 They generalize to several variables the Riemann–Liouville integrals of one variable.If 0 α"f" of a locally integrable function "f" on R"n" is the function defined by

where the constant is given by

This singular integral is well-defined provided "f" decays sufficiently rapidly at infinity, specifically if "f" ∈ L"p"(R"n") with 1 ≤ "p" 1 is classical, due to Sobolev, while for p=1 see ), the rate of decay of "f" and that of "I"α"f" are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)
where formula_3 is the vector-valued Riesz transform.More generally, the operators "I"α are well-defined for complex α such that 0 α is the locally integrable function