English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Let $n\ge 1$ and $\mathfrak{T}_{m}$ be the bilinear square Fourier multiplieroperator associated with a symbol $m$, which is defined by $$\mathfrak{T}_{m}(f_1,f_2)(x) = \biggl(\int_{0}^\infty\Big|\int_{(\mathbb{R}^n)^2} e^{2\pi ix\cdot (\xi_1 +\xi_2)}m(t\xi_1,t\xi_2) \hat{f}_{1}(\xi_1)\hat{f}_{2}(\xi_2)d\xi_1d\xi_2\Big|^2\frac{dt}{t } \biggr)^{\frac 12}. $$ Let $s$ be an integer with$s\in[n+1,2n]$ and $p_0$ be a number satisfying $2n/s\le p_0\le 2$. Supposethat $\nu_{\vec{\omega}}=\prod_{i=1}^2\omega_i^{p/ p_i}$ and each $\omega_i$ isa nonnegative function on $\mathbb{R}^n$. In this paper, we show that$\mathfrak{T}_{m}$ is bounded from $L^{p_1}(\omega_1)\times L^{p_2}(\omega_2)$to $L^p(\nu_{\vec{\omega}})$ if $p_0< p_1, p_2<\infty$ with $1/p=1/p_1+ 1/p_2$.Moreover, if $p_0>2n/s$ and $p_1=p_0$ or $p_2=p_0$, then $\mathfrak{T}_{m}$ isbounded from $L^{p_1}(\omega_1)\times L^{p_2}(\omega_2)$ to$L^{p,\infty}(\nu_{\vec{\omega}})$. The weighted end-point $L\log L$ typeestimate and strong estimate for the commutators of $\mathfrak{T}_{m}$ are alsogiven. These were done by considering the boundedness of some relatedmultilinear square functions associated with mild regularity kernels andessentially improving some basic lemmas which have been used before.

On the bilinear square Fourier multiplier operators and related multilinear square functions