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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A note on norm inequalities for integral operators on cones


Author: Ke Cheng Zhou
Journal: Trans. Amer. Math. Soc. 347 (1995), 1033-1041
MSC: Primary 47G10; Secondary 44A15, 46E99
MathSciNet review: 1249897
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Abstract: Norm inequalities for the Riemann-Liouville operator $ {R_r}f(x) = \int_{\langle 0,x\rangle } {\Delta _V^{r - 1}(x - t)f(t)dt} $ and Weyl operator $ {W_r}f(x) = \int_{\langle x,\infty \rangle } {\Delta _V^{r - 1}(t - x)f(t)dt} $ on cones in $ {R^d}$ have been obtained in the case $ r \geqslant 1$ [7]. In this note, these inequalities are further extended to the case $ r < 1$. The question of whether the Hardy operator $ Hf(x) = \int_{\langle 0,x\rangle } {f(t)dt} $ on cones is bounded from $ {L^p}(\Delta _V^\alpha (X))$ to $ {L^q}(\Delta _V^\beta (x))\;(q < p)$ is also solved.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1249897-9
PII: S 0002-9947(1995)1249897-9
Article copyright: © Copyright 1995 American Mathematical Society