Title:

Some problems associated with sum and integral inequalities

In 2 , the following extension of the higher order Rellich inequality / AV(x) 2>7(n,a,i) / /(x) 2 (1) JRn lxl JRn lxl was proven by W. Allegretto for all / G C£ (Rn {0}). The constant 7 is calculated explicitly by the author for all n > 2, a > 0 and j 6 N, giving the value of the constant in the previously unknown case n < a + 4j. Hence proving that 7 is equal to zero if and only if n < a + 4j and n a = 0 (mod 2). In this problematic case, the author finds that the higher order Rellich inequality (1) can be forced to be nontrivial if further restrictions are placed on the function in n_1. An alternative method to restricting the functional class is to look at the Rellich type inequality / AA/(x) 2 >*(n>a,4) / l/(x) 2 (2) JRn lxl JRn lxl found by W.D. Evans and R.T. Lewis in 15 for n = 2,3,4. The magnetic Laplacian is of the form Aa = (V zA)2 where in spherical coordinates H*(*i)ei ifn = 2, with e L (0,27r) and (0) = (2r). The potential A is of Aharonov Bohm type and the constant $ is dependant upon the distance of the magnetic flux to the integers Z. By finding the discrete spectrum of the Friedrichs extension of A a in L2(Sn_1), the author is able to extend the Rellich type inequality (2) to all n > 2 and a > 0. Consequently, the higher order Rellich type inequality / Ai/(x) a>n(n,a,*,j)/ l/(x) 2 (4) JRn Ix x j can be constructed. The inequality (4) is shown to be nontrivial for all n < a + and n a = 0 (mod 2), the previously problematic case. The Rellich type inequality (4) enables an analysis of the spectral properties of perturbations of the magnetic operator AA to be undertaken in L2(IRn), n > 2. Furthermore, a CLR type bound for the number of negative eigenvalues of the operator AA can be found in L2(R8), a space in which there is no CLR bound for the operator A4.
