Aistleitner, Christoph, Berkes, István, and Seip, Kristian. "GCD sums from Poisson integrals and systems of dilated functions." Journal of the European Mathematical Society 017.6 (2015): 1517-1546. <http://eudml.org/doc/277338>.
@article{Aistleitner2015,
abstract = {Upper bounds for GCD sums of the form $\sum _\{k,\{\ell \}=1\}^N\frac\{(\mathrm \{gcd\}(n_k,n_\{\ell \}))^\{2\alpha \}\}\{(n_k n_\{\ell \})^\alpha \}$ are established, where $(n_k)_\{1 \le k \le N\}$ is any sequence of distinct positive integers and $0<\alpha \le 1$; the estimate for $\alpha =1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $\alpha =1/2$. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish a Carleson–Hunt-type inequality for systems of dilated functions of bounded variation or belonging to $\mathrm \{Lip\}_\{1/2\}$, a result that in turn settles two longstanding problems on the a.e. behavior of systems of dilated functions: the a.e. growth of sums of the form $\sum _\{k=1\}^N f(n_k x)$ and the a.e. convergence of $\sum _\{k=1\}^\infty c_k f(n_kx)$ when $f$ is $1$-periodic and of bounded variation or in $\mathrm \{Lip\}_\{1/2\}$.},
author = {Aistleitner, Christoph, Berkes, István, Seip, Kristian},
journal = {Journal of the European Mathematical Society},
keywords = {GCD sums and matrices; Carleson–Hunt inequality; Poisson integral; polydisc; spectral norm; convergence of series of dilated functions; GCD sums and matrices; Carleson-Hunt inequality; Poisson integral; polydisc; spectral norm; convergence of series of dilated functions},
language = {eng},
number = {6},
pages = {1517-1546},
publisher = {European Mathematical Society Publishing House},
title = {GCD sums from Poisson integrals and systems of dilated functions},
url = {http://eudml.org/doc/277338},
volume = {017},
year = {2015},
}
TY - JOUR
AU - Aistleitner, Christoph
AU - Berkes, István
AU - Seip, Kristian
TI - GCD sums from Poisson integrals and systems of dilated functions
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 6
SP - 1517
EP - 1546
AB - Upper bounds for GCD sums of the form $\sum _{k,{\ell }=1}^N\frac{(\mathrm {gcd}(n_k,n_{\ell }))^{2\alpha }}{(n_k n_{\ell })^\alpha }$ are established, where $(n_k)_{1 \le k \le N}$ is any sequence of distinct positive integers and $0<\alpha \le 1$; the estimate for $\alpha =1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $\alpha =1/2$. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish a Carleson–Hunt-type inequality for systems of dilated functions of bounded variation or belonging to $\mathrm {Lip}_{1/2}$, a result that in turn settles two longstanding problems on the a.e. behavior of systems of dilated functions: the a.e. growth of sums of the form $\sum _{k=1}^N f(n_k x)$ and the a.e. convergence of $\sum _{k=1}^\infty c_k f(n_kx)$ when $f$ is $1$-periodic and of bounded variation or in $\mathrm {Lip}_{1/2}$.
LA - eng
KW - GCD sums and matrices; Carleson–Hunt inequality; Poisson integral; polydisc; spectral norm; convergence of series of dilated functions; GCD sums and matrices; Carleson-Hunt inequality; Poisson integral; polydisc; spectral norm; convergence of series of dilated functions
UR - http://eudml.org/doc/277338
ER -
