Weighted estimates for solutions of a Sturm–Liouville equation in the space L1(ℝ)

We consider the equationwhereUnder these conditions, (1) is correctly solvable in L1(ℝ), i.e.(i) for any function f ∈ L1(ℝ), there exists a unique solution of (1), y ∈ L1(ℝ);(ii) there is an absolute constant c1 ∈ (0, ∞) such that the solution of (1),In this work we strengthen the a priori inequality (1). We find minimal requirements for a given weight function θ ∈ Lloc1(ℝ) under which the solution of (1), y ∈ L1(ℝ), satisfies the estimatewhere c2 is some absolutely positive constant.

An equation with left and right fractional derivatives

We consider an equation with left and right fractional derivatives and with the boundary condition y(0) = lim x?0+ y(x) = 0, y(b) = lim x?b? y(x) = 0 in the space L1(0, b) and in the subspace of tempered distributions. The asymptotic behavior of solutions in the end points 0 and b have been specially analyzed by using Karamata?s regularly varying functions.