Lévy measures for Dirichlet-type spaces on the unit bidisc

Abstract

For the Dirichlet-type space 𝒟(ρ⁽¹⁾,ρ⁽²⁾) on the unit bidisc 𝔻², where ρ⁽¹⁾ and ρ⁽²⁾ are finite positive Borel measures on the closed unit disc 𝔻̅, we show that the Lévy measure ν_{(ℳ_z,1)} associated with the completely alternating multisequence {|z^α|_{𝒟(ρ⁽¹⁾,ρ⁽²⁾)}² }_{α∈ℤ₊²} admits the explicit representation dν_{(ℳ_z,1)}(x)=1/(1−x₁)d((S_*ρ⁽¹⁾)×δ₁)(x)+1/(1−x₂)d(δ₁× (S_*ρ⁽²⁾))(x) on [0,1]²backslash{(1,1)}, where S_*ρ⁽ⁱ⁾, i=1,2, denotes the pushforward measure of ρ⁽ⁱ⁾ by the map S:𝔻̅→ [0,1] defined by S(z)=|z|², z∈ 𝔻̅, and δ₁ denotes the Dirac measure at 1.

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BibTeX RIS

BibTeX

@misc{bera2026lévy,
  title = {{Lévy measures for Dirichlet-type spaces on the unit bidisc}},
  author = {Santu Bera and Shanola S. Sequeira},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20346},
  eprint = {2609.20346},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20346},
}

RIS

TY  - UNPB
TI  - Lévy measures for Dirichlet-type spaces on the unit bidisc
AU  - Santu Bera
AU  - Shanola S. Sequeira
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20346
UR  - https://arxiv.org/abs/2609.20346
AB  - For the Dirichlet-type space 𝒟(ρ^(1),ρ^(2)) on the unit bidisc 𝔻^2, where ρ^(1) and ρ^(2) are finite positive Borel measures on the closed unit disc 𝔻̅, we show that the Lévy measure ν_(ℳ_z,1) associated with the completely alternating multisequence {|z^α|_𝒟(ρ^(1),ρ^(2))^2 }_α∈ℤ_+^2 admits the explicit representation dν_(ℳ_z,1)(x)=1/(1−x_1)d((S_*ρ^(1))×δ_1)(x)+1/(1−x_2)d(δ_1× (S_*ρ^(2)))(x) on [0,1]^2backslash{(1,1)}, where S_*ρ^(i), i=1,2, denotes the pushforward measure of ρ^(i) by the map S:𝔻̅→ [0,1] defined by S(z)=|z|^2, z∈ 𝔻̅, and δ_1 denotes the Dirac measure at 1.
ER  -

Abstract

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