The analysis of integral operators whose kernels fail to be absolutely integrable (singular kernels), focusing on principal-value definitions, Calderón–Zygmund type bounds, mapping properties on function spaces, and applications to boundary value problems.

Principle

Exploit kernel cancellation, homogeneity, and smoothness away from the diagonal to define principal-value integrals and prove boundedness and continuity on Lp, Hölder and Sobolev spaces using kernel decomposition and harmonic-analytic techniques.

Demonstration

The Hilbert transform on the real line (principal-value convolution with 1/x) is bounded on Lp for 1

Misapplication

Treating singular kernels as if they were integrable without using principal-value interpretation, ignoring cancellation that yields boundedness, or applying Lp-bounds outside their valid exponent range.

Consequence

A correct singular integral framework yields robust mapping theorems, regularity transfer for solutions of PDEs, and explicit integral representations that are central to elliptic boundary problems and harmonic analysis.

Reversal

Regular integral operators with integrable kernels that are compact or smoothing; problems where singularity is absent so principal-value techniques are unnecessary.

Boundary

Applies to kernels with controlled singularities (e.g., homogenous of degree −n) on Euclidean spaces and smooth manifolds and to principal-value formulations; does not cover arbitrary distributions or purely discrete analogues without adaptation.

Semantic Tension

Tension with pseudodifferential and distributional frameworks: singular integral operators share features with Calderón–Zygmund and pseudodifferential operators but differ in emphasis on kernel cancellation and explicit principal-value constructions.

Synthesis

Singular integral theory provides the tools to define and bound nonintegrable-kernel operators through principal values and cancellation, producing mapping results and integral representations essential for boundary regularity and harmonic analysis.