## A rigorous way to define the singular integral

Suppose satisfies the following conditions

(1)

(2)

(3)

Then we can define the singular integral operator

On the face of it, this integrand may not be Lebesgue integrable. There is a way to interpret this integral

Suppose (1),(2),(3) are satisfied, define

Then with independent of . Moreover, one can prove is a cauchy sequence, then there exist , such that

in

From this, we define .

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