Estimates for Hilbert transforms along variable general curves

We consider the problem of proving L-2(R-2) boundedness and single annulus L-p(R-2) estimate for the Hilbert transform along variable general curve (t, u(1) (x(1))t + u(2) (x(1)) gamma(t)) H-u1,H-u2,H-gamma f(x(1), x(2)):-p.v. integral(infinity)(-infinity) f(x(1)-t, x(2)-u(1)(x(1))t-u(2)(x(1))gamma(t))dt/t ,for all (x(1), x(2))is an element of R-2, where p is an element of (1, infinity), gamma is a general curve on R, u(1) : R -> R and u(2) : R -> R are measurable functions. Moreover, all the bounds are independent of the measurable functions u(1) and u(2). For any given p is an element of (1, infinity), we also obtain the L-p (R) boundedness of the corresponding Carleson operator C(N1,N2,gamma)f (x):=- sup(N1,N2 is an element of R)vertical bar p. v. integral(infinity)(-infinity)e(iN1t+iN2-gamma(t)) f (x - t) dt/t vertical bar, for all x is an element of R. (C) 2020 Elsevier Inc. All rights reserved.