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**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.

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.

Reviewer: C.L.Parihar (Indore)

### MSC:

44A12 | Radon transform |