On hyper-singular integrals

The integrals ∫ ∞ − ∞ t λ − 1 + ϕ ( t ) d t and ∫ t 0 ( t − s ) λ − 1 b ( s ) d s are considered, λ ≠ 0 , − 1 , − 2 … , where ϕ ∈ C ∞ 0 ( R ) and 0 ≤ b ( s ) ∈ L 2 l o c ( R ) . These integrals are defined in this paper for λ ≤ 0 , λ ≠ 0 , − 1 , − 2 , … , although they diverge classically for λ ≤ 0 . Integral equations and inequalities are considered with the kernel ( t − s ) λ − 1 + .