CalderónZygmundSchätzungen

Calderón–Zygmund theory is a framework in harmonic analysis that studies singular integral operators of Calderón–Zygmund type. These operators are linear and initially defined on smooth functions by integral kernels K(x,y) with a specific singularity along the diagonal x = y. The kernels satisfy a size condition |K(x,y)| ≤ C/|x−y|^n and smoothness or Hörmander-type conditions that control how K varies with x and y away from the diagonal.

One of the central results is that such operators extend to bounded operators on Lp(R^n) for 1 < p < ∞. This Lp-boundedness is established using the Calderón–Zygmund decomposition, a real-variable technique that splits an L1 function into a global "good" part and a localized "bad" part, enabling weak-type (1,1) and Lp estimates. The theory also encompasses a broad class of convolution-type and non-convolution-type operators, including those with kernels that are not smooth at the origin but satisfy appropriate cancellation and smoothness conditions.

Historical and influence notes: the theory is named for Alberto Calderón and Antoni Zygmund, who laid its foundations in the 1950s. Subsequent developments include the T(1) theorem of Journé, extensions to spaces of homogeneous type, and connections to Hardy spaces and BMO. Calderón–Zygmund theory has wide-ranging applications in partial differential equations, potential theory, and modern analysis, serving as a central tool in understanding regularity and boundedness properties of singular integrals.