever f and g are smooth and have disjoint support.
An operator is said to be weakly bounded if there is a constant C such that for all normalised bumps φ and ψ.