arXiv subjects

Zengyan Si

Publications and source records attributed to Zengyan Si.

10 recordsLinked to original sources

arXiv2026

Multilinear extrapolation of compactness on mixed-norm spaces

In this paper, we develop the Rubio de Francia extrapolation theorem for the multilinear compactness on mixed-norm Lebesgue spaces. More precisely, if a multilinear operator is bounded on weighted product spaces, then its compactness can be extrapolated from unweighted product spaces to the full range of weighted mixed-norm spaces. This result is mainly based on a multilinear interpolation theorem of compactness on weighted mixed-norm spaces, for which we present a characterization of compactness on mixed-norm spaces and a multilinear interpolation theorem of boundedness on multi-mixed-norm spaces. As applications of the extrapolation theorem, we obtain compactness results for several kinds of bi-parameter operators on weighted mixed-norm spaces, including multilinear bi-parameter Calderón-Zygmund operators, multilinear bi-parameter dyadic paraproducts, bilinear bi-parameter continuous paraproducts, and bilinear bi-parameter pseudo-differential operators.

math.CA↗

arXiv2024

A characterization of compactness via bilinear $T1$ theorem

In this paper we solve a long standing problem about the bilinear $T1$ theorem to characterize the (weighted) compactness of bilinear Calderón-Zygmund operators. Let $T$ be a bilinear operator associated with a standard bilinear Calderón-Zygmund kernel. We prove that $T$ can be extended to a compact bilinear operator from $L^{p_1}(w_1^{p_1}) \times L^{p_2}(w_2^{p_2})$ to $L^p(w^p)$ for all exponents $\frac1p = \frac{1}{p_1} + \frac{1}{p_2}>0$ with $p_1, p_2 \in (1, \infty]$ and for all weights $(w_1, w_2) \in A_{(p_1, p_2)}$ if and only if the following hypotheses hold: (H1) $T$ is associated with a compact bilinear Calderón-Zygmund kernel, (H2) $T$ satisfies the weak compactness property, and (H3) $T(1,1), T^{*1}(1,1), T^{*2}(1,1) \in \mathrm{CMO}(\mathbb{R}^n)$. This is also equivalent to the endpoint compactness: (1) $T$ is compact from $L^1(w_1) \times L^1(w_2)$ to $L^{\frac12, \infty}(w^{\frac12})$ for all $(w_1, w_2) \in A_{(1, 1)}$, or (2) $T$ is compact from $L^{\infty}(w_1^{\infty}) \times L^{\infty}(w_2^{\infty})$ to $\mathrm{CMO}_λ(w^{\infty})$ for all $(w_1, w_2) \in A_{(\infty, \infty)}$. Besides, any of these properties is equivalent to the fact that $T$ admits a compact bilinear dyadic representation. Our main approaches consist of the following new ingredients: (i) a resulting representation of a compact bilinear Calderón-Zygmund operator as an average of some compact bilinear dyadic shifts and paraproducts; (ii) extrapolation of endpoint compactness for bilinear operators; and (iii) compactness criterion in weighted Lorentz spaces. Finally, to illustrate the applicability of our result, we demonstrate the hypotheses (H1)-(H3) through examples including bilinear continuous/dyadic paraproducts, bilinear pseudo-differential operators, and bilinear commutators.

math.CA↗

arXiv2022

Limited range extrapolation with quantitative bounds and applications

In recent years, sharp or quantitative weighted inequalities have attracted considerable attention on account of $A_2$ conjecture solved by Hytönen. Advances have greatly improved conceptual understanding of classical objects such as Calderón-Zygmund operators. However, plenty of operators do not fit into the class of Calderón-Zygmund operators and fail to be bounded on all $L^p(w)$ spaces for $p \in (1, \infty)$ and $w \in A_p$. In this paper we develop Rubio de Francia extrapolation with quantitative bounds to investigate quantitative weighted inequalities for operators beyond the (multilinear) Calderón-Zygmund theory. We mainly establish a quantitative multilinear limited range extrapolation in terms of exponents $p_i \in (\mathfrak{p}_i^-, \mathfrak{p}_i^+)$ and weights $w_i^{p_i} \in A_{p_i/\mathfrak{p}_i^-} \cap RH_{(\mathfrak{p}_i^+/p_i)'}$, $i=1, \ldots, m$, which refines a result of Cruz-Uribe and Martell. We also present an extrapolation from multilinear operators to the corresponding commutators. Additionally, our result is quantitative and allows us to extend special quantitative estimates in the Banach space setting to the quasi-Banach space setting. Our proof is based on an off-diagonal extrapolation result with quantitative bounds. Finally, we present various applications to illustrate the utility of extrapolation by concentrating on quantitative weighted estimates for some typical multilinear operators such as bilinear Bochner-Riesz means, bilinear rough singular integrals, and multilinear Fourier multipliers. In the linear case, based on the Littlewood-Paley theory, we include weighted jump and variational inequalities for rough singular integrals.

math.CA↗

arXiv2020

Weak and strong types estimates for square functions associated with operators

Let $L$ be a linear operator in $L^2(\mathbb{R}^n)$ which generates a semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy the Gaussian upper bound. In this paper, we investigate several kinds of weighted norm inequalities for the conical square function $S_{α,L}$ associated with an abstract operator $L$. We first establish two-weight inequalities including bump estimates, and Fefferman-Stein inequalities with arbitrary weights. We also present the local decay estimates using the extrapolation techniques, and the mixed weak type estimates corresponding Sawyer's conjecture by means of a Coifman-Fefferman inequality. Beyond that, we consider other weak type estimates including the restricted weak-type $(p, p)$ for $S_{α, L}$ and the endpoint estimate for commutators of $S_{α, L}$. Finally, all the conclusions aforementioned can be applied to a number of square functions associated to $L$.

math.CA↗

arXiv2020

Weak and strong type estimates for the multilinear Littlewood-Paley operators

Let $S_α$ be the multilinear square function defined on the cone with aperture $α\geq 1$. In this paper, we investigate several kinds of weighted norm inequalities for $S_α$. We first obtain a sharp weighted estimate in terms of aperture $α$ and $\vec{w} \in A_{\vec{p}}$. By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman-Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyer's conjecture, for which a Coifman-Fefferman inequality with the precise $A_{\infty}$ norm is proved. Finally, we present the local decay estimates using the extrapolation techniques and dyadic analysis respectively. All the conclusions aforementioned hold for the Littlewood-Paley $g^*_λ$ function. Some results are new even in the linear case.

math.FA↗

arXiv2019

Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

Let $0<α<n$ and $M_α$ be the fractional maximal function. The nonlinear commutator of $M_α$ and a locally integrable function $b$ is given by $[b,M_α](f)=bM_α(f)-M_α(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_α]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or $BMO(\rn)$ spaces, by which some new characterizations for certain subclasses of Lipschitz and $BMO(\rn)$ spaces are obtained.

math.CA↗

arXiv2016

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

Let $n\ge 1$ and $\mathfrak{T}_{m}$ be the bilinear square Fourier multiplier operator associated with a symbol $m$, which is defined by $$ \mathfrak{T}_{m}(f_1,f_2)(x) = \biggl( \int_{0}^\infty\Big|\int_{(\mathbb{R}^n)^2} e^{2πix\cdot (ξ_1 +ξ_2) }m(tξ_1,tξ_2) \hat{f}_{1}(ξ_1)\hat{f}_{2}(ξ_2)dξ_1 dξ_2\Big|^2\frac{dt}{t } \biggr)^{\frac 12}. $$ Let $s$ be an integer with $s\in[n+1,2n]$ and $p_0$ be a number satisfying $2n/s\le p_0\le 2$. Suppose that $ν_{\vecω}=\prod_{i=1}^2ω_i^{p/ p_i}$ and each $ω_i$ is a nonnegative function on $\mathbb{R}^n$. In this paper, we show that $\mathfrak{T}_{m}$ is bounded from $L^{p_1}(ω_1)\times L^{p_2}(ω_2)$ to $L^p(ν_{\vecω})$ 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 $\mathfrak{T}_{m}$ is bounded from $L^{p_1}(ω_1)\times L^{p_2}(ω_2)$ to $L^{p,\infty}(ν_{\vecω})$. The weighted end-point $L\log L$ type estimate and strong estimate for the commutators of $\mathfrak{T}_{m}$ are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.

math.CA↗

arXiv2015

On General multilinear square function with non-smooth kernels

In this paper, we obtain some boundedness of the following general multilinear square functions $T$ with non-smooth kernels, which extend some known results significantly. $$ T(\vec{f})(x)=\big( \int_{0}^\infty \big|\int_{(\mathbb{R}^n)^m}K_v(x,y_1,\dots,y_m) \prod_{j=1}^mf_{j}(y_j)dy_1,\dots,dy_m\big|^2\frac{dv}{v}\big)^{\frac 12}. $$ The corresponding multilinear maximal square function $T^*$ was also introduced and weighted strong and weak type estimates for $T^*$ were given.

math.CA↗

arXiv2012

Commutator Theorems for Fractional Integral Operators on Weighted Morrey Spaces

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(R^n)$ with Gaussican kernel bounds, and let $L^{-α/2}$ be the fractional integrals of $L$ for $0<α<n.$ For any locally integrable function $b$, The commutators associated with $L^{-α/2}$ are defined by $[b,L^{-α/2}](f)(x)=b(x)L^{-α/2}(f)(x)-L^{-α/2}(bf)(x)$. When $b\in BMO(ω)$(weighted $BMO$ space) or $b\in BMO$, the author obtain the necessary and sufficient conditions for the boundedness of $[b,L^{-α/2}]$ on weighted Morrey spaces respectively.

math.FA↗

arXiv2012

Necessary and sufficient conditions for boundedness of commutators of the general fractional integral operators on weighted Morrey spaces

We prove that $b$ is in $Lip_{\bz}(\bz)$ if and only if the commutator $[b,L^{-α/2}]$ of the multiplication operator by $b$ and the general fractional integral operator $L^{-α/2}$ is bounded from the weighed Morrey space $L^{p,k}(ω)$ to $L^{q,kq/p}(ω^{1-(1-α/n)q},ω)$, where $0<β<1$, $0<α+β \frac{1-k}{p/q-k},$ and here $r_ω$ denotes the critical index of $ω$ for the reverse Hölder condition.

math.FA↗