Construction of a certain superharmonic majorant

Koosis, Paul. "Construction of a certain superharmonic majorant." Annales de l'institut Fourier 44.3 (1994): 729-766. <http://eudml.org/doc/75080>.

@article{Koosis1994,
abstract = {Given a function $f(t)\ge 0$ on $\{\Bbb R\}$ with $\int ^\infty _\{-\infty \} (f(t)/(1+t^2))dt&lt; \infty $ and $\vert f(t)-f(t^\{\prime \})\vert \le l\vert t-t^\{\prime \}\vert $, a procedure is exhibited for obtaining on $\{\Bbb C\}$ a (finite) superharmonic majorant of the function\begin\{\} F(z):\{1\over \pi \}\int ^\infty \_\{-\infty \}\{\vert \{\frak J\}z\vert \over \vert z-t\vert ^2\} f(t)dt-Al\vert \{\frak J\}z\vert ,\end\{\}where $A$ is a certain (large) absolute constant. This leads to fairly constructive proofs of the two main multiplier theorems of Beurling and Malliavin. The principal tool used is a version of the following lemma going back almost surely to Beurling: suppose that $f(t)$, positive and bounded away from 0 on $\{\Bbb R\}$, is such that $\int ^\infty _\{-\infty \}(f(t)/(1+t^2)dt&lt; \infty $ and denote, for any constant $\alpha &gt;0$ and each $x\in \{\Bbb R\}$, the unique value $&gt;0$ of $y$ making\begin\{\}\{1\over \pi \}\int ^\infty \_\{-\infty \}\{yf(t)\over (x-t)^2+y^2\}dt=\alpha y\end\{\}by $Y_\alpha (x)$; then $\int ^\infty _\{-\infty \}(Y_\alpha (x)/(1+ x^2))dx&lt; \infty $.},
author = {Koosis, Paul},
journal = {Annales de l'institut Fourier},
keywords = {Poisson integrals; superharmonic functions; superharmonic majorants; multipliers; multiplier theorems of Beurling and Malliavin},
language = {eng},
number = {3},
pages = {729-766},
publisher = {Association des Annales de l'Institut Fourier},
title = {Construction of a certain superharmonic majorant},
url = {http://eudml.org/doc/75080},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Koosis, Paul
TI - Construction of a certain superharmonic majorant
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 3
SP - 729
EP - 766
AB - Given a function $f(t)\ge 0$ on ${\Bbb R}$ with $\int ^\infty _{-\infty } (f(t)/(1+t^2))dt&lt; \infty $ and $\vert f(t)-f(t^{\prime })\vert \le l\vert t-t^{\prime }\vert $, a procedure is exhibited for obtaining on ${\Bbb C}$ a (finite) superharmonic majorant of the function\begin{} F(z):{1\over \pi }\int ^\infty _{-\infty }{\vert {\frak J}z\vert \over \vert z-t\vert ^2} f(t)dt-Al\vert {\frak J}z\vert ,\end{}where $A$ is a certain (large) absolute constant. This leads to fairly constructive proofs of the two main multiplier theorems of Beurling and Malliavin. The principal tool used is a version of the following lemma going back almost surely to Beurling: suppose that $f(t)$, positive and bounded away from 0 on ${\Bbb R}$, is such that $\int ^\infty _{-\infty }(f(t)/(1+t^2)dt&lt; \infty $ and denote, for any constant $\alpha &gt;0$ and each $x\in {\Bbb R}$, the unique value $&gt;0$ of $y$ making\begin{}{1\over \pi }\int ^\infty _{-\infty }{yf(t)\over (x-t)^2+y^2}dt=\alpha y\end{}by $Y_\alpha (x)$; then $\int ^\infty _{-\infty }(Y_\alpha (x)/(1+ x^2))dx&lt; \infty $.
LA - eng
KW - Poisson integrals; superharmonic functions; superharmonic majorants; multipliers; multiplier theorems of Beurling and Malliavin
UR - http://eudml.org/doc/75080
ER -