Integral Equations and Operator Theory

, Volume 29, Issue 3, pp 261-268

First online:

On singular integrals along surfaces related to black spaces

  • Lung-Kee ChenAffiliated withDepartment of Mathematics, Oregon State University
  • , Dashan FanAffiliated withDepartment of Mathematical Sciences, University of Wisconsin-Milwaukee

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


Leth(t) be an arbitrary bounded radial function and let Γ(x) be a real measurable and radial function defined onR n−1. Forx, yR 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.

1991 Mathematics Subject Classification

42B20 42B25