The local versions of H p ( ℝ n ) spaces at the origin

Shan Lu; Da Yang

Studia Mathematica (1995)

  • Volume: 116, Issue: 2, page 103-131
  • ISSN: 0039-3223

Abstract

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Let 0 < p ≤ 1 < q < ∞ and α = n(1/p - 1/q). We introduce some new Hardy spaces H K ̇ q α , p ( ℝ n ) which are the local versions of H p ( ℝ n ) spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizations in M. Frazier and B. Jawerth’s sense. We also prove an interpolation theorem for operators on H K ̇ q α , p ( ℝ n ) and discuss the H K ̇ q α , p ( ℝ n ) -boundedness of Calderón-Zygmund operators. Similar results can also be obtained for the non-homogeneous Hardy spaces H K q α , p ( ℝ n ) .

How to cite

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Lu, Shan, and Yang, Da. "The local versions of $H^p(ℝ^n)$ spaces at the origin." Studia Mathematica 116.2 (1995): 103-131. <http://eudml.org/doc/216223>.

@article{Lu1995,
abstract = {Let 0 < p ≤ 1 < q < ∞ and α = n(1/p - 1/q). We introduce some new Hardy spaces $HK̇_q^\{α,p\}(ℝ^n)$ which are the local versions of $H^p(ℝ^n)$ spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizations in M. Frazier and B. Jawerth’s sense. We also prove an interpolation theorem for operators on $HK̇_q^\{α,p\}(ℝ^n)$ and discuss the $HK̇_q^\{α,p\}(ℝ^n)$-boundedness of Calderón-Zygmund operators. Similar results can also be obtained for the non-homogeneous Hardy spaces $HK_q^\{α,p\}(ℝ^n)$.},
author = {Lu, Shan, Yang, Da},
journal = {Studia Mathematica},
keywords = {Herz spaces; Hardy spaces; local versions; atomic and molecular decompositions; -transform; Calderón-Zygmund operators},
language = {eng},
number = {2},
pages = {103-131},
title = {The local versions of $H^p(ℝ^n)$ spaces at the origin},
url = {http://eudml.org/doc/216223},
volume = {116},
year = {1995},
}

TY - JOUR
AU - Lu, Shan
AU - Yang, Da
TI - The local versions of $H^p(ℝ^n)$ spaces at the origin
JO - Studia Mathematica
PY - 1995
VL - 116
IS - 2
SP - 103
EP - 131
AB - Let 0 < p ≤ 1 < q < ∞ and α = n(1/p - 1/q). We introduce some new Hardy spaces $HK̇_q^{α,p}(ℝ^n)$ which are the local versions of $H^p(ℝ^n)$ spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizations in M. Frazier and B. Jawerth’s sense. We also prove an interpolation theorem for operators on $HK̇_q^{α,p}(ℝ^n)$ and discuss the $HK̇_q^{α,p}(ℝ^n)$-boundedness of Calderón-Zygmund operators. Similar results can also be obtained for the non-homogeneous Hardy spaces $HK_q^{α,p}(ℝ^n)$.
LA - eng
KW - Herz spaces; Hardy spaces; local versions; atomic and molecular decompositions; -transform; Calderón-Zygmund operators
UR - http://eudml.org/doc/216223
ER -

References

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  1. [1] A. Baernstein II and E. T. Sawyer, Embedding and multiplier theorems for H p ( ℝ n ) , Mem. Amer. Math. Soc. 318 (1985). 
  2. [2] Y. Z. Chen and K. S. Lau, On some new classes of Hardy spaces, J. Funct. Anal. 84 (1989), 255-278. Zbl0677.30030
  3. [3] C. Fefferman and E. M. Stein, H p spaces of several variables, Acta Math. 129 (1972), 137-193. Zbl0257.46078
  4. [4] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799. Zbl0551.46018
  5. [5] M. Frazier and B. Jawerth, The φ-transform and applications to distribution spaces, in: Lecture Notes in Math. 1302, Springer, 1988, 223-246. Zbl0648.46038
  6. [6] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34-170. Zbl0716.46031
  7. [7] J. García-Cuerva, Hardy spaces and Beurling algebras, J. London Math. Soc. (2) 39 (1989), 499-513. 
  8. [8] S. Z. Lu and F. Soria, On the Herz spaces with power weights, in: Fourier Analysis and Partial Differential Equations, J. Garcí a-Cuerva, E. Hernández, F. Soria and J. L. Torrea (eds.), Stud. Adv. Math., CRC Press, Boca Raton, 1995, 227-236. Zbl1039.42503
  9. [9] S. Z. Lu and D. C. Yang, The Littlewood-Paley function and φ-transform characterizations of a new Hardy space H K 2 associated with the Herz space, Studia Math. 101 (1992), 285-298. Zbl0811.42005
  10. [10] S. Z. Lu and D. C. Yang, Some new Hardy spaces associated with the Herz spaces and their wavelet characterizations, J. Beijing Normal Univ. (Natural Sci.) 29 (1993), 10-19. Zbl0782.42019
  11. [11] M. H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Astérisque 77 (1980), 67-149. Zbl0472.46041
  12. [12] M. H. Taibleson and G. Weiss, Certain function spaces associated with a.e. convergence of Fourier series, in: Conference in Honor of A. Zygmund, Vol. I, Wadsworth, 1983, 95-113. 

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