In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator
T ( f ) ( x ) = ∫ K ( x , y ) f ( y ) d y , {\displaystyle T(f)(x)=\int K(x,y)f(y)\,dy,}
whose kernel function K : R n × R n → R {\displaystyle K:\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} } is singular along the diagonal x = y {\displaystyle x=y} . Specifically, the singularity is such that | K ( x , y ) | {\displaystyle |K(x,y)|} is of size | x − y | − n {\displaystyle |x-y|^{-n}} asymptotically as | x − y | → 0 {\displaystyle |x-y|\to 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 | → ϵ {\displaystyle |y-x|\to \epsilon } as ϵ → 0 {\displaystyle \epsilon \to 0} , but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp spaces, for example L p ( R n ) {\displaystyle L^{p}(\mathbb {R} ^{n})} .
The Hilbert transform
The archetypal singular integral operator is the Hilbert transform H {\displaystyle H} . It is given by convolution against the kernel K ( x ) = 1 / ( π x ) {\displaystyle K(x)=1/(\pi x)} for x {\displaystyle x} in R {\displaystyle \mathbb {R} } . More precisely,
H ( f ) ( x ) = 1 π lim ε → 0 ∫ | x − y | > ε 1 x − y f ( y ) d y . {\displaystyle H(f)(x)={\frac {1}{\pi }}\lim _{\varepsilon \to 0}\int _{|x-y|>\varepsilon }{\frac {1}{x-y}}f(y)\,dy.}
The most straightforward higher dimension analogues of these are the Riesz transforms, which replace K ( x ) = 1 / x {\displaystyle K(x)=1/x} with
K i ( x ) = x i | x | n + 1 {\displaystyle K_{i}(x)={\frac {x_{i}}{|x|^{n+1}}}}
where i = 1 , . . . , n {\displaystyle i=1,...,n} and x i {\displaystyle x_{i}} is the i {\displaystyle i} -th component of x {\displaystyle x} in R n {\displaystyle \mathbb {R} ^{n}} . All of these operators are bounded on L p {\displaystyle L^{p}} and satisfy weak-type ( 1 , 1 ) {\displaystyle (1,1)} estimates.
Singular integrals of convolution type
A singular integral of convolution type is an operator T {\displaystyle T} defined by convolution with a kernel K {\displaystyle K} that is locally integrable on R n ∖ { 0 } {\displaystyle R^{n}\setminus \{0\}} , in the sense that
Suppose that the kernel satisfies:
The size condition on the Fourier transform of K {\displaystyle K}
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