Explicit calculation of singular integrals of tensorial polyadic kernels

The Riesz transform of u u : alttext="script upper S left-parenthesis double-struck R Superscript n Baseline right-parenthesis right-arrow script prime right-parenthesis"> S ( mathvariant="double-struck">R n stretchy="false">) stretchy="false">→ ′ encoding="application/x-tex">\mathcal {S}(\mathbb {R}^n) \rightarrow \mathcal {S’}(\mathbb {R}^n) is defined as a convolution by singular kernel, and can be conveniently expressed using the Fourier simple multiplier. We extend this analysis to higher order transforms, i.e. some type integrals that contain tensorial polyadic kernels define an integral for functions times ellipsis ×<!-- × <mml:mo>…<!-- … {R}^{ \times \dots n})</mml:annotation> . show transformed kernel also tensor, propose general method compute explicitely mutliplier. Analytical results are given, well recursive algorithm, coefficients kernel. compare result direct numerical evaluation, discuss case alttext="n equals 2"> = 2 encoding="application/x-tex">n=2 , with application image analysis.