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Complete low-cut filter and the best constant of Sobolev inequality Hiroyuki Yamagishi, Yoshinori Kametaka, Atsushi Nagai, Kohtaro Watanabe and Kazuo Takemura 1 Tokyo Metropolitan College of Industrial Technology, 1-10-40 Higashi-ooi, Shinagawa, Tokyo 140-0011, Japan 2 Osaka University, 1-3 Machikaneyama-cho, Toyonaka 560-8531, Japan 3 Nihon University, 2-11-1 Shinei, Narashino 275-8576, Japan 4 National Defense Academy, 1-10-20 Yokosuka 239-8686, Japan