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A Remark on Certain Integral Operators of Fractional Type
Published online by Cambridge University Press: 20 November 2018
Abstract
For $m,n\,\in \,\mathbb{N},\,1\,<\,m\le \,n$, we write $n={{n}_{1}}+\cdots +{{n}_{m}}$ where $\{{{n}_{1}},\ldots \,,{{n}_{m}}\}\,\subset \,\mathbb{N}$. Let ${{A}_{1}}\,,\,\ldots \,,\,{{A}_{m}}$ be $n\,\times \,n$ singular real matrices such that
where ${{N}_{j}}\,=\,\{x\,:\,{{A}_{j}}x\,=\,0\},\,\text{dim(}{{N}_{j}}\text{)}\,=\,n\,-\,{{n}_{j}}$, and ${{A}_{1}}\,,\,\ldots \,,\,{{A}_{m}}$ is invertible. In this paper we study integral operators of the form
${{n}_{1}}\,+\,\cdots \,+\,{{n}_{m}}\,=\,n$ , $\frac{{{\alpha }_{1}}}{{{n}_{1}}}\,=\,\cdots \,=\,\frac{{{\alpha }_{m}}}{{{n}_{m}}}\,=\,r$, $0\,<\,r\,<\,1$, and the matrices ${{A}_{i}}\text{ }\!\!'\!\!\text{ s}$ are as above. We obtain the ${{H}^{p}}({{\mathbb{R}}^{n}})\,-\,{{L}^{q}}({{\mathbb{R}}^{n}})$ boundedness of ${{T}_{r}}$ for $0\,<\,p\,<\,\frac{1}{r}$ and $\frac{1}{q}\,=\,\frac{1}{p}\,-\,r$.
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- Copyright © Canadian Mathematical Society 2018
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