Oscillatory hyper Hilbert transforms along variable curves

摘要

For n = 2 or 3 and x is an element of Double-struck capital R-n; we study the oscillatory hyper Hilbert transformT alpha,beta f(x) along an appropriate variable curve Gamma(t; x) in Double-struck capital R-n (namely, Gamma(t; x) is a curve in Double-struck capital R-n for each fixed x), where alpha > beta > 0: We obtain some L-p boundedness theorems of T-alpha,T-beta, under some suitable conditions on alpha and beta. These results are extensions of some earlier theorems. However, T(alpha,beta)f(x) is not a convolution in general. Thus, we only can partially employ the Plancherel theorem, and we mainly use the orthogonality principle to prove our main theorems.