Rough maximal bilinear singular integrals

• Autores: Eva Buriánková, Petr Honzík
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 70, Fasc. 3, 2019, págs. 431-446
• Idioma: inglés
• DOI: 10.1007/s13348-019-00239-4
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• Resumen

  • We study the rough maximal bilinear singular integral \begin{aligned} T^{*}_\varOmega (f,g)(x)=\! \sup _{\varepsilon >0}\left| \int _{\mathbb {R}^{n}\setminus B\left( 0,\varepsilon \right) }\! \int _{\mathbb {R}^{n}\setminus B\left( 0,\varepsilon \right) }\!\frac{ \varOmega ((y,z)/|(y,z)|)}{ |(y,z)|^{2n}}f(x-y)g(x-z) dydz\right| , \end{aligned} where \varOmega is a function in L^\infty (\mathbb S^{2n-1}) with vanishing integral. We prove it is bounded from L^p\times L^q\rightarrow L^r, where 1 p,q \infty and 1/r=1/p+1/q. We also discuss results for \varOmega \in L^s(\mathbb S^{2n-1}), 1 s \infty .