摘要
For n = 2 or 3 and x ∈ Rn, we study the oscillatory hyper Hilbert transform Tα,β f(x)=∫Rf(x-Г(t,x))e^-i|t|-β|t|^-1-α dt along an appropriate variable curveГ(t, x) in R^n (namely,Г(t, x) is a curve in R^n for each fixed x), where α〉β〉0. We obtain some LP boundedness theorems of Tα,β, under some suitable conditions on α and β. These results are extensions of some earlier theorems. However, Tα,β 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.
基金
This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11671363, 11371316, 11771388).
