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A note on Hwp-boundedness of Riesz transforms and θ -Calderón-Zygmund operators through molecular characterization

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Let 0 < p ≤ 1 and w in the Muckenhoupt class A1. Recently, by using the weighted atomic decomposition and molecular characterization, Lee, Lin and Yang[11] established that the Riesz transforms Rj, j = 1,2, …,n, are bounded on Hwp(Rn). In this note we extend this to the general case of weight w in the Muckenhoupt class A through molecular characterization. One difficulty, which has not been taken care in [11], consists in passing from atoms to all functions in Hwp(Rn). Furthermore, the Hwp-boundedness of θ-Calderón-Zygmund operators are also given through molecular characterization and atomic decomposition.

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Correspondence to Luong Dang Ky.

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Ky, L.D. A note on Hwp-boundedness of Riesz transforms and θ -Calderón-Zygmund operators through molecular characterization. Anal. Theory Appl. 27, 251–264 (2011). https://doi.org/10.1007/s10496-011-0251-z

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Key words

  • Muckenhoupt weight
  • Riesz transform
  • Calderón-Zygmund operator

AMS (2010) subject classification

  • 42B20
  • 42B25
  • 42B30