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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Multilinear convolutions defined by measures on spheres

Author: Daniel M. Oberlin
Journal: Trans. Amer. Math. Soc. 310 (1988), 821-835
MSC: Primary 42A85; Secondary 42B15
MathSciNet review: 943305
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Abstract: Let $ \sigma $ be Lebesgue measure on $ {\Sigma _{n - 1}}$ and write $ \sigma = ({\sigma _1}, \ldots ,{\sigma _n})$ for an element of $ {\Sigma _{n - 1}}$. For functions $ {f_1}, \ldots ,{f_n}$ on $ {\mathbf{R}}$, define

$\displaystyle T({f_1}, \ldots ,{f_n})(x) = \int_{{\Sigma _{n - 1}}} {{f_1}(x - {\sigma _1}) \cdots {f_n}(x - {\sigma _n})\,d\sigma ,\qquad x \in {\mathbf{R}}.} $

This paper partially answers the question: for which values of $ p$ and $ q$ is there an inequality

$\displaystyle \vert\vert T({f_1}, \ldots ,{f_n})\vert{\vert _q} \leqslant C\vert\vert{f_1}\vert{\vert _p} \cdots \vert\vert{f_n}\vert{\vert _p}?$

References [Enhancements On Off] (What's this?)

  • [1] J. Bergh and J. Löfström, Interpolation spaces, Springer-Verlag, Berlin, 1976.
  • [2] R. R. Coifman and Y. Meyer, Fourier analysis of multilinear convolutions, Caldéron's theorem, and analysis on Lipschitz curves, Euclidean Harmonic Analysis (College Park, Md., 1979), Lecture Notes in Math., vol. 779, Springer-Verlag, Berlin, 1980, pp. 104-122. MR 576041 (81g:42021)
  • [3] M. Murray, Multilinear convolutions and transference, Michigan Math. J. 31 (1984), 321-330. MR 767611 (86c:42003)
  • [4] D. Oberlin, A multilinear Young's inequality, Canad. Math. Bull. (to appear). MR 956371 (90b:43003)
  • [5] E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. MR 0290095 (44:7280)

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