Singular Integral
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High Quality Content by WIKIPEDIA articles! In mathematics, singular integrals are central to abstract harmonic analysis and are intimately connected with the study of partial differential equations. integrals are central to abstract harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operatorwhose kernel function K : Rn×Rn Rn is singular along the diagonal x = y. Specifically, the singularity is such that K(x,y) is of size x y n asymptotically as x y 0. Since such integrals may not in general be…mehr

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High Quality Content by WIKIPEDIA articles! In mathematics, singular integrals are central to abstract harmonic analysis and are intimately connected with the study of partial differential equations. integrals are central to abstract harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operatorwhose kernel function K : Rn×Rn Rn is singular along the diagonal x = y. Specifically, the singularity is such that K(x,y) is of size x y n asymptotically as x y 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over y x as 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp(Rn).Observe that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.