Oscillatory hyper-Hilbert transform along curves on modulation spaces

Abstract

We consider the boundedness of the n-dimension oscillatory hyper-Hilbert transform along homogeneous curves on the α-modulation spaces, including the inhomogeneous Besov spaces and the classical modulation spaces. The main theorems signicantly improve some known results.

Keywords

Oscillatory hyper-Hilbert transform / inhomogeneous Besov spaces / -modulation spaces / homogeneous curves

References

[1] Borup L, Nielsen M. Boundedness for pseudodifferential operators on multivariate α-modulation spaces. Ark Mat, 2006, 44: 241–259

[2] Chandarana S. L^p-bounds for hypersingular integral operators along curves. Pacific J Math, 1996, 175(2): 389–416

[3] Chen J C, Fan D S, Wang M, Zhu X R. L^p bounds for oscillatory Hyper-Hilbert transform along curves. Proc Amer Math Soc, 2008, 136(9): 3145–3153

[4] Chen J C, Fan D S, Zhu X R. Sharp L² boundedness of the oscillatory hyper-Hilbert transform along curves. Acta Math Sin (Engl Ser), 2010, 26(4): 653–658

[5] Cheng M F. Hypersingular integral operators on modulation spaces for 0<p<1. J Inequal Appl, 2012, 165,

[6] Cheng M F, Zhang Z Q. Hypersingular integrals along homogeneous curves on modulation spaces. Acta Math Sinica (Chin Ser), 2010, 53(3): 531–540 (in Chinese)

[7] Gröbner P. Banachraume glatter funtionen and zerlegungsmethoden. Ph D Thesis, Univ of Vienna, Austria, 1992

[8] Guo W C, Fan D S, Wu H X, Zhao G P. Sharpness of complex interpolation on α-modulation spaces. J Fourier Anal Appl, 2016, 22(2): 427–461

[9] Han J S, Wang B X.α modulation spaces (I) embedding, interpolation and algebra properties. J Math Soc Japan, 2014, 66(4): 1315–1373

[10] Huang Q, Chen J C. Cauchy problem for dispersive equations in α-modulation spaces. Electron J Differential Equations, 2014, (158): 1–10

[11] Huang Q, Fan D S, Chen J C. Critical exponent for evolution equations in modulation spaces. J Math Anal Appl, 2016, 443: 230–242

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

[13] Wang B X, Huo Z H, Hao C C, Guo Z H. Harmonic Analysis Method for Nonlinear Evolution Equations. I. Hackensack: World Scientific, 2011

[14] Wu X M, Chen J C. Boundedness of fractional integral operators on α-modulation spaces. Appl Math J Chinese Univ, 2014, 29(3): 339–351

[15] Wu X M, Yu X. Strongly singular integrals along curves on α-modulation spaces. J Inequal Appl, 2017, 185,

[16] Zhao G P, Chen J C, Fan D S, Guo W C. Unimodular Fourier multipliers on homogeneous Besov spaces. J Math Anal Appl, 2015, 425: 536–547

[17] Zielinski M. Highly oscillatory singular integrals along curves. Ph D Thesis, Univ of Wisconsin-Madison, WI, USA, 1985

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature