## Journal of the Mathematical Society of Japan

### Boundedness of sublinear operators on product Hardy spaces and its application

#### Abstract

Let $p\in(0,\,1]$. In this paper, the authors prove that a sublinear operator $T$ (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spaces $H^{p}(\mbi{R}^{n}\times\mbi{R}^{m})$ to some quasi-Banach space $\mathcal{B}$ if and only if $T$ maps all $(p,\,2,\,s_{1},\,s_{2})$-atoms into uniformly bounded elements of $\mathcal{B}$. Here $s_{1}\ge\lfloor n(1/p-1)\rfloor$ and $s_{2}\ge\lfloor m(1/p-1)\rfloor$. As usual, $\lfloor n(1/p-1)\rfloor$ denotes the maximal integer no more than $n(1/p-1)$. Applying this result, the authors establish the boundedness of the commutators generated by Calderón-Zygmund operators and Lipschitz functions from the Lebesgue space $L^{p}(\mbi{R}^{n}\times\mbi{R}^{m})$ with some $p>1$ or the Hardy space $H^{p}(\mbi{R}^{n}\times\mbi{R}^{m})$ with some $p\le1$ but near 1 to the Lebesgue space $L^{q}(\mbi{R}^{n}\times\mbi{R}^{m})$ with some $q>1$.

#### Article information

Source
J. Math. Soc. Japan, Volume 62, Number 1 (2010), 321-353.

Dates
First available in Project Euclid: 5 February 2010

https://projecteuclid.org/euclid.jmsj/1265380433

Digital Object Identifier
doi:10.2969/jmsj/06210321

Mathematical Reviews number (MathSciNet)
MR2648225

Zentralblatt MATH identifier
1195.42060

#### Citation

CHANG, Der-Chen; YANG, Dachun; ZHOU, Yuan. Boundedness of sublinear operators on product Hardy spaces and its application. J. Math. Soc. Japan 62 (2010), no. 1, 321--353. doi:10.2969/jmsj/06210321. https://projecteuclid.org/euclid.jmsj/1265380433

#### References

• T. Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo, 18 (1942), 588–594.
• M. Bownik, Boundedness of operators on Hardy spaces via atomic decompositions, Proc. Amer. Math. Soc., 133 (2005), 3535–3542.
• S. A. Chang and R. Fefferman, A continuous version of duality of $H^{1}$ with BMO on the bidisc, Ann. of Math. (2), 112 (1980), 179–201.
• S. A. Chang and R. Fefferman, The Calderón-Zygmund decomposition on product domains, Amer. J. Math., 104 (1982), 455–468.
• S. A. Chang and R. Fefferman, Some recent developments in Fourier analysis and $H^{p}$-theory on product domains, Bull. Amer. Math. Soc. (N. S.), 12 (1985), 1–43.
• W. Chen, Y. Han and C. Miao, Bi-commutators of fractional integrals on product spaces, Math. Nachr., 281 (2008), 1108–1118.
• R. R. Coifman, A real variable characterization of $H^{p}$, Studia Math., 51 (1974), 269–274.
• R. Fefferman, Singular integrals on product domains, Bull. Amer. Math. Soc. (N. S.), 4 (1981), 195–201.
• R. Fefferman, Singular integrals on product $H^{p}$ spaces, Rev. Mat. Iberoamericana, 1 (1985), 25–31.
• R. Fefferman, Calderón-Zygmund theory for product domains: $H^{p}$ spaces, Proc. Nat. Acad. Sci. U. S. A., 83 (1986), 840–843.
• R. Fefferman, Harmonic analysis on product spaces, Ann. of Math. (2), 126 (1987), 109–130.
• R. Fefferman and E. M. Stein, Singular integrals on product spaces, Adv. in Math., 45 (1982), 117–143.
• S. H. Ferguson and M. T. Lacey, A characterization of product BMO by commutators, Acta Math., 189 (2002), 143–160.
• M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Mathematics, 79, the American Mathematical Society, Providence, R. I., 1991.
• J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, 1985.
• L. Grafakos, Classical Fourier Analysis, Second Edition, Graduate Texts in Math., 249, Springer, New York, 2008.
• Y. Han, A problem in $H^{p}(\mbi{R}^{2}_{+}\times \mbi{R}^{2}_{+})$ spaces, Chinese Sci. Bull., 34 (1989), 617–622.
• Y. Han and D. Yang, $H^{p}$ boundedness of Calderón-Zygmund operators on product spaces, Math. Z., 249 (2005), 869–881.
• G. Hu, D. Yang and Y. Zhou, Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type, Taiwanese J. Math., 13 (2009), 91–135.
• J.-L. Journé, Calderón-Zygmund operators on product spaces, Rev. Mat. Iberoamericana, 1 (1985), 55–91.
• R. H. Latter, A characterization of $H^{p}(\mbi{R}^{n})$ in terms of atoms, Studia Math., 62 (1978), 93–101.
• Y. Meyer and R. R. Coifman, Wavelets. Calderón-Zygmund and multilinear operators, Cambridge University Press, Cambridge, 1997.
• Y. Meyer, M. Taibleson and G. Weiss, Some functional analytic properties of the spaces $B_{q}$ generated by blocks, Indiana Univ. Math. J., 34 (1985), 493–515.
• J. Pipher, Journé's covering lemma and its extension to higher dimensions, Duke Math. J., 53 (1986), 683–690.
• E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N. J., 1970.
• M. H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Astérisque, 77 (1980), 67–149.
• K. Yabuta, A remark on the $(H^{1}, L^{1})$ boundedness, Bull. Fac. Sci. Ibaraki Univ. Ser. A, 25 (1993), 19–21.
• D. Yang and Y. Zhou, Boundedness of Marcinkiewicz integrals and their commutators in $H^{1}(\mbi{R}^{n}\times\mbi{R}^{m})$, Sci. China Ser. A, 49 (2006), 770–790.
• D. Yang and Y. Zhou, A boundedness criterion via atoms for linear operators in Hardy spaces, Constr. Approx., 29 (2009), 207–218.
• D. Yang and Y. Zhou, Boundedness of sublinear operators in Hardy spaces on RD-spaces via atoms, J. Math. Anal. Appl., 339 (2008), 622–635.