Holomorphic potentials and multipliers for Hardy-Sobolev spaces

Cascante, Carme; Fàbrega, Joan; Ortega, J. M.

doi:10.1007/s00605-014-0688-1

Cited by 3 publications

(8 citation statements)

References 10 publications

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“…With the help of Theorem 3.5, we can establish the desired multiplier norm estimate of 𝑓 𝜇 . It can be regarded as a quantitative version of the result of Cascante, Fàbrega and Ortega [9] in the Hilbert space setting. Proposition 3.6.…”

Section: Direct Proof Of Theorem 11mentioning

confidence: 97%

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Totally null sets and capacity in Dirichlet type spaces

Chalmoukis

Hartz

2022

Journal of London Math Soc

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In the context of Dirichlet type spaces on the unit ball of ℂ 𝑑 , also known as Hardy-Sobolev or Besov-Sobolev spaces, we compare two notions of smallness for compact subsets of the unit sphere. We show that the functional analytic notion of being totally null agrees with the potential theoretic notion of having capacity zero. In particular, this applies to the classical Dirichlet space on the unit disc and logarithmic capacity. In combination with a peak interpolation result of Davidson and the second named author, we obtain strengthening of boundary interpolation theorems of Peller and Khrushchëv and of Cohn and Verbitsky.M S C ( 2 0 2 0 ) 46E22 (primary), 31B15 (secondary)

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Section: Direct Proof Of Theorem 11mentioning

confidence: 97%

“…In [9], Cascante, Fàbrega and Ortega showed that if 0 < 𝑎 < 1 and if the holomorphic potential 𝑓 𝜇 is bounded in 𝔹 𝑑 , then it is a multiplier of  𝑎 . They also proved an 𝐿 𝑝 -analogue of this statement.…”

Section: Direct Proof Of Theorem 11mentioning

confidence: 99%

Totally null sets and capacity in Dirichlet type spaces

Chalmoukis

Hartz

2022

Journal of London Math Soc

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show abstract

“…With the help of Theorem 3.5, we can establish the desired multiplier norm estimate of f µ . It can be regarded as a quantitative version of the result of Cascante, Fàbrega and Ortega [9] in the Hilbert space setting. Proposition 3.6.…”

Section: Direct Proof Of Theorem 11mentioning

confidence: 97%

“…We briefly sketch the argument in general. In [9], Cascante, Fàbrega and Ortega showed that if 0 < a < 1 and if the holomorphic potential f µ is bounded in B d , then it is a multiplier of D a . They also proved an L p -analogue of this statement.…”

Section: Direct Proof Of Theorem 11mentioning

confidence: 99%

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Totally null sets and capacity in Dirichlet type spaces

Chalmoukis¹,

Hartz²

2020

Preprint

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In the context of Dirichlet type spaces on the unit ball of C d , also known as Hardy-Sobolev or Besov-Sobolev spaces, we compare two notions of smallness for compact subsets of the unit sphere. We show that the functional analytic notion of being totally null agrees with the potential theoretic notion of having capacity zero. In particular, this applies to the classical Dirichlet space on the unit disc and logarithmic capacity. In combination with a peak interpolation result of Davidson and the second named author, we obtain strengthenings of boundary interpolation theorems of Peller and Khrushchëv and of Cohn and Verbitsky.

show abstract