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α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable On the structure of polyharmonic Bergman spaces. Universidade de Lisboa, Instituto Superior T´ ecnico Lisboa, Portugal Lu´ ıs V. Pessoa MAIN CONFERENCE: Function Spaces and Complex Analysis October 27–31, 2014 Centre International de Rencontres Math´ ematiques (CIRM) Marseille, France Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces

α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable Abstract I will present some new results on the structure of polyharmonic Bergman spaces over some domains in terms of the compression of the Beurling-Ahlfors transform. It will be explained how the results are a consequence of the validity of Dzhuraev’s formulas, i.e. how such study can be based on the fact that the compression of the Beurling-Ahlfors transform is a power partial isometry over special domains. Theorems of Paley-Wienner type for polyharmonic Bergman spaces will be given for half-spaces. The talk is partially based on a joint work with A. M. Santos. Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces

α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable Poly-Bergman spaces U ⇢ C non-empty, open and connected ; dA ( z ) = dxdy area measure ✓ ∂ ◆ ✓ ∂ ◆ ∂ z := 1 ∂ z := 1 ∂ x + i ∂ ∂ x � i ∂ , 2 ∂ y 2 ∂ y Definition (Poly-Bergman spaces) j ( U ) if f 2 L 2 ( U , dA ) , f is smooth and f 2 A 2 ∂ j z f = 0 and ∂ − j z f = 0 , respectively if j 2 Z + and j 2 Z − (1.1) if j 2 Z then f is j -polyanalytic if is smooth and satisfies (1.1) if j 2 Z − then it is also usually said that f is | j | -anti-polyanalytic if j = 0 then we have the special case A 2 0 ( U ) = { 0 } Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces

α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable Hilbert spaces of α -Polyanalytic functions, α = ( j , k ) Now we consider α := ( j , k ) a pair of non-negative integers Definition ( α -polyanalytic function) f is smooth on U and ∂ j z ∂ k z f = 0 ( j , k = 0 , 1 , . . . ) Definition ( α -polyanalytic Bergman space) α ( U ) if f 2 L 2 ( U , dA ) and f is α -polyanalytic f 2 A 2 Is A 2 α ( U ) a Hilbert space? The following two results and some analysis will allow to say Yes. First, some definitions. Define N j , k := A 2 j ( D ) \ A 2 − k ( D ) , j , k 2 Z + [see L.V.P. 14 ] Then N j , k = span { z l z n : l = 0 , 1 , . . . , k � 1; n = 0 , . . . , j � 1 } Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces

α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable Hilbert spaces of α -Polyanalytic functions, α = ( j , k ) Theorem (Yu.I. Karlovich, L.V.P. 08) The following assertions hold: i) B D , j and B D , k commute ( j , k 2 Z ) ; B D , j B D , − k is the projection of L 2 ( D , dA ) onto N j , k ii) ( j , k 2 Z + ) . Lemma Let H be a Hilbert space and let M , N 2 B ( H ) be projections. Then, P := M + N � MN is a projection i ff M and N commute. Furthermore, if P is a projection, then its range coincides with Im M + Im N . Theorem (L.V.P. 14) Let j , k = 0 , 1 , . . . and let α := ( j , k ) . Then A 2 α ( D ) is closed in L 2 ( D ) . If B D , α denotes the orthogonal projection of L 2 ( D ) onto A 2 α ( D ) , then B D , α = B D , j + B D , − k � B D , j B D , − k . Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces

α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable α -Polyanalytic functions and Singular Integral Operators The unitary Beurling-Ahlfors transform and its compression to L 2 ( U ) Z Sf ( z ) := � 1 f ( w ) ( w � z ) 2 d A ( w ) and S U := χ U S χ U π C Dzhuraev’s Operators (for j 2 Z + ) D U , j = I � ( S U ) j ( S ∗ U ) j U ) j ( S U ) j and D U , − j = I � ( S ∗ If U is bounded finitely connected, ∂ U is smooth then B U , j � D U , j 2 K ( j 2 Z ± ) . The existence of Dzhuraev’s formulas are strongly dependent on the regularity of the boundary Yu.I. Karlovich, L.V.P. 08; L.V.P 13 Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces

α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable α -Polyanalytic functions and Singular Integral Operators Theorem (Yu.I. Karlovich, L.V.P. 08; L.V.P. 14) B D , j = D D , j , B Π , j = D Π , j , B E , j = D E , j If U 2 { D , Π , E } then S U is a ⇤ -power partial isometry Theorem (L.V.P. 14) Let j and k be nonnegative integers and let α := ( j , k ) . Then, D ) j + k ( S D ) k = I � ( S ∗ D ) k ( S D ) j + k ( S ∗ B D , α = I � ( S D ) j ( S ∗ D ) j . Some results are then easily generalised to L p , 1 < p < + 1 Theorem (L.V.P. 14) Let j , k = 0 , 1 , . . . and let α := ( j , k ) . Then, B D , α defines a bounded idempotent acting on L p ( D ) , for 1 < p < + 1 . Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces

α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable α -Polyanalytic functions and Singular Integral Operators The compression of the Riesz transforms of even order ( S j = R − 2 j ) Z ( w � z ) j − 1 S D , j f ( z ) := ( � 1) j | j | ( w � z ) j +1 f ( w ) dA ( w ) , j 2 Z ± π D From results in Yu.I. Karlovich; L.V.P. 08 we known that S D , − j = ( S D ) j . D ) j S D , j = ( S ∗ and Theorem (L.V.P. 14) Let j , k = 0 , 1 , . . . and let α := ( j , k ) . Then, B D , α = I � S D , − j S D , j + k S D , − k = I � S D , k S D , − j − k S D , j . Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces

α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable α -Polyanalytic Bergman spaces are RKHS α -Polyanalytic Bergman spaces are reproducing kernel Hilbert spaces Theorem (L.V.P. 14) Let U ⇢ C be a domain, let j , k = 0 , 1 , . . . let α := ( j , k ) . Then A 2 α ( U ) is a RKHS. For every n , m = 0 , 1 , . . . and every z 2 U , one has M k f k , f 2 A 2 | ∂ n z ∂ m z f ( z ) |  α ( U ) d n + m +1 z where M is a positive constant only depending on n , m , j and k . B D , α is integral operator with kernel given by the α -polyanalytic Bergman kernel K D , α ( z , w ) , which has a non-friendly representation. Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces

α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable True Poly-Bergman Spaces and More N j , k Type Spaces For j 2 Z ± , the true poly Bergman spaces, which were introduced over half-spaces in N. Vasilevski 99 A 2 ( ± 1) ( D ) := A 2 A 2 ( j ) ( D ) := A 2 j ( D ) A 2 ± 1 ( D ) and j − sgn j ( D ) Then it is clear that B D , ( j ) = B D , j � B D , j − 1 , j > 1 and B D , ( j ) = B D , j � B D , j +1 , j < � 1 . We introduce the following spaces like in the definition of N j , k A 2 ( j ) ( D ) \ A 2 N ( j ) , k := − k ( D ) = Im B D , ( j ) B D , − k A 2 j ( D ) \ A 2 N j , ( k ) := ( − k ) ( D ) = Im B D , j B D , ( − k ) A 2 ( j ) ( D ) \ A 2 N ( j ) , ( k ) := ( − k ) ( D ) = Im B D , ( j ) B D , ( − k ) Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces

α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable Unitary Operators on True Poly Bergman Type Spaces Theorem (L.V.P. 14) Let j 2 Z + and let k 2 Z ± . The operators ( S D ) j : A 2 ( k ) ( D ) N ( k ) , j ! A 2 ( k + j ) ( D ) , k > 0 ( S D ) j : A 2 ( k ) ( D ) ! A 2 ( k + j ) ( D ) N j , ( − k − j ) , 0 < j < � k as well as the following ones D ) j : A 2 ( k ) ( D ) N j , ( − k ) ! A 2 ( S ∗ ( k − j ) ( D ) , k < 0 D ) j : A 2 ( k ) ( D ) ! A 2 ( S ∗ ( k − j ) ( D ) N ( k − j ) , j , 0 < j < k are isometric isomorphisms. Furthermore 5 D ) j = A 2 Ker ( S D ) j = A 2 Ker ( S ∗ j ( D ) and − j ( D ) . Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces

α -Polyanalyticity Polyharmonic Spaces Unitary Operators Half-Spaces More on the Structure The Real Variable Unitary Operators on Poly-Bergman Type Spaces Theorem (L.V.P. 14) Let j 2 Z + and k 2 Z ± . The operators ( S D ) j : A 2 k ( D ) N k , j ! A 2 k + j ( D ) A 2 j ( D ) , k > 0 ( S D ) j : A 2 k ( D ) A 2 − j ( D ) ! A 2 k + j ( D ) N j , − k − j , 0 < j < � k as well as the following ones D ) j : A 2 k ( D ) N j , − k ! A 2 k − j ( D ) A 2 ( S ∗ − j ( D ) , k < 0 D ) j : A 2 k ( D ) A 2 j ( D ) ! A 2 ( S ∗ k − j ( D ) N k − j , j , 0 < j < k are isometric isomorphisms. Lu´ ıs V. Pessoa On the structure of polyharmonic Bergman spaces