Hilbert transforms along double variable fractional monomials

Abstract

In this paper, we obtain the L 2 (R 2) boundedness and single annulus L p (R 2) estimate for the Hilbert transform H α,β along double variable fractional monomial u 1 (x 1)[t] α + u 2 (x 1)[t] β H α,β f (x 1 , x 2) := p. v. ∞ −∞ f (x 1 − t, x 2 − u 1 (x 1)[t] α − u 2 (x 1)[t] β) dt t with the bounds are independent of the measurable function u 1 and u 2. At the same time, we also obtain the L p (R) boundedness of the corresponding Carleson operator C α,β f (x) := sup N 1 ,N 2 ∈R p. v.

$L^p(\mathbb{R}^2)$-boundedness of Hilbert Transforms and Maximal Functions along Plane Curves with Two-variable Coefficients

“…Then, Bennett [4] extended the L 2 (R 2 ) results of [10] to the case U(x 1 , x 2 ) = P(x 1 ), where P(x 1 ) is a polynomial. Some other related results about the one-variable coefficient case, we refer to [8,30,13,39].…”