For a Dirichlet series symbol g(s)=∑n≥1bnn−s, the associated Volterra operator Tg acting on a Dirichlet series f(s)=∑n≥1ann−s is defined by the integral



f↦−∫+∞sf(w)g′(w)dw.

We show that Tg is a bounded operator on the Hardy space Hp of Dirichlet series with 00 whenever Tg is bounded. In particular, such g belong to Hp for every p<∞. We relate the boundedness of Tg to several other BMO-type spaces: BMOA in half-planes, the dual of H1, and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for m-homogeneous symbols as well as for general symbols. Finally, we consider the action of Tg on reproducing kernels for appropriate sequences of subspaces of H2. Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbols g

Brevig, Ole Fredrik

Perfekt, Karl-Mikael

Seip, Kristian

arXiv

For a Dirichlet series symbol g.s/ D P n 1 bnn s , the associated Volterra operator Tg acting on a Dirichlet series f .s/ D P n 1 ann s is defined by the integral f 7! Z C1 s f .w/g0 .w/ dw: We show that Tg is a bounded operator on the Hardy space Hp of Dirichlet series with 0  0 whenever Tg is bounded. In particular, such g belong to Hp for every p < 1. We relate the boundedness of Tg to several other BMO-type spaces: BMOA in half-planes, the dual of H1 , and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for m-homogeneous symbols as well as for general symbols. Finally, we consider the action of Tg on reproducing kernels for appropriate sequences of subspaces of H2 . Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbolspublishedVersion© De Gruyter 2019

NTNU Open (Norwegian University of Science and Technology)

English

Volterra operators on Hardy spaces of Dirichlet series

For a Dirichlet series symbol g.s/ D P n 1 bnn s , the associated Volterra operator Tg acting on a Dirichlet series f .s/ D P n 1 ann s is defined by the integral f 7! Z C1 s f .w/g0 .w/ dw: We show that Tg is a bounded operator on the Hardy space Hp of Dirichlet series with 0  0 whenever Tg is bounded. In particular, such g belong to Hp for every p < 1. We relate the boundedness of Tg to several other BMO-type spaces: BMOA in half-planes, the dual of H1 , and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for m-homogeneous symbols as well as for general symbols. Finally, we consider the action of Tg on reproducing kernels for appropriate sequences of subspaces of H2 . Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbol

NORA - Norwegian Open Research Archives

Abstract
               For a Dirichlet series symbol 
                     
                        
                           
                              
                                 g
                                 ⁢
                                 
                                    (
                                    s
                                    )
                                 
                              
                              =
                              
                                 
                                    ∑
                                    
                                       n
                                       ≥
                                       1
                                    
                                 
                                 
                                    
                                       b
                                       n
                                    
                                    ⁢
                                    
                                       n
                                       
                                          -
                                          s
                                       
                                    
                                 
                              
                           
                        
                        
                        {g(s)=\sum_{n\geq 1}b_{n}n^{-s}}
                     
                  , the associated Volterra operator 
                     
                        
                           
                              𝐓
                              g
                           
                        
                        
                        {\mathbf{T}_{g}}
                     
                   acting on a Dirichlet series 
                     
                        
                           
                              
                                 f
                                 ⁢
                                 
                                    (
                                    s
                                    )
                                 
                              
                              =
                              
                                 
                                    ∑
                                    
                                       n
                                       ≥
                                       1
                                    
                                 
                                 
                                    
                                       a
                                       n
                                    
                                    ⁢
                                    
                                       n
                                       
                                          -
                                          s
                                       
                                    
                                 
                              
                           
                        
                        
                        {f(s)=\sum_{n\geq 1}a_{n}n^{-s}}
                     
                   is defined by the integral
               
                  
                     
                        
                           
                              
                                 f
                                 ↦
                                 
                                    -
                                    
                                       
                                          ∫
                                          s
                                          
                                             +
                                             ∞
                                          
                                       
                                       
                                          f
                                          ⁢
                                          
                                             (
                                             w
                                             )
                                          
                                          ⁢
                                          
                                             g
                                             ′
                                          
                                          ⁢
                                          
                                             (
                                             w
                                             )
                                          
                                          ⁢
                                          
                                             𝑑
                                             w
                                          
                                       
                                    
                                 
                              
                              .
                           
                        
                        
                        {f\mapsto-\int_{s}^{+\infty}f(w)g^{\prime}(w)\,dw}.
                     
                  
               
               We show that 
                     
                        
                           
                              𝐓
                              g
                           
                        
                        
                        {\mathbf{T}_{g}}
                     
                   is a bounded operator on the Hardy space 
                     
                        
                           
                              ℋ
                              p
                           
                        
                        
                        {\mathcal{H}^{p}}
                     
                   of Dirichlet series with 
                     
                        
                           
                              0
                              &lt;
                              p
                              &lt;
                              ∞
                           
                        
                        
                        {0&lt;p&lt;\infty}
                     
                   if and only if the symbol g satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of 
                     
                        
                           
                              BMOA
                              ⁡
                              
                                 (
                                 𝔻
                                 )
                              
                           
                        
                        
                        {{\operatorname{BMOA}}(\mathbb{D})}
                     
                  . A further analogy with classical 
                     
                        
                           BMO
                        
                        
                        {{\operatorname{BMO}}}
                     
                   is that 
                     
                        
                           
                              exp
                              ⁡
                              
                                 (
                                 
                                    c
                                    ⁢
                                    
                                       |
                                       g
                                       |
                                    
                                 
                                 )
                              
                           
                        
                        
                        {\exp(c|g|)}
                     
                   is integrable (on the infinite polytorus) for some 
                     
                        
                           
                              c
                              &gt;
                              0
                           
                        
                        
                        {c&gt;0}
                     
                   whenever 
                     
                        
                           
                              𝐓
                              g
                           
                        
                        
                        {\mathbf{T}_{g}}
                     
                   is bounded. In particular, such g belong to 
                     
                        
                           
                              ℋ
                              p
                           
                        
                        
                        {\mathcal{H}^{p}}
                     
                   for every 
                     
                        
                           
                              p
                              &lt;
                              ∞
                           
                        
                        
                        {p&lt;\infty}
                     
                  . We relate the boundedness of 
                     
                        
                           
                              𝐓
                              g
                           
                        
                        
                        {\mathbf{T}_{g}}
                     
                   to several other 
                     
                        
                           BMO
                        
                        
                        {{\operatorname{BMO}}}
                     
                  -type spaces: 
                     
                        
                           BMOA
                        
                        
                        {{\operatorname{BMOA}}}
                     
                   in half-planes, the dual of 
                     
                        
                           
                              ℋ
                              1
                           
                        
                        
                        {\mathcal{H}^{1}}
                     
                  , and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for m-homogeneous symbols as well as for general symbols. Finally, we consider the action of 
                     
                        
                           
                              𝐓
                              g
                           
                        
                        
                        {\mathbf{T}_{g}}
                     
                   on reproducing kernels for appropriate sequences of subspaces of 
                     
                        
                           
                              ℋ
                              2
                           
                        
                        
                        {\mathcal{H}^{2}}
                     
                  . Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbols g.</jats:p

Volterra operators on Hardy spaces of Dirichlet series

Abstract

Similar works

Full text

Available Versions

NTNU Open (Norwegian University of Science and Technology)

NORA - Norwegian Open Research Archives

Crossref

Central Archive at the University of Reading