50 Years with Hardy Spaces pp 467-484 | Cite as

# 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} **G***σ*) *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 *⊆* ℝ^{n} with a positive Green function.

## Keywords

Weighted norm inequalities sublinear elliptic equations Green’s function weak maximum principle fractional Laplacian## Mathematics Subject Classification (2010)

Primary 35J61 Secondary 31B15 42B37 42B25## Preview

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