On the H p -convergence of Dirichlet series

Transcription

1 On the H p -convergence of Dirichlet series Antonio Pérez Hernández joint work with Andreas Defant ICMAT Bilbao March 2018

2 Dirichlet series a n n s n N

3 Dirichlet series vs Power series a n n s n N c k z k k 0 z 0 D(0, z 0 ) := { z < z 0 }

4 Dirichlet series vs Power series a n n s n N c k z k k 0 s 0 z 0 C s0 := { Re s > Re s 0 } D(0, z 0 ) := { z < z 0 }

5 Radius of convergence f = k 0 c kz k R(f ) : = sup {r > 0: f converges pointwise on D(0, r)} = sup {r > 0: f converges uniformly on D(0, r)} = sup {r > 0: f converges absolutely on D(0, r)}

6 Abscissas of convergence D = n a nn s σc σ c (D) := inf {σ : D converges pointwise on C σ }

7 Abscissas of convergence D = n a nn s σc σu σ c (D) := inf {σ : D converges pointwise on C σ } σ u (D) := inf {σ : D converges uniformly on C σ }

8 Abscissas of convergence D = n a nn s σc σu σa σ c (D) := inf {σ : D converges pointwise on C σ } σ u (D) := inf {σ : D converges uniformly on C σ } σ a (D) := inf {σ : D converges absolutely on C σ }

9 Abscissas of convergence 1 D = n a nn s 1 σc σu σa? σ c (D) := inf {σ : D converges pointwise on C σ } σ u (D) := inf {σ : D converges uniformly on C σ } σ a (D) := inf {σ : D converges absolutely on C σ }

10 Abscissas of convergence 1 D = n a nn s 1 σc σu σa? Theorem (Bohr 1914, Bohnenblust, Hille, 1931) sup {σ a (D) σ u (D)} = 1 D 2

11 Absolute vs uniform convergence: a deeper insight For each x 1, let S(x) be the smallest constant satisfying a n S(x) sup a n (an Re (s)>0 n s ) in C

12 Absolute vs uniform convergence: a deeper insight For each x 1, let S(x) be the smallest constant satisfying a n S(x) sup a n (an Re (s)>0 n s ) in C Queffélec (1995); Konyagin, Queffélec (2002); Balasubramanian, Calado, Queffélec (2006); de la Bretèche (2007); Maurizi, Queffélec (2010); Defant, Frerick, Ortega-Cerdá, Ounaïes, Seip (2012); Brevig (2014) Theorem (Defant, Frerick, Ortega-Cerdá, Ounaïes, Seip, 2012) S(x) = [( x exp 1 ) log ) ] + o(1) x log log x 2

13 H p -norm Let D(s) = n a nn s be a Dirichlet polynomial D Hp := ( 1 T lim D(it) ) 1 p p dt T + 2T T (1 p < + ) it it The notation H p refers to the so-called Hardy spaces of Dirichlet series introduced by Hedenmalm, Lindqvist, Seip (1997) and Bayart (2002).

14 H p -norm Let D(s) = n a nn s be a Dirichlet polynomial D Hp := ( 1 T ) 1 p lim D(it) p dt T + 2T T (1 p < + ) σ Hp σ u σ a

15 H p -norm Let D(s) = n a nn s be a Dirichlet polynomial D Hp := ( 1 T ) 1 p lim D(it) p dt T + 2T T (1 p < + ) σ Hp = σ Hq p, q (Bayart, 2002) σ u σ a

16 Let 1 p < q < and x 1. Define (q, p, x) as the best (i.e. smallest) constant satisfying a n Hq n s (q, p, x) a n Hp n s for every (a n ) in C. Problem Estimate (q, p, x) asymptotically on x +.

17 Particular case: q = 2, p = 1 a n H2 n s???? a n H1 n s

18 Particular case: q = 2, p = 1 a ( n H2 ) 1 n s = a n 2 2 (Carlson, 1922) a n H2 n s???? a n H1 n s

19 Particular case: q = 2, p = 1 a ( n H2 ) 1 n s = a n 2 2 (Carlson, 1922) [ a n 2 ] 1 2 d(n) a n H1 n s (Helson, 2006) a n H2 n s???? a n H1 n s

20 Particular case: q = 2, p = 1 a ( n H2 ) 1 n s = a n 2 2 (Carlson, 1922) [ a n 2 ] 1 2 d(n) a n H1 n s (Helson, 2006) a n H2 n s max d(n) a n H1 n s

21 Particular case: q = 2, p = 1 a ( n H2 ) 1 n s = a n 2 2 (Carlson, 1922) [ a n 2 ] 1 2 d(n) a n H1 n s (Helson, 2006) [ log x ( ) ] max d(n) exp log 2 + o(1) (Wigert, 1907) log log x a [ n H2 log x n s exp log log x ( ) ] log 2 + o(1) a n H1 n s

22 Bohr s point of view {Dirichlet series} a n n s n Prime factorization: n = p α := p α 1 1 pα

23 Bohr s point of view {Dirichlet series} α a n n s n a p α(p s 1 )α 1 (p s 2 )α 2... Prime factorization: n = p α := p α 1 1 pα

24 Bohr s point of view {Dirichlet series} α a n n s n a p α(p s 1 )α 1 (p s 2 )α 2... a p α=c α {Power series} α c α z α α c α z α 1 1 zα Prime factorization: n = p α := p α 1 1 pα

25 Bohr s point of view {Dirichlet series} α a n n s n a p α(p s 1 )α 1 (p s 2 )α 2... a p α=c α {Power series} α c α z α α c α z α 1 1 zα Prime factorization: n = p α := p α 1 1 pα For every 1 p < + a n n s Hp = p α x a p αz α Lp(T N )

26 Reformulation using Bohr s correspondence Let 1 p < q < and x 1. Then, (q, p, x) is the smallest constant satisfying c α z α Lq(T (q, p, x) c α z α N ) Lp(T N ) p α x for every (c α ) α in C. p α x

27 Comparing L p and L q norms Theorem (Bayart, 2002) For every m N and every finitely supported family (c α ) α in C [ ] c α z α Lq q exp m log c α z α Lp p α =m where α := α 1 + α α =m

28 Comparing L p and L q norms Theorem (Bayart, non-homogenous version) For every m N and every finitely supported family (c α ) α in C [ ] c α z α Lq q exp m log c α z α Lp p α m where α := α 1 + α α m

29 Comparing L p and L q norms Theorem (Bayart, non-homogenous version) For every m N and every finitely supported family (c α ) α in C [ ] c α z α Lq q exp m log c α z α Lp p α m where α := α 1 + α α m Consequence: a ( n Hq log x n s exp log 2 log q ) p a n n s Hp Not good enough!! For p = 1, q = 2 we got a better estimation!

30 Decomposition method of Konyagin-Queffélec Fixed 2 y x consider L(x, y) := {n x : p n p > y} S(x, y) := {n x : p n p y}

31 Decomposition method of Konyagin-Queffélec Fixed 2 y x consider L(x, y) := {n x : p n p > y} S(x, y) := {n x : p n p y} Each n x is uniquely written as n = j k for some j S(x, y), k L(x, y)

32 Decomposition method of Konyagin-Queffélec Fixed 2 y x consider L(x, y) := {n x : p n p > y} S(x, y) := {n x : p n p y} Each n x is uniquely written as n = j k for some j S(x, y), k L(x, y) so that a n n s = j S(x,y) ( k L(x,y) a jk k s ) 1 j s

33 Combining Bayart s result and Konyagin-Queffélec decomposition, we get that for every 2 y x ( log x q ) (q, p, x) exp log y log S(x, y) p

34 Combining Bayart s result and Konyagin-Queffélec decomposition, we get that for every 2 y x ( log x q ) (q, p, x) exp log y log S(x, y) p Theorem (de Bruijn, 1966) We have, uniformly for 2 y x, log S(x, y) = Z ( ( O log y + 1 )) log log 2x where Z := log x ( log y log 1 + y ) + y ( log x log y log 1 + log x y ).

35 Combining Bayart s result and Konyagin-Queffélec decomposition, we get that for every 2 y x ( log x q ) (q, p, x) exp log y log S(x, y) p The choice leads to ( (log log x) 2 ) y := exp log log x + log log log x [ log x ( q )] (q, p, x) exp log + o(1) log log x p

36 Lower estimation There is a family of Dirichlet polynomials D x (s) = a n,x n s, x 1 satisfying D x Hq D x Hp [ log x ( q )] exp log + o(1) log log x p

37 We consider polynomials of the type: D k,m (s) = ( 1 k k j=1 p s j ) m Q k,m (z) = ( 1 k k ) m z j j=1

38 We consider polynomials of the type: D k,m (s) = ( 1 k k j=1 p s j ) m Q k,m (z) = ( 1 k k ) m z j j=1 Theorem If k, m N satisfy that 1 < [ ] mp < [ ] mq < k, then Q k,m Lq(T N ) Q k,m Lp(T N ) ( C(p, q) m 1 2q 1 2p q ) m/2e 4m2 k p

39 References F. Bayart Opérateurs de composition sur des espaces de séries de Dirichlet et problèmes d hypercyclicité simultanée, PhD thesis, L université des sciences et technologies de Lille, A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounaïes and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Annals of mathematics, (2011). A. Defant, M. Masty lo. L p -Norms and Mahler s Measure of Polynomials on the n-dimensional torus, Constructive Approximation 44.1 (2016): A. Defant, A. Pérez Optimal comparison of the p-norm of Dirichlet polynomials, Israel Journal of Mathematics, Volume 221, Issue 2, pp (2017). H. Hedenmalm, P. Lindqvist, K. Seip, Hilbert space of Dirichlet series and systems of dilated functions in L 2 (0, 1), Duke Math. J., 86, no.1, 1 37 (1997). H. Helson, Hankel forms and sums of random variables, Studia Math., 176(1) (2006). S.V. Konyagin, H. Queffélec, The Translation 1/2 in the Theory of Dirichlet Series, Real Analysis Exchange, Volume 27, Issue 1, pp (2001). H. Queffélec, M. Queffélec,Diophantine approximation and Dirichlet series, Hindustan Book Agency (2013). G. Tenenbaum, Introduction to analytic and probabilistic number theory, Vol. 46, Cambridge University Press (1995).