Consider the equation
where , . The inversion problem for (1) is called regular in Lp if, uniformly in p∈[1, ∞] for any f(x)∈ Lp(R), equation (1) has a unique solution y(x)∈ Lp(R) of the form
with . Here G(x, t) is the Green function corresponding to (1) and c is an absolute constant. For a given s∈[l, ∞], necessary and sufficient conditions are obtained for assertions (2) and (3) to hold simultaneously:
(2) the inversion problem for (1) is regular in Lp;
(3) for any f(x)∈LS(R).