Hokkaido Mathematical Journal

Certain bilinear operators on Morrey spaces

Dashan FAN and Fayou ZHAO

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Abstract

In this paper, we consider that $T(f,g)$ is a bilinear operator satisfying \begin{equation*} |T(f,g)(x)|\preceq \int_{\mathbb{R}^{n}}\frac{|f(x-ty)g(x-y)|}{|y|^{n}}dy \end{equation*} for $x$ such that $0\notin {\rm supp}~(f(x-t\cdot )) \cap {\rm supp}~(g(x+\cdot ))$. We obtain the boundedness of $T(f,g)$ on the Morrey spaces with the assumption of the boundedness of the operator $T(f,g)$ on the Lebesgues spaces. As applications, we yield that many well known bilinear operators, as well as the first Calderón commutator, are bounded from the Morrey spaces $L^{q,\lambda_{1}}\times L^{r,\lambda_{2}}$ to $L^{p,\lambda}$, where $\lambda /p={\lambda_{1}}/{q}+{\lambda_{2}}/{r}$.

Article information

Source
Hokkaido Math. J., Volume 47, Number 1 (2018), 143-159.

Dates
First available in Project Euclid: 13 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1520928063

Digital Object Identifier
doi:10.14492/hokmj/1520928063

Mathematical Reviews number (MathSciNet)
MR3773728

Zentralblatt MATH identifier
06853594

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory 42B35: Function spaces arising in harmonic analysis

Keywords
Multilinear operators bilinear Hilbert transform the first Calderón commutator Morrey spaces

Citation

FAN, Dashan; ZHAO, Fayou. Certain bilinear operators on Morrey spaces. Hokkaido Math. J. 47 (2018), no. 1, 143--159. doi:10.14492/hokmj/1520928063. https://projecteuclid.org/euclid.hokmj/1520928063


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