# Multiplier transformations on ${H}^{p}$ spaces

Studia Mathematica (1998)

- Volume: 131, Issue: 2, page 189-204
- ISSN: 0039-3223

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topChen, Daning, and Fan, Dashan. "Multiplier transformations on $H^{p}$ spaces." Studia Mathematica 131.2 (1998): 189-204. <http://eudml.org/doc/216575>.

@article{Chen1998,

abstract = {The authors obtain some multiplier theorems on $H^p$ spaces analogous to the classical $L^p$ multiplier theorems of de Leeuw. The main result is that a multiplier operator $(Tf)^(x) = λ(x)f̂(x)$$(λ ∈ C(ℝ^n))$ is bounded on $H^p(ℝ^n)$ if and only if the restriction $\{λ(εm)\}_\{m∈Λ\}$ is an $H^p(T^n)$ bounded multiplier uniformly for ε>0, where Λ is the integer lattice in $ℝ^n$.},

author = {Chen, Daning, Fan, Dashan},

journal = {Studia Mathematica},

keywords = {multiplier operator; bounded operator},

language = {eng},

number = {2},

pages = {189-204},

title = {Multiplier transformations on $H^\{p\}$ spaces},

url = {http://eudml.org/doc/216575},

volume = {131},

year = {1998},

}

TY - JOUR

AU - Chen, Daning

AU - Fan, Dashan

TI - Multiplier transformations on $H^{p}$ spaces

JO - Studia Mathematica

PY - 1998

VL - 131

IS - 2

SP - 189

EP - 204

AB - The authors obtain some multiplier theorems on $H^p$ spaces analogous to the classical $L^p$ multiplier theorems of de Leeuw. The main result is that a multiplier operator $(Tf)^(x) = λ(x)f̂(x)$$(λ ∈ C(ℝ^n))$ is bounded on $H^p(ℝ^n)$ if and only if the restriction ${λ(εm)}_{m∈Λ}$ is an $H^p(T^n)$ bounded multiplier uniformly for ε>0, where Λ is the integer lattice in $ℝ^n$.

LA - eng

KW - multiplier operator; bounded operator

UR - http://eudml.org/doc/216575

ER -

## References

top- [1] P. Auscher and M. J. Carro, On relations between operators on ${\mathbb{R}}^{n}$, ${T}^{n}$ and ${\mathbb{Z}}^{n}$, Studia Math. 101 (1990), 165-182.
- [2] D. Chen, Multipliers on certain function spaces, Ph.D. thesis, Univ. of Wisconsin-Milwaukee, 1998.
- [3] D. Fan, Hardy spaces on compact Lie groups, Ph.D. thesis, Washington University, St. Louis, 1990.
- [4] C. Fefferman and E. M. Stein, ${H}^{p}$ spaces of several variables, Acta Math. 129 (1972), 137-193. Zbl0257.46078
- [5] D. Goldberg, A local version of real Hardy spaces, ibid. 46 (1979), 27-42. Zbl0409.46060
- [6] C. Kenig and P. Thomas, Maximal operators defined by Fourier multipliers, Studia Math. 68 (1980), 79-83. Zbl0442.42013
- [7] S. Krantz, Fractional integration on Hardy spaces, ibid. 73 (1982), 87-94. Zbl0504.47034
- [8] K. de Leeuw, On ${L}_{p}$ multipliers, Ann. of Math. 91 (1965), 364-379.
- [9] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. Zbl0232.42007

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