A NOTE ON SINGULAR INTEGRALS WITH DOMINATING MIXED SMOOTHNESS IN TRIEBEL-LIZORKIN SPACES

doi:10.1016/S0252-9602(14)60087-4

Abstract

Abstract:

Let h be a measurable function defined on R⁺×R⁺. Let Ω∈ L(log L⁺)^ν_q (Sⁿ¹⁻¹×Sⁿ²⁻¹) (1 ≤ ν_q ≤2) be homogeneous of degree zero and satisfy certain cancellation condi-tions. We show that the singular integral

Tf(x₁, x₂) = p. v.∫∫_Rn1+n2Ω(y´₁, y´₂)h(|y₁|, |y₂|)/|y₁|ⁿ¹ |y₂|ⁿ²f(x₁ − y₁, x₂ − y₂)dy₁dy₂maps from S^α_¹, ^α^₂_{p, q}F(R^n₁ × R^n₂) boundedly to itself for 1 < p, q <∞, α₁, α₂ ∈R.

Key words: singular integrals, Marcinkiewicz integrals, mixed smoothness, Triebel-Lizorkin spaces

CLC Number:

42B20

Hung Viet LE. A NOTE ON SINGULAR INTEGRALS WITH DOMINATING MIXED SMOOTHNESS IN TRIEBEL-LIZORKIN SPACES[J].Acta mathematica scientia,Series B, 2014, 34(4): 1331-1344.

Trendmd

References

[1] Al-Salman A, Al-Jarrah A A. On rough singular integrals along surfaces on product domains. Turk J Math, 2005, 29: 275–285

[2] Al-Salman A, Al-Qassem H, Pan Yibiao. Singular integrals on product domains. Indiana Univ Math J, 2006, 55(1): 369–387

[3] Al-Salman A, Al-Qassem H, Cheng L C, Pan Y. Lp bounds for the function of Marcinkiewicz. Math Res Lett, 2002, 9: 697–700

[4] Al-Qassem H, Pan Yibiao. Lp boundedness for singular integrals with rough kernels on product domains. Hokkaido Mathe J, 2002, 31: 555–613

[5] Al-Qassem H, Ali M. Lp boundedness for singular integral operators with L(log L⁺)² kernels on product spaces. Kyungpook Math J, 2006, 46: 377–387

[6] Chen Jiecheng, Fan Dashan, Ying Yiming. Singular integral operators on function spaces. J Math Anal Appl, 2002, 276(2): 691–708

[7] Chen Larry, Le H V. Singular integrals with mixed homogeneity in product spaces. Math Ineq Appl, 2011, 14(1): 155–172

[8] Duoandikoetxea J. Multiple singular integrals and maximal functions along hypersurfaces. Annales de l’institut Fourier, 1986, 36(4): 185–206

[9] Duoandikoetxea J, de Francia J L R. Maximal and singular integral operators via Fourier transform estimates.
Invent Math, 1986, 84(3): 541–561

[10] Fan Dashan, Wu Huoxiong. On the generalized Marcinkiewicz integral operators with rough kernels. Canad Math Bull, 2011, 54(1): 100–112

[11] Fefferman R. Singular integrals on product spaces. Adv Math, 1982, 45: 117–143

[12] Schmeisser H -J. Recent developments in the theory of function spaces with dominating mixed smoothness//
Nonlinear Analysis, Function Spaces and Applications. Proceedings of the Spring School Held in Prague, May 30-June 6, 2006. Vol 8. Czech Academy of Sciences, Mathematical Institute, Praha, 2007: 145–204.

[13] Schmeisser H -J, Triebel H. Topics in Fourier Analysis and Function Spaces. Wiley: Chister, 1987

[14] Triebel H. Theory of Function Spaces, Monographs in Mathematics, 78. Basel: Birkh¨auser Verlag, 1983

[15] Walsh T. On the function of Marcinkiewicz. Studia Math, 1972, 78: 203–217