MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS ON PRODUCT SPACES
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概要
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"Suppose that $¥Omega(x^{¥prime}, y^{¥prime})¥in L^{1}(S^{n-1}¥times S^{m-1})$ is ahomogeneous function of degree zero $satis¥Phi ing$ the mean zero propeIty (1.1), and that $h(s,t)$ ls a bounded function on $Rx$ R. The Marcinkiewicz integral operator $m(f)$ along a continuous surface $¥gamma(u, v)$ on the product space $R^{n}xR^{m}(n¥geq 2, m¥geq 2)$ is defined by $¥iota_{O}f(¥xi, ¥eta, z)=(¥int_{R}¥int_{R}|F_{l,¥epsilon}(x, y, z)|^{2}2^{-2t-2¥epsilon}dtds)^{1/2}$ where $F_{t,e}(¥xi, ¥eta, z)$ $=¥int_{|_{¥nu}^{x}|^{:}}<2h(|x|, |y|)|x|^{-n+1}|y|^{-m+1}¥Omega(x^{¥prime}, y^{¥prime})f(¥xi-x,¥eta-y, z-¥gamma(|x|, |y|))dxdy<2$ We prove that the operator $¥nu_{¥Omega}f$ is bounded on $L^{P}(R^{n} xR^{m}xR),$ $p¥in(1, ¥infty)$ , provided that $¥Omega$ is a function in certain block space $B_{q}^{0,1}(S^{n-1}xS^{m-1})$ for some $q>1$ and that two lower dimensional maximal functions related to $¥gamma$ are bounded on $L^{p}$ . These two lower dimensional maximal functions are natural extension of a well-known maximal function along curves."
- 2001-00-00
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