Singular and maximal Radon transforms: Analysis and geometry

Michael Christ, Alexander Nagel, Elias M. Stein and Stephen Wainger

Review from Zentralblatt MATH:

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 $(x_0, 0)\in \bbfR^n\times \bbfR^k$, with range in $\bbfR^n$. It is assumed that $\gamma$ satisfies several equivalent curvature conditions.

Let $K$ be a Calderón-Zygmund kernel in $\bbfR^k$. It implies that $K\in C^1(\bbfR^n/\{0\})$ 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 $$T(f)(x)= \psi(x) pv \int_{|t|\le a} f(\gamma_t(x)) K(t) dt,\tag 1$$ where $pv \int_{|t|\le a} g(t) dt= \lim_{\varepsilon\to 0} \int_{\varepsilon\le|t|\le a} g(t) dt$. 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 $\bbfR^n$, uniformly for every $|t|\le a'$, for some constant $a'> a$. Then the operator $$f\mapsto\psi(x) \int_{a\le|t|\le a'} f(\gamma_t(x)) K(t) dt$$ 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 $L^p(\bbfR^n)$ 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 $L^p(\bbfR^n)$ to itself, for every $1< p<\infty$.

A more general formulation of these theorems is also investigated.

Reviewed by C.L.Parihar

Keywords: Radon transforms; curvature conditions; Calderón-Zygmund kernel; bounded operator; submanifold

Classification (MSC2000): 44A12

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