Fractional integrals over a function of finite type on the intersection spaces

Let ϕ be a function of finite type in [ − 1 , 1 ] . We define a fractional integral I s , ϕ over ϕ by I s , ϕ f ( x ) = ∫ [ − 1 , 1 ] f ( x − ϕ ( t ) ) d t | t | s and prove the ( L p , r , L q ) -norm inequalities, where L p , r = L p ∩ L r , 1 / q = s / r + ( 1 − s ) / p , 1 < r ⩽ p ⩽ ∞ and 0 < s < 1 . For r = 1 , we derive the weak-type norm inequality for I s , ϕ , provided ϕ ′ and ϕ ″ do not vanish.