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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
DOI: https://doi.org/10.1090/S0002-9947-1988-0943305-7
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

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


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DOI: https://doi.org/10.1090/S0002-9947-1988-0943305-7
Article copyright: © Copyright 1988 American Mathematical Society