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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 n is called ω -Zygmund if there exists a positive constant C such that | μ ( Q + ) - μ ( Q - ) | C ω ( ( Q + ) ) | Q + | for any pair Q + , Q - 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 ( ( Q + ) ) for any pair Q + , Q - n of adjacent cubes of the same size, ( Q + ) = ( Q - ) < 1 . We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic condition...

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