arXiv subjects

Guixiang Hong

Publications and source records attributed to Guixiang Hong.

60 records · Page 4Linked to original sources

arXiv2014

The behavior of the bounds of matrix-valued maximal inequality in $\mathbb{R}^n$ for large $n$

In this paper, we study the behavior of the bounds of matrix-valued maximal inequality in $\mathbb{R}^n$ for large $n$. The main result of this paper is that the $L_p$-bounds ($p>1$) can be taken to be independent of $n$, which is a generalization of Stein and Strömberg's resut in the scalar-valued case. We also show that the weak type $(1,1)$ bound has similar behavior as Stein and Stömberg's.

math.FA↗

arXiv2014

The behaviour of square functions from ergodic theory in $L^{\infty}$

In this paper, we analyze carefully the behaviour in $L^\infty(\mathbb R)$ of the square functions $S$ and $S_\mathcal I$'s, originating from ergodic theory. Firstly, we show that we can find some function $f\in L^\infty(\mathbb{R})$, such that $Sf$ equals infinity on a nonzero measure set. Secondly, we can find compact supported function $f\in L^\infty(\mathbb{R})$ and $\mathcal I$ such that $S_\mathcal{I} f$ does not belong to $BMO$ space. Finally, we show that $S$ is bounded from $L^{\infty}_c$ to $BMO$ space. As a consequence, we solve an open question posed by Jones, Kaufman, Rosenblatt and Wierdl in \cite{JKRW98}. That is, $S_\mathcal I$ are uniformly bounded in $L^p(\mathbb R)$ with respect to $\mathcal I$ for $2<p<\infty$.

math.CA↗

arXiv2014

Banach lattice-valued $q$-variation and convexity

In this paper, we show that the $q$-variation for differential operator is not bounded in $L^p(\mathbb{R};L^{\infty}(\mathbb{R}))$ for any $1<p<\infty$. As a consequence, the $q$-variation operator can not be used to characterize the Hardy-Littlewood property of the underlying Banach lattice. Moreover, for Köthe function spaces $X$ with $X^*$ norming such that $X$ is $r$-convex for some large $r$, and $X$ is not $s$-convex for any $s$, $r<s<\infty$, we obtain lower bounds of the $(L^p(\mathbb{R};X),L^p(\mathbb{R};X)$-bounds of the $q$-variation operator, which tends to $\infty$, as $r$ tends to $\infty$.

math.FA↗

arXiv2012

John-Nirenberg inequality and atomic decomposition for noncommutative martingales

In this paper, we study the John-Nirenberg inequality for BMO and the atomic decomposition for H1 of noncommutative martingales. We first establish a crude version of the column (resp. row) John-Nirenberg inequality for all 0 < p < \infty. By an extreme point property of Lp -space for 0 < p \leq 1, we then obtain a fine version of this in equality. The latter corresponds exactly to the classical John-Nirenberg inequality and enables us to obtain an exponential integrability inequality like in the classical case. These results extend and improve Junge and Musat's John-Nirenberg inequality. By duality, we obtain the corresponding q-atomic decomposition for different Hardy spaces H1 for all 1<q\leq\infty, which extends the 2-atomic decomposition previously obtained by Bekjan et al. Finally, we give a negative answer to a question posed by Junge and Musat about BMO.

math.FA↗

arXiv2012

Calderón-Zygmund operators associated to matrix-valued kernels

Calderón-Zygmund operators with noncommuting kernels may fail to be Lp-bounded for $p \neq 2$, even for kernels with good size and smoothness properties. Matrix-valued paraproducts, Fourier multipliers on group vNa's or noncommutative martingale transforms are frameworks where we find such difficulties. We obtain weak type estimates for perfect dyadic CZO's and cancellative Haar shifts associated to noncommuting kernels in terms of a row/column decomposition of the function. Arbitrary CZO's satisfy $H_1 \to L_1$ type estimates. In conjunction with $L_\infty \to BMO$, we get certain row/column Lp estimates. Our approach also applies to noncommutative paraproducts or martingale transforms with noncommuting symbols/coefficients. Our results complement recent results of Junge, Mei, Parcet and Randrianantoanina.

math.CA↗

arXiv2011

Wavelet approach to operator-valued Hardy spaces

This paper is devoted to the study of operator-valued Hardy spaces via wavelet method. This approach is parallel to that in noncommutative martingale case. We show that our Hardy spaces defined by wavelet coincide with those introduced by Tao Mei via the usual Lusin and Littlewood-Paley square functions. As a consequence, we give an explicit complete unconditional basis of the Hardy space H1(R) when H1(R) is equipped with an appropriate operator space structure.

math.FA↗