Mixed-norm spaces and interpolation

Joaquín Ortega; Joan Fàbrega

Studia Mathematica (1994)

  • Volume: 109, Issue: 3, page 233-254
  • ISSN: 0039-3223

Abstract

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Let D be a bounded strictly pseudoconvex domain of ℂ n with smooth boundary. We consider the weighted mixed-norm spaces A δ , k p , q ( D ) of holomorphic functions with norm ∥ f ∥ p , q , δ , k = ( ∑ | α | ≤ k ʃ 0 r 0 ( ʃ ∂ D r | D α f | p d σ r ) q / p r δ q / p - 1 d r ) 1 / q . We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces A δ , k p ( D ) and we give results about real and complex interpolation between them. We apply these results to prove that A δ , k p , q ( D ) is the intersection of a Besov space B s p , q ( D ) with the space of holomorphic functions on D. Further, we obtain several properties of the mixed-norm spaces.

How to cite

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Ortega, Joaquín, and Fàbrega, Joan. "Mixed-norm spaces and interpolation." Studia Mathematica 109.3 (1994): 233-254. <http://eudml.org/doc/216072>.

@article{Ortega1994,
abstract = {Let D be a bounded strictly pseudoconvex domain of $ℂ^n$ with smooth boundary. We consider the weighted mixed-norm spaces $A^\{p,q\}_\{δ,k\}(D)$ of holomorphic functions with norm $∥f∥_\{p,q,δ,k\} = (∑_\{|α|≤k\} ʃ_\{0\}^\{r_0\} (ʃ_\{∂D_\{r\}\} |D^\{α\} f|^p dσ_\{r\})^\{q/p\} r^\{δq/p-1\} dr)^\{1/q\}$. We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces $A^\{p\}_\{δ,k\}(D)$ and we give results about real and complex interpolation between them. We apply these results to prove that $A^\{p,q\}_\{δ,k\}(D)$ is the intersection of a Besov space $B^\{p,q\}_\{s\}(D)$ with the space of holomorphic functions on D. Further, we obtain several properties of the mixed-norm spaces.},
author = {Ortega, Joaquín, Fàbrega, Joan},
journal = {Studia Mathematica},
keywords = {analytic functions; mixed-norm spaces; real interpolation; complex interpolation; Besov spaces of holomorphic functions; Bergman-Sobolev space; weighted mixed-norm spaces; holomorphic functions; pseudoconvex domain; interpolation; Besov space},
language = {eng},
number = {3},
pages = {233-254},
title = {Mixed-norm spaces and interpolation},
url = {http://eudml.org/doc/216072},
volume = {109},
year = {1994},
}

TY - JOUR
AU - Ortega, Joaquín
AU - Fàbrega, Joan
TI - Mixed-norm spaces and interpolation
JO - Studia Mathematica
PY - 1994
VL - 109
IS - 3
SP - 233
EP - 254
AB - Let D be a bounded strictly pseudoconvex domain of $ℂ^n$ with smooth boundary. We consider the weighted mixed-norm spaces $A^{p,q}_{δ,k}(D)$ of holomorphic functions with norm $∥f∥_{p,q,δ,k} = (∑_{|α|≤k} ʃ_{0}^{r_0} (ʃ_{∂D_{r}} |D^{α} f|^p dσ_{r})^{q/p} r^{δq/p-1} dr)^{1/q}$. We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces $A^{p}_{δ,k}(D)$ and we give results about real and complex interpolation between them. We apply these results to prove that $A^{p,q}_{δ,k}(D)$ is the intersection of a Besov space $B^{p,q}_{s}(D)$ with the space of holomorphic functions on D. Further, we obtain several properties of the mixed-norm spaces.
LA - eng
KW - analytic functions; mixed-norm spaces; real interpolation; complex interpolation; Besov spaces of holomorphic functions; Bergman-Sobolev space; weighted mixed-norm spaces; holomorphic functions; pseudoconvex domain; interpolation; Besov space
UR - http://eudml.org/doc/216072
ER -

References

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