Existence of the Gauge for Fractional Laplacian Schrödinger Operators.

Let Ω ⊆ R n be an open set, where n ≥ 2 . Suppose ω is a locally finite Borel measure on Ω . For α ∈ (0 , 2) , define the fractional Laplacian (- ▵) α / 2 via the Fourier transform on R n , and let G be the corresponding Green's operator of order α on Ω . Define T (u) = G (u ω). If ‖ T ‖ L 2 (ω) → L 2 (ω) < 1 , we obtain a representation for the unique weak solution u in the homogeneous Sobolev space L 0 α / 2 , 2 (Ω) of (- ▵) α / 2 u = u ω + ν on Ω , u = 0 on Ω c , for ν in the dual Sobolev space L - α / 2 , 2 (Ω) . If Ω is a bounded C 1 , 1 domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when ν = χ Ω . These estimates are used to study the existence of a solution u 1 (called the "gauge") of the integral equation u 1 = 1 + G (u 1 ω) corresponding to the problem (- ▵) α / 2 u = u ω on Ω , u ≥ 0 on Ω , u = 1 on Ω c. We show that if ‖ T ‖ < 1 , then u 1 always exists if 0 < α < 1 . For 1 ≤ α < 2 , a solution exists if the norm of T is sufficiently small. We also show that the condition ‖ T ‖ < 1 does not imply the existence of a solution if 1 < α < 2 . The condition ‖ T ‖ ≤ 1 is necessary for the existence of u 1 for all 0 < α ≤ 2 . [ABSTRACT FROM AUTHOR]