Polyanalytic Hardy decomposition of higher order Lipschitz functions

Higher order Hardy inequalities

This note deals with the inequality (∫ b a |u(x)|w0(x)dx )1/q ≤ C (∫ b a |u(x)|wk(x)dx )1/p , (1) more precisely, with conditions on the parameters p > 1, q > 0 and on the weight functions w0, wk (measurable and positive almost everywhere) which ensure that (1) holds for all functions u from a certain class K with a constant C > 0 independent of u. Here −∞ ≤ a < b ≤ ∞ and k ∈ N and we will cons...

Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space

We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can be decomposed f into a finite number of BiLipschitz functions f |Fi so that the k-Hausdorff content of f([0, 1] r ∪Fi) is small. We thus generalize a theorem of P. Jones [Jon88] from the setting of R to the setting of a general metric spa...

On a Higher-order Hardy Inequality

The Hardy inequality ∫ Ω |u(x)|pd(x)−p dx c ∫ Ω |∇u(x)|p dx with d(x) = dist(x, ∂Ω) holds for u ∈ C∞ 0 (Ω) if Ω ⊂ n is an open set with a sufficiently smooth boundary and if 1 < p < ∞. P.Haj lasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend...

Higher Order Dynamic Mode Decomposition

Double Sine Series and Higher Order Lipschitz Classes of Functions (communicated by Hüseyin Bor)

Let ω(h, k) be a modulus of continuity, that is, ω(h, k) is a continuous function on the square [0, 2π] × [0, 2π], nondecreasing in each variable, and possessing the following properties: ω(0, 0) = 0, ω(t1 + t2, t3) ≤ ω(t1, t3) + ω(t2, t3), ω(t1, t2 + t3) ≤ ω(t1, t2) + ω(t1, t3). Yu ([3]) introduced the following classes of functions: HH := {f(x, y) : ‖f(x, y)− f(x+ h, y)− f(x, y + k) + f(x+ h,...