For d=2,3, the paper proves the sharp bound I_{p,d}(u;N) << N^epsilon(N^{p-d(d+1)/2}+N^{p/2-u}) for 0<u<=1 and all p>0, with a conditional extension to d>=4.
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An extended Vinogradov's mean value theorem
For d=2,3, the paper proves the sharp bound I_{p,d}(u;N) << N^epsilon(N^{p-d(d+1)/2}+N^{p/2-u}) for 0<u<=1 and all p>0, with a conditional extension to d>=4.