A SUBLINEAR VERSION OF SCHUR'S LEMMA AND ELLIPTIC PDE

作者：Quinn Stephen^*; Verbitsky Igor E

来源：Analysis & PDE, 2018, 11(2): 439-466.

DOI：10.2140/apde.2018.11.439

• weighted norm inequalities
• sublinear elliptic equations
• Green's function
• weak maximum principle
• fractional Laplacian

We study the weighted norm inequality of (1, q )-type,
parallel to G nu parallel to(q)(L)(Omega, d sigma) <= C parallel to nu parallel to for all nu is an element of M+ (Omega),
along with its weak-type analogue, for 0 < q < 1, where G is an integral operator associated with the nonnegative kernel G on Omega x Omega. Here M+ (Omega) denotes the class of positive Radon measures in Omega; sigma, nu is an element of M+ (Omega), and parallel to nu parallel to = nu(Omega).
For both weak-type and strong-type inequalities, we provide conditions which characterize the measures sigma for which such an embedding holds. The strong-type (1, q)-inequality for 0 < q < 1 is closely connected with existence of a positive function u such that u >= G (u(q) sigma), i.e., a supersolution to the integral equation
u - G(u(q)sigma) = 0, u is an element of L-loc(q) (Omega, sigma).
This study is motivated by solving sublinear equations involving the fractional Laplacian,
(-Delta)(alpha/2)u - u(q)sigma = 0,
in domains Omega subset of R-n which have a positive Green function G for 0 < alpha < n.

• 出版日期2018