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Symmetric and Zygmund measures in several variables

Evgueni Doubtsov, Artur Nicolau (2002)

Annales de l’institut Fourier

Let ω : ( 0 , ∞ ) → ( 0 , ∞ ) be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure μ ∈ ℝ n is called ω -Zygmund if there exists a positive constant C such that | μ ( Q + ) - μ ( Q - ) | ≤ C ω ( ℓ ( Q + ) ) | Q + | for any pair Q + , Q - ⊂ ℝ n of adjacent cubes of the same size. Similarly, μ is called an ω - symmetric measure if there exists a positive constant C such that | μ ( Q + ) / μ ( Q - ) - 1 | ≤ C ω ( ℓ ( Q + ) ) for any pair Q + , Q - ⊂ ℝ n of adjacent cubes of the same size, ℓ ( Q + ) = ℓ ( Q - ) < 1 . We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic condition...

Vitali Lemma approach to differentiation on a time scale

Chuan Jen Chyan, Andrzej Fryszkowski (2004)

Studia Mathematica

A new approach to differentiation on a time scale is presented. We give a suitable generalization of the Vitali Lemma and apply it to prove that every increasing function f: → ℝ has a right derivative f₊’(x) for μ Δ -almost all x ∈ . Moreover, ∫ [ a , b ) f ₊ ' ( x ) d μ Δ ≤ f ( b ) - f ( a ) .

Currently displaying 81 – 100 of 106