Fourier multipliers for Sobolev spaces on the Heisenberg group |
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Authors: | S Jitendriya R Radha D Venku Naidu |
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Institution: | 1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India
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Abstract: | In this paper, it is shown that the class of right Fourier multipliers for the Sobolev space W k,p (H n ) coincides with the class of right Fourier multipliers for L p (H n ) for k ∈ ?, 1 < p < ∞. Towards this end, it is shown that the operators R j $ \bar R $ j ??1 and $ \bar R $ j R j ??1 are bounded on L p (H n ), 1 < p < ∞, where $$ R_j = \frac{\partial } {{\partial z_j }} - \frac{i} {4}\bar z_j \frac{\partial } {{\partial t}}, \bar R_j = \frac{\partial } {{\partial \bar z_j }} + \frac{i} {4}z_j \frac{\partial } {{\partial t}} $$ and ? is the sublaplacian on H n . This proof is based on the Calderon-Zygmund theory on the Heisenberg group. It is also shown that when p = 1, the class of right multipliers for the Sobolev space W k,1(H n ) coincides with the dual space of the projective tensor product of two function spaces. |
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