Hilbert spaces of Dirichlet series Joaquim Ortega-Cerd`a Puerto de la Cruz, March 5,6,8,9, 2012 Overview I Ordinary Dirichlet series. I Regions of convergence. I Definition and reproducing kernel. I Local behaviour and the discrete Hilbert transform. I Almost periodicity. I Pointwise convergence. I Bohr correspondence. p I L variants. I Multipliers and applications. I Composition operators. I Other weights. I Estimates of coefficients. I Open problems. Ordinary Dirichlet series We will consider series of the form 1 X −s f (s) = ann ; n=1 where s = σ + it 2 C is a complex variable. To make things interesting we assume that the series converges at least in a point s = s0. Actually the region of convergence will be a half plane by the Theorem (Abel-Dedekind-Dirichlet) P If the series n an has bounded partial sums and the sequence bn P is of bounded variation and limn bn = 0 then the series n anbn converges. Replacing f (s) by g(s) = f (s + s0) we may assume s0 = 0 without loss of generality. Since jn−s − (n + 1)−s j = O(n−1−Re(s)), then −s + n is of bounded variation in C0 = fz 2 C; <z > 0g. Regions of convergence Actually the convergence is uniform in a cone with vertex s0 as in the picture. Thus f (s) is an holomorphic function in the half plane to the right of s0. In contrast to the power series the regions of convergence differ when we consider pointwise convergence, uniform convergence or absolute convergence. Regions of convergence Given f (s) we can define (at least) three abcissa σC ; σU ; σA. σ is the smallest σ such that f (s) is convergent in + . I C CσC I σU is the smallest σ such that f (s) converges uniformly in + for any " > 0. CσU +" I σA is the smallest σ such that f (s) converges absolutely in + . CσA Clearly σC ≤ σU ≤ σA. It is also easy that σA − σC ≤ 1 (it can be anything in between 0 and 1). Remark The function f (s) may continue analytically in a region bigger than + . CσC Keep in mind the following example: X g(s) = (−1)nn−s : g(s) = 2ζ(s)(2−s − 1=2) n≥1 This is an entire function such that σC = 0 and σA = σU = 1. Regions of convergence It remains to clarify what is the relation between σU and σA. Bohr studied another abscissa σB or abscissa of boundedness I σB is the smallest σ such that f (s) converges at some point s and it is bounded in + for any " > 0. CσB +" And he proved Theorem (Bohr) σB = σU . In particular this means that whenever we have an analytic + function f that coincides with a Dirichlet series in a half plane Ca + and it is holomorphic and bounded up to Cb with b < a, then the + Dirichlet series converges to f up to Cb . Regions of convergence Proof (Maurizi, Queff´elec). We only need to prove that σU ≤ σB . Suppose that P −s + f (s) = ann defines a bounded analytic function in C0 and let kf k1 = sup + jf (s)j. Denote by C0 PN −s DN (s) = DN [f ](s) = n=1 ann . Then Proposition (Balasubramanian, Calado, Queff´elec) kDN k1 ≤ C log Nkf k1: Now summing by parts: N N−1 X −(s+") X −" −" −" ann = Dn(s)(n − (n + 1) ) + N DN (s): n=1 n=1 C" log n and the new series is dominated by n"+1 kf k1. Regions of Convergence Theorem σA − σB ≤ 1=2 (Bohr) and 1=2 is sharp (Bohnenblust-Hille). We will prove a refined version. Theorem (DFOOS) The series 1 X − 1 n p o janj n 2 exp c log n log log n n=1 P −s + is convergentp for every n ann bounded Dirichletp series in C0 if c < 1= 2 and it may be divergent if c > 1= 2. Definition and reproducing kernel Definition The space H2 will be the Hilbert space of analytic functions defined by 1 X −s f (s) = ann n=1 2 P 2 with kf k := janj < 1. The functions in the space converge in σ > 1=2. In fact by Cauchy-Schwartz any function in H satisfies σA ≤ 1=2. 2 + There are function on H that are only defined in C1=2. The values f (s) determine the coefficients and of course the norm, but it will be interesting to express the norm directly in terms of the growth of f (s). Carlson's theorem P1 −s 2 Let f (s) = n=1 ann 2 H , then for any σ > σU we have Z T X 2 −2σ 1 2 janj n = lim jf (σ + it)j dt: T !1 T n 0 In particular H2 is the closure of the Dirichlet polynomials PN −s p(s) = n=1 ann under the norm 1 Z T kpk2 = lim jp(it)j2 dt: T !1 T 0 This way of computing the norm is not valid for all f 2 H2 because in general f is not defined in the imaginary axis. The reproducing kernel Definition Since n−s is an orthonormal basis and point evaluation is bounded for σ > 1=2, the function 1 X n−s n−w¯ = ζ(s +w ¯ ) n=1 2 + is the reproducing kernel for H , i.e, for any w 2 C1=2 f (w) = hf (s); ζ(s +w ¯ )i p p and the norm of the pointwisep evaluation is k(s; s) = ζ(2σ) which blows up as 1= 2σ as we approach the boundary of C1=2. Local behaviour Theorem For any f 2 H the following holds: Z T 2 2 jf (σ + it)j dt ≤ C(T )kf kH 8σ ≥ 1=2: 0 PN −s Proof: Let D(s) = n=1 ann . The statement follows from: Z T N N 2 X 2 X 2 jD(it)j dt = T janj + O( njanj ); 0 n=1 n=1 which trivially implies Z T N 2 N X janj X n jD(σ + it)j2 dt = T + O( ja j2): n2σ n2σ n 0 n=1 n=1 Local behaviour R T 2 PN 2 PN 2 0 jD(it)j dt = T n=1 janj + O( n=1 njanj ); Z T N Z T 2 X i(log n−log m)t jD(it)j dt = an¯am e dt = 0 n;m=1 0 N i log nT i log mT X X an¯am X ane ¯ame = T ja j2 + i − i : n log n − log m log n − log m n n6=m n6=m The last two terms are similar and can be controlled if we prove: 2 2 2 X xmyn X jxmj X jynj ≤ C ; λm − λn dm dn m6=n m n where δn = minm6=n jλm − λnj. In our case λn = log n and dn = 1=n Local behaviour and the discrete Hilbert transform In order to prove: 2 2 2 X xmyn X jxmj X jynj ≤ C ; λm − λn dm dn m6=n m n P consider the measure µ = n dnδλn . The points λn 2 R thus the curvature of µ = 0 and by the definition of dn it has linear growth (it is doubling). R f (w) 2 2 Thus C(f )(z) = z−w dµ(w) is bounded from L (dµ) to L (dµ). Therefore there is a constant C > 0 such that for all an; bm we have 2 X amdm ¯ X 2 X 2 bndn ≤ C jamj dm jbnj dn; λm − λn m6=n m n Finally taking am = xm=dm and bn =y ¯n=dn we get the result. Immediate consequences This gives several immediate consequences: Proposition If f 2 H2 then 2 + I g(s) = f (s)=s 2 H (C1=2). I The zeros of f satisfy the Blaschke condition on the halfplane + C1=2. I For almost every t 2 R there exists the non-tangential limit of f at the point 1=2 + it And some not so immediate Proposition A compactly supported measure µ is a Carleson measure for H2 R 2 2 2 (i.e. + jf j dµ ≤ Ckf k for all f 2 H ) if and only if µ is a C1=2 2 + Carleson measure for the Hardy space H (C1=2). Almost periodicity + Let f be holomorphic in Ca . Let " > 0. A number τ is an translation number of f if sup jf (s + iτ) − f (s)j ≤ ": + s2Ca Definition + A function f is uniformly almost periodic in Ca if, for every " > 0, there is an M > 0 such that every interval of length M contains at least one translation number of f . One can prove that: Theorem Suppose that f (s) is a Dirichlet series and a > σU , then f is + uniformly almost periodic in Ca . Almost periodicity This is a consequence of Kronecker's theorem that says Theorem Let p1 = 2; p2 = 3; p3 = 5;::: be the primes, then n f(t log p1; t log p2;:::; t log pn) ⊂ T ; t 2 Rg n is dense in T (the flow is equidistributed). Thus for any " > 0 we can find a sequence Tn ! 1 relatively dense such that d(Tn log pj ; 2πZ) < " for all j = 1;:::; n and therefore jdn(s − iTn) − dn(s)j ≤ C" for a given Dirichlet polynomial of degree n. By uniform approximation of the function f by Dirichlet + polynomials in Ca we get the desired result. In particular if f 2 H2 and f (a) = 0 <(a) > 1=2, then in any strip <(a) − " < <f < <(a) + " we can find infinitely many zeros.