Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

被引:17
|
作者
Bayart, Frederic [1 ]
Defant, Andreas [2 ]
Frerick, Leonhard [3 ]
Maestre, Manuel [4 ]
Sevilla-Peris, Pablo [5 ]
机构
[1] Univ Blaise Pascal Campus Cezeaux, Lab Math, F-63177 Clermont Ferrand, France
[2] Carl von Ossietzky Univ Oldenburg, Inst Math, D-26111 Oldenburg, Germany
[3] Univ Trier, Fachbereich Math 4, D-54294 Trier, Germany
[4] Univ Valencia, Dept Anal Matemat, Fac Matemat, E-46100 Burjassot, Spain
[5] Univ Politecn Valencia, IUMPA, E-46022 Valencia, Spain
关键词
HARDY-SPACES; CONVERGENCE; INEQUALITY; POLYDISK;
D O I
10.1007/s00208-016-1511-1
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Let H-infinity be the set of all ordinary Dirichlet series D = Sigma(n) a(n)(n-1) ann-s representing bounded holomorphic functions on the right half plane. A completely multiplicative sequence (b(n)) of complex numbers is said to be an l(1)-multiplier for H-infinity whenever Sigma(n vertical bar)a(n)b(n vertical bar) < infinity for every D is an element of H-infinity. We study the problem of describing such sequences (b(n)) in terms of the asymptotic decay of the subsequence (b(pj)), where p(j) denotes the j th prime number. Given a completely multiplicative sequence b = (b(n)) we prove (among other results): b is an l(1)-multiplier for H-infinity provided vertical bar b(pj)vertical bar < 1 for all j and (lim(n)) over bar 1/log(n) Sigma(n)(j=1) b(p j)*(2) < 1, and conversely, if b is an l(1)-multiplier for H-infinity, then vertical bar b(pj)vertical bar < 1 for all j and (lim(n)) over bar 1/log(n) Sigma(n)(j=1) b(p j)*(2) <= 1 (here b* stands for the decreasing rearrangement of b). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences z in the infinite dimensional polydisk D-infinity (the open unit ball of l(infinity)) for which every bounded and holomorphic function f on D-infinity has an absolutely convergent monomial series expansion Sigma(alpha) partial derivative alpha f (0)/alpha! z alpha. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus T-infinity.
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页码:837 / 876
页数:40
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