Weighted Norm Inequalities of (1, q)-Type for Integral and Fractional Maximal Operators

Abstract

We study weighted norm inequalities of (1, q)-type for 0 < q < 1,along with their weak-type counterparts, where ν = ν(Ω), and G is an integral operator with nonnegative kernel,These problems are motivated by sublinear elliptic equations in a domain Ω ⊂ R n with non-trivial Green's function G(x, y) associated with the Laplacian, fractional Laplacian, or more general elliptic operator.We also treat fractional maximal operators M α (0 ≤ α < n) on R n , and characterize strong-and weak-type (1, q)-inequalities for M α and more general maximal operators, as well as (1, q)-Carleson measure inequalities for Poisson integrals.2010 Mathematics Subject Classification. Primary 35J61, 42B37; Secondary 31B15, 42B25.

Pointwise estimates of solutions to semilinear elliptic equations and inequalities

“…See also [5], [6] for matching lower and upper bounds of solutions to the equation (−∆) α 2 u = u q σ (0 < α < n) on R n in the case 0 < q < 1. We remark that necessary and sufficient conditions for the existence of a positive solution in the case 0 < q < 1 to the homogeneous equation u = G(u q dσ) in Ω for quasi-symmetric kernels G which satisfy the weak maximum principle are given in [21] (see also [20]).…”

Quasilinear elliptic equations with sub-natural growth terms in bounded domains

“…Recently, Verbitsky and his colleagues studied the problem of existence of solutions to elliptic equations related to (1.1) and presented some criteria (see [8,9,10,17,16,31,32,33,34,37,38]). In their study, they treated the cases of Ω = R n , or p = 2.…”

Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms

“…In the classical case α = 1, our approach is employed to obtain the existence and uniqueness of a positive finite energy solution u ∈ Ẇ 1,2 0 (Ω), such that Ω |∇u| 2 dx < +∞ (see Definition 2.1 in the case p = 2), to the equation (1.3) − ∆u = σu q + µ in Ω, where 0 < q < 1 and Ω ⊂ R n is an arbitrary domain (possibly unbounded) which possesses a positive Green's function. The existence of positive weak solutions to (1.3), not necessarily of finite energy, is discussed in [23], [24]. We would like to point out that the existence and uniqueness of bounded solutions to (1.3) on Ω = R n in the case where µ is a nonnegative constant was characterized in [6].…”

“…The existence of positive weak solutions to (1.3), not necessarily of finite energy, is discussed in [23], [24].…”