tailieunhanh - Báo cáo toán học: "Hardy and Rellich inequalities in non-integral dimension "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hardy và Rellich sự bất bình đẳng trong chiều kích không thể tách rời. | Copyright by INCREST 1985 J. OPERATOR THEORY 9 1983 143-146 HARDY AND RELLICH INEQUALITIES IN NON-INTEGRAL DIMENSION BARRY SIMON In 1 Davies asked the following question. Let Ho A on L2 R3 . Let 1 F x with c 0 a 0. H Ho V can always be defined as a self-adjoint operator via a form sum 4 5 When is it true that 2 D H D H Ọ D V 1 Davies found the striking answer 2 holds if a 3 2 or a 2 but fails if 3 2 a 2. For the borderline case a 2 he found that 2 is false if 0 c 3 4 and true if c 3 2. He left open the case a 2 3 4 c 3 2. Our original goal was to fill in this gap and we will give an analysis of all the a 2 situations. We will concentrate on the question of whether the operator sum Hữ V is a closed operator on D 7 o n D K . That D o n D V is a core for H in the cases in this paper where it is closed is known see . 5 vol. II p. 161 so 2 will be established by this closure result. It is an observation of Glimm Jaffe 2 that Hữ V is closed on D He n D V if and only if for some a b 0 3 IIFrpil Il- in fact for the case 4 y cỊxị-2 H A- -V we will prove 3 with optimal constants Theorem 1. On R3 3 holds with V H given by 4 if and only if c 3 4 and in that case it holds with b 0 and a -- c c 3 4 -1 and is false with any smaller constant. The first observation is that V and FT have the same properties under scaling this will be critical later also. That is if w 2 x 2 ỉ 44 BARRY SIMON then m z KmU -1 - Z-2K ii . Hh z - - Thus if 3 holds with any b it also holds with the same a b for 2-2F and z-27f O - equivalently for the same a but for b replaced by bX . Taking z to zero we find that if 3 holds it holds for b 0. Thus we are reduced to seeing if 3 WK w. . But since H has no kernel this is equivalent to asking if 3 ỊịFỉf-1ịi a . we must see if VH-1 is bounded and if so to compute its norm. Since J7 -1 computes with rotations and dilations this will be easy. In the usual way VHis a direct sum of operators on subspaces of the form rg r y m ge L2 0 oo dr . Thus we .