Igor E. Verbitsky

We study quasilinear elliptic equations of the type - Δ p ⁢ u = σ ⁢ u q + μ {-\Delta_{p}u=\sigma u^{q}+\mu} in ℝ n {\mathbb{R}^{n}} in the case 0 < q < p - 1 {0p-Laplacian. Similar equations with more general local and nonlocal operators in place of Δ p {\Delta_{p}} are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions u: u ⁢ ( x ) ≈ ( 𝐖 p ⁢ σ ⁢ ( x ) ) p - q p - q - 1 + 𝐊 p , q ⁢ σ ⁢ ( x ) + 𝐖 p ⁢ μ ⁢ ( x ) , x ∈ ℝ n , u(x)\approx({\mathbf{W}}_{p}\sigma(x))^{\frac{p-q}{p-q-1}}+{\mathbf{K}}_{p,q}% \sigma(x)+{\mathbf{W}}_{p}\mu(x),\quad x\in\mathbb{R}^{n}, where 𝐖 p {{\mathbf{W}}_{p}} and 𝐊 p , q {{\mathbf{K}}_{p,q}} are, respectively, the Wolff potential and the intrinsic Wolff potential, with the constants of equivalence depending only on p, q, and n. The contributions of μ and σ in these pointwise estimates are totally separated, which is a new phenomenon even when p = 2 {p=2} .

Let σ and ω be locally finite Borel measures on ℝ^d, and let $p\in (1,\infty )$ p ∈ ( 1 , ∞ ) and $q\in (0,\infty )$ q ∈ ( 0 , ∞ ) . We study the two-weight norm inequality $ \lVert T(f\sigma ) \rVert _{L^{q}(\omega )}\leq C \lVert f \rVert _{L^{p}(\sigma )}, \quad \text {for all} f \in L^{p}(\sigma ), $ ∥ T ( f σ ) ∥ L q ( ω ) ≤ C ∥ f ∥ L p ( σ ) , for all f ∈ L p ( σ ) , for both the positive summation operators T = T_λ(⋅σ) and positive maximal operators T = M_λ(⋅σ). Here, for a family {λ_Q} of non-negative reals indexed by the dyadic cubes Q, these operators are defined by $ T_{\lambda }(f\sigma ):={\sum }_{Q} \lambda _{Q} \langle {f}\rangle ^{\sigma }_{Q} 1_{Q} \quad \text { and } \quad M_{\lambda }(f\sigma ):=\sup _{Q} \lambda _{Q} \langle {f}\rangle ^{\sigma }_{Q} 1_{Q}, $ T λ ( f σ ) : = ∑ Q λ Q 〈 f 〉 Q σ 1 Q and M λ ( f σ ) : = sup Q λ Q 〈 f 〉 Q σ 1 Q , where $\langle {f}\rangle ^{\sigma }_{Q}:=\frac {1}{\sigma (Q)} {\int \limits }_{Q} |f| d \sigma $ 〈 f 〉 Q σ : = 1 σ ( Q ) ∫ Q | f | d σ . We obtain new characterizations of the two-weight norm inequalities in the following cases: (1) For T = T_λ(⋅σ) in the subrange q < p. Under the additional assumption that σ satisfies the $A_{\infty }$ A ∞ condition with respect to ω, we characterize the inequality in terms of a simple integral condition. The proof is based on characterizing the multipliers between certain classes of Carleson measures. (2) For T = M_λ(⋅σ) in the subrange q < p. We introduce a scale of simple conditions that depends on an integrability parameter and show that, on this scale, the sufficiency and necessity are separated only by an arbitrarily small integrability gap. (3) For the summation operators T = T_λ(⋅σ) in the subrange 1 < q < p. We characterize the inequality for summation operators by means of related inequalities for maximal operators T = M_λ(⋅σ). This maximal-type characterization is an alternative to the known potential-type characterization. The subrange of the exponents q < p appeared recently in applications to nonlinear elliptic PDE with $\lambda _{Q} = \sigma (Q) |Q|^{\frac {\alpha }{d}-1}$ λ Q = σ ( Q ) | Q | α d − 1 , α ∈ (0, d). In this important special case T_λ is a discrete analogue of the Riesz potential $I_{\alpha }=(-{\Delta })^{-\frac {\alpha }{2}}$ I α = ( − Δ ) − α 2 , and M_λ is the dyadic fractional maximal operator.