Two-weight Lp → Lq bounds for positive dyadic operators in the case 0 < q < 1 ≤ p < ∞

Let sigma, omega be measures on R-d, and let {lambda(Q)}(Q is an element of D) be a family of non-negative reals indexed by the collection D of dyadic cubes in R-d. We give necessary and sufficient conditions for the twoweight norm inequality parallel to T-lambda(f sigma)parallel to(Lq(omega)) for the positive dyadic operator T-lambda(f sigma):= Sigma(Q is an element of D) lambda(Q)(1/sigma(Q) integral(Q) fd sigma) 1(Q) in the difficult range 0 <q <1 Furthermore, we introduce a scale of discrete Wolff potential conditions that depends monotonically on an integrability parameter, and prove that such conditions are necessary (but not sufficient) for small parameters, and sufficient (but not necessary) for large parameters. Our characterization applies to Riesz potentials I-alpha(f sigma) = (-Delta)(-alpha/2) (f sigma), 0 <alpha <d, since it is known that they can be controlled by model dyadic operators. The weighted norm inequality for Riesz potentials in this range of p, q has been characterized previously only in the special case where sigma is a Lebesgue measure.Peer reviewe

Local and global behaviour of nonlinear equations with natural growth terms

This paper concerns a study of the pointwise behaviour of positive solutions to certain quasi-linear elliptic equations with natural growth terms, under minimal regularity assumptions on the underlying coefficients. Our primary results consist of optimal pointwise estimates for positive solutions of such equations in terms of two local Wolff's potentials.Comment: In memory of Professor Nigel Kalto

Weighted Trace Inequalities for Fractional Integrals and Applications to Semilinear Equations

AbstractWe show that the two-weight trace inequality for the Riesz potentials Iα, ||Iαf||Lp(w) ≤ C|| f||Lp(v), holds if Iαw ∈ Lp′loc (σ) and Iα[(Iαw)p′ σ] ≤ CIαw a.e. Here w and v are non-negative weight functions on Rn, and σ = v1-p′. The converse is also also true under some mild restrictions on w and v. We also consider more general inequalities for measures which are not necessarily absolutely continuous with respect to Lebesgue measure. In contrast to the known characterizations of the trace inequality, this "pointwise" condition is stated explicitly in terms of potentials of w and σ, rather than measures of some subsets of Rn. Applications to the problem of the existence of positive solutions for the semilinear elliptic equation −Δu = σ(x) uq + w(x) (1 < q < ∈) on Rn are given

Factorization of Tent Spaces and Hankel Operators

AbstractIt is shown that the factorization Tpq=Tp∞·T∞q for tent spaces proved by R. R. Coifman et al. (1985, J. Funct. Anal.62, 304–335) for p>q, q=2, holds true for all 0<p, q<∞. From this certain strong factorization theorems are derived for spaces Hps of fractional derivatives of Hp functions, and more general Triebel spaces. In particular, it is proved that Hps=Hp·BMOAs . Applications considered include characterizations of symbols of bounded Hankel operators Hφ: Hp→Hqs, complex interpolation of tent spaces, and Carleson measure theorems for derivatives of Hp functions

Accretivity of the General Second Order Linear Differential Operator

For the general second order linear differential operatorL0=∑j,k=1najk∂j∂k+∑j=1nbj∂j+c with complex-valued distributional coefficients a j,k , b j , and c in an open set Ω ⊆ ℝ n (n ≥ 1), we present conditions which ensure that −L is accretive, i.e., Re ⟨−Lϕ, ϕ⟩≥0 for all φ ∈ C 0 ∞ (Ω). © 2019, Springer-Verlag GmbH Germany & The Editorial Office of AMS