Poly-Bergman Projections and Orthogonal Decompositions of L 2-spaces Over Bounded Domains

Karlovich, Yuri I.; Pessoa, Luís V.

doi:10.1007/978-3-7643-8684-9_12

Cited by 18 publications

(23 citation statements)

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“…Since

B_{U, j} = C {\overset{B}{true}}_{U, j} C,

one concludes that is necessary and sufficient for . It has been shown [, Theorem ], for every positive integer j , that in the unit disk case and remain valid with

K_{j} = 0

and

{\overset{K}{true}}_{j} = 0 .

Theorem For every positive integer j , the following formulas hold :

\begin{matrix} left B_{D, j} & left = & left I - {(S_{D})}^{j} {(S_{double-struckD}^{*})}^{j}, \\ left B_{D, - j} & left = & left I - {(S_{double-struckD}^{*})}^{j} {(S_{D})}^{j} . \end{matrix}

Furthermore, if k is a positive integer then

B_{D, j} B_{D, - k} = P_{j k},

where

P_{j k}

is the orthogonal projection of

L^{2} (double-struckD)

onto the

j k

finite dimensional space given by

span \{z^{l} {\bar{z}}^{s} : l = 0, ..., k - 1; s = 0, ..., j - 1\} .

…”

Section: Poly‐bergman Spaces and Singular Integral Operatorsmentioning

confidence: 99%

“…Let U denotes a non‐empty domain and define the singular integral operators acting on the space

L^{2} (U)

S_{U, j} f (z) : = \frac{{(- 1)}^{j} | j |}{π} \int_{U} \frac{{(w - z)}^{j - 1}}{{(\bar{w} - \bar{z})}^{j + 1}} f (w) d A (w), j \in {double-struckZ}_{\pm} .

In [, Corollary ] and in [, Corollary 2.9.1] it is proved that

S_{D, - j} = {(S_{double-struckD})}^{j} and S_{D, j} = {(S_{D}^{*})}^{j}, j \in {double-struckZ}_{+} .

For every non‐zero integer j , denote by

S_{j}

the singular integral operator

S_{C, j} .

By Lebesgue's dominated convergence Theorem, for every positive integer j , one obtains that

S_{- j} = {s ‐ lim}_{n} S_{n D, - j} = {s ‐ lim}_{n} {(S_{n D})}^{j} = S^{j} and S_{j} = S_{- j}^{*} = {(S^{*})}^{j} .

It's well known that the operator S is unitary, e.g. the latter follows from the Mikhilin symb...…”

Section: Poly‐bergman Spaces and Singular Integral Operatorsmentioning

confidence: 99%

“…For the case of the sectors

Π^{m} : = {z : Im z^{m} > 0}, m = 2, ...,

we find in [, Theorem 5.3] equalities for the Bergman and anti‐Bergman projections different of those in and , more precisely one has that

\begin{matrix} left B_{Π^{m}} & left = & left I - (() S_{Π^{m}} + R_{Π^{m}}) {(S_{Π^{m}} + R_{Π^{m}})}^{*}, \\ left {\overset{B}{true}}_{Π^{m}} & left = & left I - {(S_{Π^{m}} + R_{Π^{m}})}^{*} (() S_{Π^{m}} + R_{Π^{m}}) . \end{matrix}

Here

R_{Π^{m}}

for

m = 2, ...

is the bounded operator defined by

R_{Π^{m}} = \sum_{k = 1}^{m - 1} R_{m, k} and R_{m, k} = χ_{Π^{m}} W_{ɛ_{m}}^{2 k} S χ_{Π^{m}} I (k = 1, ..., m - 1),

where

W_{ɛ_{m}} f (z) : = ɛ_{m} f (ɛ_{m} z)

for

ɛ_{m} : = e^{i π / m}

and

S_{U}

is the Beurling operator of the domain U , i.e.

S_{U}

is the two‐...…”

Section: Introductionmentioning

confidence: 97%

“…A proof of Dzhuraev's formulas in the case of multi connected domains with a smooth boundary , the unit disk and the upper half‐plane can be found in [, Chapter 3], [, Theorem and Corollary ] and [, Theorem 1.1], respectively. The representation of the poly‐Bergman projection via the Beurling operator in the case of upper half‐plane and the unit disk is proved in [, Theorem 3.7] and [, Theorem ], respectively. The variation of the domain allows us to obtain exact Dzhuraev's formulas , and the poly‐Bergman kernels for the upper half‐plane from the corresponding ones for the unit disk.…”

Section: Introductionmentioning

confidence: 99%

“…The dependence on the boundary regularity of equalities between Bergman kind projections and operators on the *‐algebra generated by singular integral operators is not yet completely understood. Equalities between the poly‐Bergman projections and operators in the *‐algebra generated by the Beurling operator

S_{U}

were an important step in ,, ,, , whenever U is the unit disk, a half‐plane or a bounded multi connected domain with a smooth boundary.…”

Section: Introductionmentioning

confidence: 99%

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The method of variation of the domain for poly‐Bergman spaces

Pessoa

2013

Mathematische Nachrichten

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If Un,n∈N, is an increasing sequence (well ordered by inclusion) of domains then the sequence of poly‐Bergman projections on the domains Un strongly converges to the poly‐Bergman projection on the limit domain. As a corollary some properties of the poly‐Bergman spaces on the half‐planes are deduced from the corresponding ones in the unit disk. We obtain explicit representation of the poly‐Bergman projections in terms of the two‐dimensional singular integral operators SΠ,j,j∈double-struckZ±, likewise explicit formulas for the poly‐Bergman kernels. We prove that the poly‐Bergman projections on the sectors with a non‐smooth boundary do not admit the usual representations by the two‐dimensional singular integral operators. The variation of the domain and the latter peculiarity of the poly‐Bergman projections allow us to furnish a larger class of domains not admitting Dzhuraev's formulas.

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“…Since

B_{U, j} = C {\overset{B}{true}}_{U, j} C,

one concludes that is necessary and sufficient for . It has been shown [, Theorem ], for every positive integer j , that in the unit disk case and remain valid with