A condition for a two-weight norm inequality for singular integral operators

Approximation and quasicontinuity of Besov and Triebel–Lizorkin functions

“…For 0 < γ ≤ 1/2, the γ-median of a measurable function u : X → R over a set A of finite measure is m γ u (A) = inf a ∈ R : µ({x ∈ A : u(x) > a}) < γµ(A) . Median values and median maximal functions have already been studied and used in different problems of analysis for quite some time, see [12], [14], [15], [20], [23], [24], [25], [31], [32], [38], [43] and [46].…”

Generalized Lebesgue Points for Hajłasz Functions

“…Operator M γ = M γ ∞ and its variants have turned out to be useful in harmonic analysis and in the theory of function spaces, see for example [7], [8], [9], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [27], [28], [29], [32], [36], [37], [39]. Theorem 1.1.…”

A median approach to differentiation bases

“…Here, B(x, r) = {y ∈ R n : |x − y| < r} is an open ball of radius r > 0 centered at x and |A| denotes the Lebesgue measure of set A ⊂ R n . If the integral averages in (1.1) are replaced by medians, or more generally, by γ-medians, m γ f (A) = inf a ∈ R : |{x ∈ A : f (x) > a}| < γ|A| , where A ⊂ R n is a bounded and measurable set and 0 < γ < 1, then for every measurable function f : R n → [−∞, ∞], with |f (x)| < ∞ for almost every x ∈ R n , we have (1.2) lim r→0 m γ f (B(x, r)) = f (x) for almost every x ∈ R n , see [4], [28]. Let L 0 (R n ) denote the set of all measurable functions f : R n → [−∞, ∞] such that |f (x)| < ∞ for almost every x ∈ R n .…”

“…where f : R n → [−∞, ∞] is a measurable function with |f (x)| < ∞ for almost every x ∈ R n . Medians and related maximal functions have turned out to be useful in harmonic analysis and function spaces, see [3], [4], [5], [8], [9], [11], [13], [14], [15], [16], [17], [18], [24], [26], [27], [28], [30], [31], [33]. The main advantage of a median over an integral average is that it applies also when the function is not necessarily locally integrable.…”