A Remark on Certain Integral Operators of Fractional Type

Rocha, Pablo Alejandro

Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 370-375

Résumé

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 $$\underset{i=1\,}{\overset{m}{\mathop{\oplus }}}\,\underset{1\le j\ne i\le m}{\mathop \bigcap }\,\,{{N}_{j}}\,=\,{{\mathbb{R}}^{n}},$$ 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 $${{T}_{r}}f(x)\,=\,{{\int }_{{{\mathbb{R}}^{n}}}}|x\,-\,{{A}_{1y}}{{|}^{-{{n}_{1}}+{{\alpha }_{{{1}_{\ldots }}}}}}|x\,-\,{{A}_{m}}y{{|}^{-{{n}_{m}}+{{\alpha }_{m}}}}f(y)dy,$$ ${{n}_{1}}\,+\,\cdots \,+\,{{n}_{m}}\,=\,n$ , $\frac{{{\alpha }_{1}}}{{{n}_{1}}}\,=\,\cdots \,=\,\frac{{{\alpha }_{m}}}{{{n}_{m}}}\,=\,r$ , $0\,<\,r\,<\,1$ , 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 $\frac{1}{q}\,=\,\frac{1}{p}\,-\,r$ .

Détail

DOI : 10.4153/CMB-2017-043-6

Mots-clés : 42B20, 42B30, integral operator, Hardy space