## Singular and maximal Radon transforms: Analysis and geometry.(English)Zbl 0960.44001

The paper is devoted to prove the $$L^p$$ boundedness of singular Radon transforms and their maximal analogues. Let $$\gamma$$ be a $$C^\infty$$ mapping $$(x,t)\mapsto \gamma(x,t)= \gamma_t(x)$$ defined in a neighbourhood of the point $$(x_0, 0)\in \mathbb{R}^n\times \mathbb{R}^k$$, with range in $$\mathbb{R}^n$$. It is assumed that $$\gamma$$ satisfies several equivalent curvature conditions.
Let $$K$$ be a Calderón-Zygmund kernel in $$\mathbb{R}^k$$. It implies that $$K\in C^1(\mathbb{R}^n/\{0\})$$ is homogeneous of degree $$k$$ and satisfies $$\int_{|t|=1} K(T) d\sigma(t)= 0$$. A nonnegative $$C^\infty$$ cut-off function $$\psi$$, supported near $$x_0$$, and a small positive constant $$a$$ are also chosen. Then the singular Radon transform $$T$$, defined initially for compactly supported $$C^1$$ functions by $T(f)(x)= \psi(x) pv \int_{|t|\leq a} f(\gamma_t(x)) K(t) dt,\tag{1}$ where $$pv \int_{|t|\leq a} g(t) dt= \lim_{\varepsilon\to 0} \int_{\varepsilon\leq|t|\leq a} g(t) dt$$. It is also assumed that the map $$x\mapsto \gamma_t(x)$$ is a diffeomorphism from a neighborhood of the support of $$\psi$$ to an open subset of $$\mathbb{R}^n$$, uniformly for every $$|t|\leq a'$$, for some constant $$a'> a$$. Then the operator $f\mapsto\psi(x) \int_{a\leq|t|\leq a'} f(\gamma_t(x)) K(t) dt$ is bounded on $$L^p$$ for every $$p\in [1,\infty]$$. The author proves the following theorems:
Theorem 11.1: Suppose that $$\gamma$$ satisfies the curvature conditions and that $$K$$ is as above. Then the operator $$T$$ defined by (1) extends to a bounded operator from $$L^p(\mathbb{R}^n)$$ to itself, for every $$1< p< \infty$$.
Theorem 11.2: Suppose that $$\gamma$$ satisfies the curvature conditions in a neighbourhood of the support of $$\psi$$. Then the corresponding maximal operator $M(f)(x)= \sup_{r> 0}{1\over r^k} \Biggl|\int_{M_x\cap B(x,r)} f(y) d\sigma_x(y)\Biggr|,$ where $$B(x,r)$$ is the ball of radius $$r$$ centered at $$x$$, and $$M_x$$ is a smooth $$k$$-dimensional submanifold with $$x\in M_x$$, extends to a bounded operator from $$L^p(\mathbb{R}^n)$$ to itself, for every $$1< p<\infty$$.
A more general formulation of these theorems is also investigated.