Sharp L log α L inequalities for conjugate functions

Matts Essén[1]; Daniel F. Shea[2]; Charles S. Stanton[3]

  • [1] University of Uppsala, Department of Mathematics, Box 480, 751 06 Uppsala (Suède)
  • [2] University of Wisconsin, Department of Mathematics, Madison WI 53706-1313 (USA)
  • [3] California State University at San Bernardino, Department of Mathematics, San Bernardino CA 92407 (USA)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 2, page 623-659
  • ISSN: 0373-0956

Abstract

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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 ≤ α < ∞ . 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 proof of the M. Riesz’s inequality for functions in L p , ( 1 < p < 2 ) , also with sharp constant. All these inequalities are special cases of a general sharp inequality for conjugate functions (cf. Theorem 6).

How to cite

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Essén, Matts, Shea, Daniel F., and Stanton, Charles S.. "Sharp $L\;{\rm log}^\alpha L$ inequalities for conjugate functions." Annales de l’institut Fourier 52.2 (2002): 623-659. <http://eudml.org/doc/115989>.

@article{Essén2002,
abstract = {We give a method for constructing functions $\phi $ and $\psi $ for which $H(x,y) = \phi (x)-\psi (y)$ 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 $L\log ^\{\alpha \}L$, for $-1\le \alpha &lt;\infty $. In particular, the case $\alpha = 1$ 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 proof of the M. Riesz’s inequality for functions in $L^p$, $(1&lt;p&lt;2)$, also with sharp constant. All these inequalities are special cases of a general sharp inequality for conjugate functions (cf. Theorem 6).},
affiliation = {University of Uppsala, Department of Mathematics, Box 480, 751 06 Uppsala (Suède); University of Wisconsin, Department of Mathematics, Madison WI 53706-1313 (USA); California State University at San Bernardino, Department of Mathematics, San Bernardino CA 92407 (USA)},
author = {Essén, Matts, Shea, Daniel F., Stanton, Charles S.},
journal = {Annales de l’institut Fourier},
keywords = {conjugate functions; norm estimates; minimal thinness; inequality},
language = {eng},
number = {2},
pages = {623-659},
publisher = {Association des Annales de l'Institut Fourier},
title = {Sharp $L\;\{\rm log\}^\alpha L$ inequalities for conjugate functions},
url = {http://eudml.org/doc/115989},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Essén, Matts
AU - Shea, Daniel F.
AU - Stanton, Charles S.
TI - Sharp $L\;{\rm log}^\alpha L$ inequalities for conjugate functions
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 2
SP - 623
EP - 659
AB - We give a method for constructing functions $\phi $ and $\psi $ for which $H(x,y) = \phi (x)-\psi (y)$ 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 $L\log ^{\alpha }L$, for $-1\le \alpha &lt;\infty $. In particular, the case $\alpha = 1$ 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 proof of the M. Riesz’s inequality for functions in $L^p$, $(1&lt;p&lt;2)$, also with sharp constant. All these inequalities are special cases of a general sharp inequality for conjugate functions (cf. Theorem 6).
LA - eng
KW - conjugate functions; norm estimates; minimal thinness; inequality
UR - http://eudml.org/doc/115989
ER -

References

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