Some improved Caffarelli-Kohn-Nirenberg inequalities

For 1 < p < N and -infinity < gamma < N-p/p we obtain the following improved Hardy-Sobolev Inequalities integral(Omega)vertical bar del phi vertical bar(p)vertical bar x vertical bar(-p gamma)dx - (N - p(gamma + 1)/p)(P) integral(Omega) vertical bar phi vertical bar(p) /vertical bar x vertical bar(p(gamma+1)) dx >= C(p, q, r, gamma, vertical bar Omega vertical bar) (integral(Omega) vertical bar del phi vertical bar(q) vertical bar x vertical bar(-r gamma)dx) (p/q), where 1 < q < p and q <= r < infinity if gamma <= 0.1 <= r < p + p(N, p, q, gamma) if gamma > 0, for some positive constant p(N, p, q, gamma). Also we give an alternative proof of the optimal improved inequality for p = 2 by Wang-Willem in [16].