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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, Calderón’s theorem, and analysis of Lipschitz curves, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 104–122. MR 576041
  • [3] Margaret A. M. Murray, Multilinear convolutions and transference, Michigan Math. J. 31 (1984), no. 3, 321–330. MR 767611, 10.1307/mmj/1029003076
  • [4] Daniel M. Oberlin, A multilinear Young’s inequality, Canad. Math. Bull. 31 (1988), no. 3, 380–384. MR 956371, 10.4153/CMB-1988-054-0
  • [5] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095

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Article copyright: © Copyright 1988 American Mathematical Society