Must the Fourier series of an L function converge pointwise almost everywhere? In the 1960’s, Carleson answered this question in the affirmative, by studying a particular type of maximal singular integral operator, which has since become known as a Carleson operator. In the past 40 years, a number of important results have been proved for generalizations of the original Carleson operator. In th...

Abstract This paper characterises the boundedness and compactness of Agler–McCarthy monomial operators by reducing them to weighted composition deriving explicit Carleson measure criteria on half-plane. The results are illustrated examples.

The present paper establishes the correspondence between properties of solutions a class PDEs and geometry sets in Euclidean space. We settle question whether (quantitative) absolute continuity elliptic measure with respect to surface uniform rectifiability boundary are equivalent, an optimal divergence form operators satisfying suitable Carleson condition domains Ahlfors regular boundaries. re...

Let $$p(\cdot ):\ {\mathbb {R}^n}\rightarrow (0,\infty )$$ be a variable exponent function satisfying the globally log-Hölder continuous condition and A general expansive matrix on $${\mathbb {R}^n}$$ . In this article, authors introduce anisotropic Campanato-type spaces give some applications. Especially, using known atomic finite characterizations of Hardy space $$H_A^{p(\cdot )}(\mathbb {R}^...

Let e be a homogeneous subset of R in the sense of Carleson. Let μ be a finite positive measure on R and Hμ(x) its Hilbert transform. We prove that if limt→∞ t|e ∩ {x | |Hμ(x)| > t}| = 0, then μs(e) = 0, where μs is the singular part of μ.

is a Carleson measure in a Lipschitz domain Ω ⊂ R, n ≥ 1, (here δ (X) = dist (X, ∂Ω)). If the harmonic measure dωL0 ∈ A∞, then dωL1 ∈ A∞. This is an analog to Theorem 2.17 in [8] for divergence form operators. As an application of this, a new approximation argument and known results we obtain: Let L be an elliptic operator with coefficients A and drift term b; L can be in divergence or nondiver...