Consider the oscillatory hyper-Hilbert transform Hn,α,βf（x）=∫0^1 f（x-Г（t））e^it-βt^-1-α dt along the curve P（t） = （tp1, tP2,..., tpn）, where β 〉 α ≥ 0 and 0 〈 p1 〈 p2 〈 ... 〈 Pn. We prove that H n,α,β is bounded on L2 if and only if β ≥ （n ＋ 1）α. Our work extends and improves some known results.
