Sublinear Equations and Schur’s Test for Integral Operators

Abstract

We study weighted norm inequalities of (p, r)-type, ||G(f dσ)||L^r(Ω,dσ) ≤ C|f|_Lp(Ω,σ), for all f ∈L^p(σ), for 0 < r < p and p = 1, where G(fdσ)(x) = ⎰_Ω G(x, y)f(y)dσ(y) is an integral operator associated with a nonnegative kernel G(x, y) on Ω×Ω, and σ is a locally finite positive measure in Ω. We show that this embedding holds if and only if ⎰_Ω Gσ) pr p–r dσ < +∞, provided G is a quasi-symmetric kernel which satisfies the weak maximum principle. In the case p = r q, where 0 < q < 1, we prove that this condition characterizes the existence of a non-trivial solution (or supersolution) u ∈ L^r(Ω, σ), for r = q, to the sublinear integral equation u –G(u^q dσ) = 0, u≥ 0. We also give some counterexamples in the end-point case p = 1, which corresponds to solutions u ∈ L^q(Ω, σ) of this integral equation studied recently in [19, 20]. These problems appear in the investigation of weak solutions to the sublinear equation involving the (fractional) Laplacian, (-Δ)αu – σ uq = 0, u≥ 0, for 0 < q < 1 and 0 < α < n 2 in domains Ω ⊆ ℝⁿ with a positive Green function.

Dedicated to the memory of V.P. Havin