Weighted bounded mean oscillation and the Hilbert transform
Weighted multidimensional inequalities for monotone functions
We discuss the characterization of the inequality (RN+ fq u)1/q C (RN+ fp v )1/p, 0<q, p <, for monotone functions and nonnegative weights and and . We prove a new multidimensional integral modular inequality for monotone functions. This inequality generalizes and unifies some recent results in one and several dimensions.
Weighted norm inequalities relating the and the area functions
Well behaved asymptotical convex functions
Whitney arcs and 1-critical arcs
A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that for every x ∈ γ; γ is 1-critical if there exists an f ∈ C¹(ℝⁿ) such that f’(x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in ℝ² each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This...
Whitney covers and quasi-isometry of -averaging domains.
Whitney type inequality, pointwise version
The main result of the paper estimates the asymptotic behavior of local polynomial approximation for functions at a point via the behavior of μ-differences, a generalization of the kth difference. The result is applied to prove several new and extend classical results on pointwise differentiability of functions including Marcinkiewicz-Zygmund’s and M. Weiss’ theorems. In particular, we present a solution of the problem posed in the 30s by Marcinkiewicz and Zygmund.
Why Means in Two Arguments are Special.