On singular integrals along surfaces related to black spaces

Abstract

Leth(t) be an arbitrary bounded radial function and let Γ(x) be a real measurable and radial function defined onR ⁿ⁻¹. Forx, y∈R ⁿ⁻¹, we establish that the singular integral along surfacex → (x, Γ(x)):

$$Tf(x,x_n ) = p.\upsilon .\smallint h(y)\frac{{\Omega (y)}}{{|y|^{n - 1} }}f(x - y,x_n - \Gamma (y))dy,$$

and the associated maximal singular integral are bounded inL ^p(R ⁿ) for 1<p<∞,n≥3, provided that the maximal operator

$${\rm M}_\Gamma g(x_n ) = \mathop {\sup }\limits_r \frac{1}{r}\smallint _{r/2< |t| \leqslant r} |f(x_n - \Gamma (t))|dt$$

is bounded onL ^p (R) for all 1<p.