arXiv subjects

Himali Dabhi

Publications and source records attributed to Himali Dabhi.

2 recordsLinked to original sources

arXiv2026

A variation norm Carleson theorem in higher dimensions

In 1971, C. Fefferman established a higher dimensional extension of the celebrated Carleson--Hunt theorem which gives pointwise almost everywhere convergence of the partial Fourier sums of functions in $L^p(\mathbb T), 1 < p < \infty.$ More precisely, Fefferman proved a maximal function bound for polygonal Fourier partial sums of functions in $L^p(\mathbb T^d), p>1.$ In this note, we extend Fefferman's maximal function bound to strong $r$-variation norm bounds whenever $r>2$ as well as uniform $2$-oscillation bounds. Furthermore, for functions in $L^2(\mathbb T^d)$, we establish $r$-variational and $2$-oscillation bounds for partial Fourier sums over nested rectangles whenever $r>2$.

math.CA↗

arXiv2025

Multiparameter extensions of the Christ-Kiselev maximal theorem: strong variational bounds

For a linear operator $T$ bounded from $L^p(Y)$ to $L^q(X)$, the Christ-Kiselev theorem gives $L^p \to L^q$ bounds for the maximal function $T^{*}$ associated to filtrations on $Y$. This result has been extended by establishing bounds for the maximal function associated to a product of filtrations, also known as the multiparameter extension of the Christ-Kiselev theorem. In this note, we strengthen the multiparameter theorem by proving the $r$-variational bounds for the multiparameter trunctations when $r>p$. Furthermore, we replace $T$ by a multilinear operator to obtain a strong variational, multilinear, multiparameter extension of the Christ-Kiselev theorem.

math.CA↗