On the growth of vector-valued Fourier series
Javier ParcetFernando SoriaQuanhua Xu
42A2046G10Growth of Fourier seriesUMD Banach spaces
Let $f:\mathbb{T}\to\mathrm{X}$ satisfy
\[
\int_{\mathbb{T}}\|f(x)\|_{\mathrm{X}}(\log^{+}\|f(x)\|_{\mathrm{X}})^{1+\delta\
}\dx<\infty,
\]
where $\mathrm{X}$ is a UMD Banach space and $\delta>0$. Then, we prove that
\[
\Big\|\sum_{|k|\le n}\hat{f}(k)e^{2\pi ikx}\Big\|_{\mathrm{X}}=o(\log\log n)\quad\mbox{for almost every }x\in\mathbb{T}.
\]
In other words, the "little Carleson theorem"${}$ holds for UMD-valued functions.
Indiana University Mathematics Journal
2013
text
pdf
10.1512/iumj.2013.62.5135
10.1512/iumj.2013.62.5135
en
Indiana Univ. Math. J. 62 (2013) 1765 - 1784
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