Carleson measures for the Dirichlet space on the polydisc P. - PowerPoint PPT Presentation

Carleson measures for the Dirichlet space on the polydisc P. Mozolyako with N. Arcozzi, K.-M. Perfekt and G. Sarfatti CAFT 2018 July 23, 2018

Dirichlet space D ( D ) We consider spaces of analytic functions in the unit disc a n z n = � � ˆ f ( n ) z n f ( z ) = n ≥ 0 n ≥ 0 with the norm � f � 2 � | ˆ f | 2 ( n )( n + 1) a , α = a ∈ R . n ≥ 0 For a = 0 we get the Hardy space, and a = 1 corresponds to the Dirichlet space, � � | f ( e it ) | 2 dt | f ′ ( z ) | 2 dA ( z ) + � f � 2 D ( D ) = 2 π , D T where A ( · ) is the normalized surface measure on D . Yet another way to look at the Dirichlet space is to consider analytic functions f : D → C such that the area (counting multiplicities) of f ( D ) is finite.

Carleson measures Let H be a Hilbert space of analytic functions on the domain Ω. A measure µ on ¯ Ω is called a Carleson measure, if the imbedding H �→ L 2 (¯ Ω , d µ ) is bounded, � f � 2 Ω , d µ ) � � f � 2 H . L 2 (¯ Theorem (A general one-dimensional ’theorem’) n ≥ 0 | ˆ Let f ∈ H a ( D ) , where � f � 2 f | 2 ( n )( n + 1) a . H a = � Then µ is Carleson for H a if and only if �� � �� � µ S ( I j ) � κ a I j , where { I j } is a finite collection of disjoint intervals on T . For a = 0 (i.e. for H 2 ) κ a is the Lebesgue measure, and for a = 1 (Dirichlet space) κ is the logarithmic capacity.

Another description Theorem (Local charge/energy) Assume that supp µ ⊂ T (otherwise we just push it to the boundary). Then µ is Carleson for the Dirichlet space on D iff for any dyadic interval I ⊂ T one has ( µ ( J )) 2 � µ ( I ) . � J ⊂ I

Dirichlet space D ( D 2 ) As before, we consider analytic functions on the bidisc m , n ≥ 0 a m , n z m w n . The (unweighted) Dirichlet space on D 2 f ( z , w ) = � consists of analytic functions f satisfying ( m + 1)( n + 1) | a m , n | 2 < + ∞ . � � f � 2 D ( D 2 ) = m , n ≥ 0 An equivalent definition is � � � | ∂ z f ( z , e i θ ) | 2 dA ( z ) d θ D 2 | ∂ zw f ( z , w ) | 2 dA ( z ) dA ( w ) + � f � 2 D ( D 2 ) = 2 π + D T | ∂ w f ( e it , w ) | 2 dt T 2 | f ( e it , e i θ ) | 2 dt d θ � � � 2 π dA ( w ) + 2 π . 2 π T D

Suggestion for the general two-dimensional theorem m , n ≥ 0 | ˆ Let f ∈ H a , b ( D 2 ), where � f � 2 f | 2 ( m , n )( m + 1) a ( n + 1) b . H a , b = � Then µ is Carleson for H a , b if and only if � N � N � � � � S ( I k ) × S ( J k ) ≤ C µ κ a , b I k × J k µ , k =1 k =1 where { I k } , { J k } are finite collections of disjoint intervals on T . As before, for a = b = 0 (i.e. for H 2 ( D 2 )) κ a , b is the Lebesgue measure, and for a = b = 1 (Dirichlet space) κ a , b is the bi-logarithmic capacity.

Local charge/energy for the bidisc Theorem Assume that supp µ ⊂ T 2 (again there is an argument that allows us to do so). Then µ is Carleson for the Dirichlet space on D 2 iff for any finite collection of dyadic rectangles I k × J k ⊂ T 2 , E = � N k =1 I k × J k one has ( µ ( R )) 2 � µ ( E ) . � R ⊂ E

A plan of sorts ◮ Candidate: subcapacitary measures ◮ Preliminary work: duality trick. ◮ We start with boundedness of the imbedding ◮ Modification: remove the derivative through RKHS properties ◮ Modification: remove the analytic structure ◮ Discretize the problem – replace a polydisc D d by a ”polytree” T d ◮ Discrete Setting. ◮ Develop appropriate potential theory on T d ◮ Maz’ya approach: reduce the problem to a potential-theoretic statement ◮ Reduce the potential-theoretic statement to a combinatorial one ◮ Solve the discrete problem and move it back to the polydisc ◮ Some possibly related problems.

Potential theory: basics ◮ Let X , Y be measure spaces, and let K : Y × X → R be a kernel function (subject to some basic conditions). We define � V µ ( x ) := K ( y , x ) d µ ( y ) . Y ◮ Newton and Riesz potentials � d µ ( y ) U µ ( x ) = | x − y | R 3 � d µ ( y ) I µ α ( x ) = | x − y | N − α . R N

A discrete model of the bidisc There is a standard way to discretize the unit disc via the Carleson boxes. A resulting discrete object is a uniform dyadic tree T . The same approach for the bidisc D × D produces the bitree T × T . A convenient way to represent the dyadic tree T is to consider the system ∆ of dyadic subintervals of the unit interval I 0 = [0 , 1). Respectively, the bitree corresponds to the system ∆ 2 of dyadic rectangles in Q 0 = [0 , 1) 2 (and the order relation is again given by inclusion).

Potential theory on the bitree: bilogarithmic potential We consider measures concentrated on the distinguished boundary ( ∂ T ) 2 (no loss of generality here), and all the graphs are finite (say of depth N ). Then ( ∂ T ) 2 can be identified as a collection of squares [ j 2 − N , ( j + 1)2 − N ) × [ k 2 − N , ( k + 1)2 − N ). Let µ be a non-negative Borel measure on ( ∂ T ) 2 . We define the (bilogarithmic) potential of µ to be � α ∈ ¯ V µ ( α ) := T 2 , ( ∂ T ) 2 K ( α, ω ) d µ ( ω ) , T 2 : γ ≥ α, γ ≥ ω } . where K ( α, ω ) = ♯ { γ ∈ ¯ Rectangular representation: � V µ ( Q ) = [0 , 1) 2 K ( Q , x ) d µ ( x ) , where Q is a dyadic rectangle, K is as above, and µ has a piecewise constant density on 2 − N -sized squares.

Potential theory on the bitree: capacity In particular, if y = y ( Q ) is a centerpoint of Q , then 1 1 K ( y , x ) ∼ log | y 1 − x 1 | log | y 2 − x 2 | , if x and y are ”far” enough from each other. Now let E be a compact subset of the unit square Q 0 = [0 , 1) 2 , we define Cap E := inf {E [ µ ] : V µ ( x ) ≥ 1 , x ∈ E } , where � V µ d µ E [ µ ] = is the energy of µ . By the general theory there exists a unique minimizer µ E — the equilibrium measure of the set E , such that Cap E = E [ µ E ] and V µ E ≡ 1 on supp µ E ⊂ E (we consider finite bitrees, so no need to deal with q.a.e.).

Potential theory on the bitree: capacitary strong inequality Now let µ ≥ 0, for λ > 0 consider E λ := { x ∈ Q 0 : V µ ( x ) ≥ λ } . It follows that � µ = 1 � Cap E λ ≤ E λ 2 E [ µ ] , λ since µ λ is admissible for E λ . Is it true that � ∞ λ Cap E λ d λ ≤ C E [ µ ] , 0 for some absolute constant C ? Maximum Principle: sup x ∈ supp µ V µ ( x ) � sup V µ ( x ) , x ∈ Q 0 then YES (Maz’ya, Adams, Hansson).

Potential theory on the bitree: capacitary strong inequality PROBLEM: there exists µ ≥ 0 on T 2 : x ∈ supp µ V µ ( x ) < sup V µ ( x ) = ∞ . 1 = sup x ∈ Q 0 SOLUTION (Quantitative MP): if supp x ∈ supp µ V µ ≤ 1 and λ ≥ 1, then 1 Cap E λ � λ 2 · λ E [ µ ] . Equivalent mixed energy estimate: let F ⊂ E , then � V µ E d µ F � ( E [ µ E ]) 1 2 − 1 1 2 + 1 6 ( E [ µ F ]) 6 . E [ µ E , µ F ] =

Further questions ◮ Possible extensions: 1 ≤ p < ∞ , weighted spaces. ◮ Explore the connections to the multiparameter martingales. ◮ Related problem — is there a Bellman function technique for the bitree? ◮ An example. Assume that µ is a probability measure on Q 0 . Given f ∈ L 2 ( Q 0 , d µ ) and Q ∈ ∆ 2 let � f � Q = 1 � Q f d µ . Define µ ( Q ) Mf ( x ) = sup Q ∋ x � f � Q to be the dyadic maximal function. We are interested in the inequality � � | Mf | 2 d µ ≤ C | f | 2 d µ, Q 0 Q 0 what conditions one could impose on µ for this inequality to hold?