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

Abstract

In this paper, for general plane curves γ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus L p (R 2 )-boundedness of the Hilbert transforms H ∞ U,γ along the variable plane curves (t, U(x 1 , x 2 )γ(t)) and the L p (R 2 )-boundedness of the corresponding maximal functions M ∞ U,γ , where p > 2 and U is a measurable function. The range on p is sharp. Furthermore, for 1 < p ≤ 2, under the additional conditions that U is Lipschitz and making a ε 0 -truncation with γ(2ε 0 ) ≤ 1/4 U Lip , we also obtain similar boundedness for these two operators H ε 0 U,γ and M ε 0 U,γ .

Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition

“…Street [Str17] later studied the boundedness in Sobolev spaces of the "multi-parameter" singular Radon transforms. Some other related works include [NRSW12], [RS92], [MRS95], [VW20], [Gre16], [BT13], [Wan20], [Vit17], [She18], [Roo17], [LSY20], [LY18], [Hu20], [HL18], [GHLR17], [Guo17b], [Guo15b], [Guo17a], [Guo15a], [DLZ19], [DL20], [Chr20], [CL18], [Ou17], [DPP18], [Bat13], [APP19], [PS97], [BLMW18], [CHKY08], [FGW08], [Var76], [SW78], [NRS01], [ZK19], [Lie20], [Gre07a], [Hel66], [Gre01b], [Gre01a], [NRW76], [TW03], [PS86], [See98], [Sto11], [Sto14], [Gre19b], ...…”