On the existence of nontangential boundary values of pseudocontinuable functions

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On the existence of nontangential boundary values of pseudocontinuable functions. / Александров, Алексей Борисович.

In: Journal of Mathematical Sciences , Vol. 87, No. 5, 1997, p. 3781-3787.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{4552b73c902f41de8ba60ddaf85cabb3,

title = "On the existence of nontangential boundary values of pseudocontinuable functions",

abstract = "Let θ be an inner function, let θ*(H2) = H2 Θ θH2, and let p be a finite Borel measure on the unit circle double-struck T sign. Our main purpose is to prove that, if every function f ∈ θz.ast;(H2) can be defined μ-almost everywhere on double-struck T sign in a certain (weak) natural sense, then every function f ∈ θ*(H2) has finite angular boundary values μ-almost everywhere on double-struck T sign. A similar result is true for the Lp-analog of θ*(H2) (p > 0).",

author = "Александров, {Алексей Борисович}",

note = "Funding Information: Proof. Without loss of generality, we may assume that/z = IIE.a~, where lIE is the characteristic function of a Borel set E C T. It was proved in \[15\t]h at the identity embedding operator 0*(H 2) C 0*(H P) is compact. Consequently, so is the embedding operator 0*(/'/2) ~ Lo(/z). From the Clark theorem \[8\]q uoted above we deduce that the operator f ~-* f\[E acting from L2(a~) to L~ is compact. Now it is clear that # is discrete. \[\] I am grateful to Max-Planck-Arbeitsgruppe {"}Algebraische Geometric und Zahlentheorie,{"} during the stay at which this work was finished. This research was supported in part by the Russian Foundation for Fundamental Studies, grant 94-01-0132, and by the International Science Foundation, grant R3M000.",

year = "1997",

doi = "10.1007/BF02355824",

language = "English",

volume = "87",

pages = "3781--3787",

journal = "Journal of Mathematical Sciences",

issn = "1072-3374",

publisher = "Springer Nature",

number = "5",

}

RIS

TY - JOUR

T1 - On the existence of nontangential boundary values of pseudocontinuable functions

AU - Александров, Алексей Борисович

N1 - Funding Information: Proof. Without loss of generality, we may assume that/z = IIE.a~, where lIE is the characteristic function of a Borel set E C T. It was proved in \[15\t]h at the identity embedding operator 0*(H 2) C 0*(H P) is compact. Consequently, so is the embedding operator 0*(/'/2) ~ Lo(/z). From the Clark theorem \[8\]q uoted above we deduce that the operator f ~-* f\[E acting from L2(a~) to L~ is compact. Now it is clear that # is discrete. \[\] I am grateful to Max-Planck-Arbeitsgruppe "Algebraische Geometric und Zahlentheorie," during the stay at which this work was finished. This research was supported in part by the Russian Foundation for Fundamental Studies, grant 94-01-0132, and by the International Science Foundation, grant R3M000.

PY - 1997

Y1 - 1997

N2 - Let θ be an inner function, let θ*(H2) = H2 Θ θH2, and let p be a finite Borel measure on the unit circle double-struck T sign. Our main purpose is to prove that, if every function f ∈ θz.ast;(H2) can be defined μ-almost everywhere on double-struck T sign in a certain (weak) natural sense, then every function f ∈ θ*(H2) has finite angular boundary values μ-almost everywhere on double-struck T sign. A similar result is true for the Lp-analog of θ*(H2) (p > 0).

AB - Let θ be an inner function, let θ*(H2) = H2 Θ θH2, and let p be a finite Borel measure on the unit circle double-struck T sign. Our main purpose is to prove that, if every function f ∈ θz.ast;(H2) can be defined μ-almost everywhere on double-struck T sign in a certain (weak) natural sense, then every function f ∈ θ*(H2) has finite angular boundary values μ-almost everywhere on double-struck T sign. A similar result is true for the Lp-analog of θ*(H2) (p > 0).

UR - http://www.scopus.com/inward/record.url?scp=53249115556&partnerID=8YFLogxK

U2 - 10.1007/BF02355824

DO - 10.1007/BF02355824

M3 - Article

AN - SCOPUS:53249115556

VL - 87

SP - 3781

EP - 3787

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 5

ER -

ID: 87312768