New inequalities for the jacobian

L. Greco; T. Iwaniec

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Absolute values of BMOA functions.

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The paper contains a complete characterization of the moduli of BMOA functions. These are described explicitly by a certain Muckenhoupt-type condition involving Poisson integrals. As a consequence, it is shown that an outer function with BMO modulus need not belong to BMOA. Some related results are obtained for the Bloch space.

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Sharp $L \log^{α} L$ inequalities for conjugate functions

Matts Essén, Daniel F. Shea, Charles S. Stanton (2002)

Annales de l’institut Fourier

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We give a method for constructing functions $φ$ and $ψ$ for which $H (x, y) = φ (x) - ψ (y)$ has a specified subharmonic minorant $h (x, y)$ . By a theorem of B. Cole, this procedure establishes integral mean inequalities for conjugate functions. We apply this method to deduce sharp inequalities for conjugates of functions in the class $L {log}^{α} L$ , for $- 1 \leq α < \infty$ . In particular, the case $α = 1$ yields an improvement of Pichorides’ form of Zygmund’s classical inequality for the conjugates of functions in $L log L$ . We also apply the method to produce a new...