Hardy spaces via distribution spaces

Abstract

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that $\int_{\mathbb{R}^n } {x^\gamma \varphi (x)dx = 0} $ for all multiindices γ. Let (ℐ(ℝⁿ))′ and (ℐ_∞(ℝⁿ))′ be their dual spaces, respectively. In this paper, it is proved that atomic Hardy spaces defined via (ℐ(ℝⁿ))′ and (ℐ_∞(ℝⁿ))′ coincide with each other in some sense. As an application, we show that under the condition that the Littlewood-Paley function of f belongs to L^p(ℝⁿ) for some p ∊ (0,1], the condition f ∊ (ℐ_∞(ℝⁿ))′ is equivalent to that f ∊ (ℐ(ℝⁿ))′ and f vanishes weakly at infinity. We further discuss some new classes of distributions defined via ℐ(ℝⁿ) and ℐ_∞(ℝⁿ), also including their corresponding Hardy spaces.

Keywords

Hardy space / atom / Schwartz class / distribution / Littlewood-Paley function

References

[1] Bownik M., Ho K.-P. Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces. Trans Amer Math Soc, 2006, 358(4): 1469-1510.

[2] Chang S. Y. A., Fefferman R. A continuous version of duality of H¹ with BMO on the bidisc. Ann of Math, 1980, 112(2): 179-201.

[3] Coifman R. R. A real variable characterization of H^p. Studia Math, 1974, 51: 269-274.

[4] Fefferman C., Stein E. M. H^p spaces of several variables. Acta Math, 1972, 129: 137-193.

[5] Folland G. B., Stein E. M. Hardy Spaces on Homogeneous Groups, 1982, Princeton: Princeton University Press.

[6] Frazier M., Jawerth B., Weiss G. Littlewood-Paley Theory and the Study of Function Spaces. CBMS Regional Conference Series in Mathematics, 79, 1991, Providence: Amer Math Soc.

[7] Friedlander G., Joshi M. Introduction to the Theory of Distributions, 1998 2 Cambridge: Cambridge University Press.

[8] Grafakos L. Classical and Modern Fourier Analysis, 2004, Upper Saddle River: Prentice Hall.

[9] Han Y., Müler D., Yang D. Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math Nachr, 2006, 279: 1505-1537.

[10] Han Y., Yang D. Some new spaces of Besov and Triebel-Lizorkin type on homogeneous spaces. Studia Math, 2003, 156: 67-97.

[11] Harvin V. P., Hruscev S. V., Nikol’skii N. K. Linear and Complex Variables Problem Book, 1984, Berlin: Springer.

[12] Latter R. H. A decomposition of H^p(ℝⁿ) in terms of atoms. Studia Math, 1977, 62: 92-101.

[13] Lu S. Z. Four Lectures on Real H^p Spaces, 1995, Singapore: World Scientific.

[14] Stein E. M. On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Trans Amer Math Soc, 1958, 88: 430-466.

[15] Stein E. M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 1993, Princeton: Princeton University Press.

[16] Stein E. M., Weiss G. On the theory of harmonic functions of several variables, I. The theory of H^p spaces. Acta Math, 1960, 103: 25-62.

[17] Stein E. M., Weiss G. Introduction to Fourier Analysis on Euclidean Spaces, 1971, Princeton: Princeton University Press.

[18] Torchinsky A. Real-variable Methods in Harmonic Analysis, 1986, Orlando: Academic Press.

[19] Uchiyama A. On the radial maximal function of distributions. Pacific J Math, 1986, 121(2): 467-483.