We prove a “local” Tb Theorem for square functions, in which we assume only Lq control of the pseudo-accretive system, with q > 1. We then give an application to variable coefficient layer potentials for divergence form elliptic operators with bounded measurable non-symmetric coefficients.

In quantum mechanics, predictions are made by way of calculating expectation values of observables, which take the form of Hermitian operators. Non-Hermitian operators, however, are not necessarily devoid of physical significance, and they can play a crucial role in the characterization of quantum states. Here we show that the expectation values of a particular set of non-Hermitian matrices, wh...

in this paper, we determine the structure of the space of multipliers of the range of a composition operator cφ that induces by the conditional expectation between two lp() spaces.

We consider the Bézier variant of Chlodovsky–Durrmeyer operators Dn,α for functions f measurable and locally bounded on the interval [0,∞). By using the Chanturia modulus of variation we estimate the rate of pointwise convergence of (Dn,αf) (x) at those x > 0 at which the one-sided limits f(x+), f(x−) exist. In the special case α = 1 the recent result of [14] concerning the Chlodovsky–Durrmeyer...

We prove a Harnack inequality for distributional solutions to a type of degenerate elliptic PDEs in N dimensions. The differential operators in question are related to the Kolmogorov operator, made up of the Laplacian in the last N−1 variables, a first-order term corresponding to a shear flow in the direction of the first variable, and a bounded measurable potential term. The first-order coeffi...

We prove a boundary Harnack inequality for nonlocal elliptic operators L in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if Lu1 = Lu2 = 0 in Ω ∩ B1, u1 = u2 = 0 in B1 \ Ω, and u1, u2 ≥ 0 in R, then u1 and u2 are comparable in B1/2. The result applies to arbitrary open sets Ω. When Ω is Lipschitz, we show that the quotient u1/u2 is Hölder con...

In this article, we highlight the role of Carleson measures in elliptic boundary value prob5 lems, and discuss some recent results in this theory. The focus here is on the Dirichlet problem, with 6 measurable data, for second order elliptic operators in divergence form. We illustrate, through selected 7 examples, the various ways Carleson measures arise in characterizing those classes of operat...

This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard, in the space X c of Lebesgue measurable functions f on R+ = (0,∞) such that ∞ ∫ 0 ∣∣ucf (u)∣∣p du u <∞ (1 p <∞), ess sup u>0 [ u ∣∣f (u)∣∣]<∞ (p =∞), for c ∈ R = (−∞,∞), in particular in the space Lp(0,∞) (1 p ∞). Formulas for the Mellin transforms ...

In this paper, we prove that the topological dual of the Banach space of bounded measurable functions with values in the space of nuclear operators, furnished with the natural topology, is isometrically isomorphic to the space of finitely additive linear operator-valued measures having bounded variation in a Banach space containing the space of bounded linear operators. This is then applied to ...