Sharpness of Complex Interpolation on $$\alpha $$-Modulation Spaces |
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Authors: | Weichao Guo Dashan Fan Huoxiong Wu Guoping Zhao |
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Institution: | 1.School of Mathematical Sciences,Xiamen University,Xiamen,People’s Republic of China;2.Department of Mathematics,University of Wisconsin-Milwaukee,Milwaukee,USA;3.Department of Mathematics,Zhejiang University,Hangzhou,People’s Republic of China |
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Abstract: | In this paper, we solve a long standing problem on the modulation spaces, \(\alpha \)-modulation spaces and Besov spaces. We establish sharp conditions for the complex interpolation between these function spaces. We show that no \(\alpha \)-modulation space \(M_{p,q}^{s,\alpha }\) can be regarded as the interpolation space between \(M_{p_1,q_1}^{s_1,\alpha _1}\) and \(M_{p_2,q_2}^{s_2,\alpha _2}\), unless \(\alpha _1\) is equal to \(\alpha _2\), essentially. Especially, our results show that the \(\alpha \)-modulation spaces can not be obtained by complex interpolation between modulation spaces and Besov spaces. |
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