arXiv subjects

Mingming Cao

Publications and source records attributed to Mingming Cao.

At least 19 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 Calder\'{o}n-Zygmund operators, multilinear bi-parameter dyadic paraproducts, bilinear bi-parameter continuous paraproducts, and bilinear bi-parameter pseudo-differential operators.

math.CA↗

arXiv2025

Wi-Fi Sensing Tool Release: Gathering 802.11ax Channel State Information from a Commercial Wi-Fi Access Point

Wi-Fi sensing has emerged as a powerful technology, leveraging channel state information (CSI) extracted from wireless data packets to enable diverse applications, ranging from human presence detection to gesture recognition and health monitoring. However, CSI extraction from commercial Wi-Fi access point lacks and out of date. This paper introduces ZTECSITool,a toolkit designed to capture high-resolution CSI measurements from commercial Wi-Fi 6 (802.11ax) access points, supporting bandwidths up to 160 MHz and 512 subcarriers. ZTECSITool bridges a critical gap in Wi-Fi sensing research, facilitating the development of next-generation sensing systems. The toolkit includes customized firmware and open-source software tools for configuring, collecting, and parsing CSI data, offering researchers a robust platform for advanced sensing applications. We detail the command protocols for CSI extraction, including band selection,STA filtering, and report configuration, and provide insights into the data structure of the reported CSI. Additionally, we present a Python-based graphical interface for real-time CSI visualization and analysis

eess.SP↗

arXiv2025

Parabolic Extrapolation and Its Applications to Characterizing Parabolic BMO Spaces via Parabolic Fractional Commutators

In this article, we establish the parabolic version of the celebrated Rubio de Francia extrapolation theorem. As applications, we obtain new characterizations of parabolic BMO-type spaces in terms of various commutators of parabolic fractional operators with time lag. The key tools to achieve these include to establish the appropriate form in the parabolic setting of the parabolic Rubio de Francia iteration algorithm, the Cauchy integral trick, and a modified Fourier series expansion argument adapted to the parabolic geometry. The novelty of these results lies in the fact that, for the first time, we not only introduce a new class of commutators associated with parabolic fractional integral operators with time lag, but also utilize them to provide a characterization of the parabolic BMO-type space in the high-dimensional case.

math.FA↗

arXiv2024

Dyadic analysis of compactness on product spaces

We develop the compactness theory of multilinear singular integrals on product spaces using a modern point of view. The first main result is a compact $T1$ theorem for multilinear Calder\'{o}n--Zygmund operators on product spaces. More specifically, we prove that a multilinear singular integral operator $T$ on product spaces can be extended to a compact multilinear operator from $L^{p_1}(w_1^{p_1}) \times \cdots \times L^{p_m}(w_m^{p_m})$ to $L^p(w^p)$ for all exponents $\frac1p = \sum_{j=1}^m \frac{1}{p_j}>0$ with $p_1, \ldots, p_m \in (1, \infty]$ and for all weights $\vec{w} \in A_{\vec{p}}(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})$ if the following hypotheses are satisfied: (H1) $T$ admits a compact full kernel representation, (H2) $T$ admits a compact partial kernel representation, (H3) $T$ satisfies the weak compactness property, (H4) $T$ satisfies the diagonal $\mathrm{CMO}$ condition, and (H5) $T$ satisfies the product $\mathrm{CMO}$ condition. This is a multilinear compact extension of Journ\'{e}'s $T1$ theorem on product spaces. The second main result establishes the mean continuity of commutators $[\boldsymbol{b}, T]_{\boldsymbol{\alpha}}$ on weighted Lebesgue spaces as above, which can be viewed as a substitution of compactness because the compactness of $[\boldsymbol{b}, T]_{\boldsymbol{\alpha}}$ is equivalent to $\boldsymbol{b} \equiv \text{constant}$ when $T$ is a non-degenerate bi-parameter singular integral. Our main tools include multilinear bi-parameter dyadic representation, multilinear extrapolation, multilinear interpolation, and Kolmogorov--Riesz compactness criterion.

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 Calder\'{o}n-Zygmund operators. Let $T$ be a bilinear operator associated with a standard bilinear Calder\'{o}n-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 Calder\'{o}n-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 $\mathrm{CMO}_{\lambda}(w^{\infty})$ 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 Calder\'{o}n-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↗

arXiv2023

Elliptic operators in rough sets, and the Dirichlet problem with boundary data in Hölder spaces

In this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in Hölder spaces. Our context is that of open sets $Ω\subset \mathbb{R}^{n+1}$, $n \ge 2$, satisfying the capacity density condition, without any further topological assumptions. Our main result states that if $Ω$ is either bounded, or unbounded with unbounded boundary, then the corresponding Dirichlet boundary value problem is well-posed; when $Ω$ is unbounded with bounded boundary, we establish that solutions exist, but they fail to be unique in general. These results are optimal in the sense that solvability of the Dirichlet problem in Hölder spaces is shown to imply the capacity density condition. As a consequence of the main result, we present a characterization of the Hölder spaces in terms of the boundary traces of solutions, and obtain well-posedness of several related Dirichlet boundary value problems. All the results above are new even for 1-sided chord-arc domains, and can be extended to generalized Hölder spaces associated with a natural class of growth functions.

math.AP↗

arXiv2023

A compact $T1$ theorem for Calder\'{o}n-Zygmund operators associated with Zygmund dilations

We develop a compact version of $T1$ theorem for singular integrals of Zygmund type on $\mathbb{R}^3$. More specifically, if a $(D_{\theta}, \delta_1, \delta_{2, 3})$-Calder\'{o}n-Zygmund operator $T$ associated with Zygmund dilations admits the compact full and partial kernel representations, and satisfies the weak compactness property and the cancellation condition, then $T$ can be extended to a compact operator on $L^p(w)$ whenever (i) $p \in (1, \infty)$, $w \in A_{p, \mathcal{R}}$, and $\theta, \delta_1, \delta_{2, 3} \in (0, 1]$, or (ii) $p \in (1, \infty)$, $w \in A_{p, \mathcal{Z}}$, $\theta = \delta_1 = 1$, and $\delta_{2, 3} \in (0, 1]$. Here $A_{p, \mathcal{R}}$ and $A_{p, \mathcal{Z}}$ respectively denote the class of of strong $A_p$ weights and the class of Zygmund $A_p$ weights. Beyond that, under similar bilinear assumptions, we prove bilinear Calder\'{o}n-Zygmund operators associated with Zygmund dilations are compact from $L^{p_1}(\mathbb{R}^3) \times L^{p_2}(\mathbb{R}^3)$ to $L^p(\mathbb{R}^3)$ for all $p_1, p_2 \in (1, \infty)$, where $\frac1p = \frac{1}{p_1} + \frac{1}{p_2}$. The core of the proof is a compact dyadic representation, which asserts that under the hypotheses above, a (bilinear) Calder\'{o}n-Zygmund operator associated with Zygmund dilations can be represented an average of some compact (bilinear) dyadic shifts of Zygmund nature. This further deepens our understanding of the compactness of singular integral operators.

math.CA↗

arXiv2023

Probing Complex-energy Topology via Non-Hermitian Absorption Spectroscopy in a Trapped Ion Simulator

Non-Hermitian systems generically have complex energies, which may host topological structures, such as links or knots. While there has been great progress in experimentally engineering non-Hermitian models in quantum simulators, it remains a significant challenge to experimentally probe complex energies in these systems, thereby making it difficult to directly diagnose complex-energy topology. Here, we experimentally realize a two-band non-Hermitian model with a single trapped ion whose complex eigenenergies exhibit the unlink, unknot or Hopf link topological structures. Based on non-Hermitian absorption spectroscopy, we couple one system level to an auxiliary level through a laser beam and then experimentally measure the population of the ion on the auxiliary level after a long period of time. Complex eigenenergies are then extracted, illustrating the unlink, unknot or Hopf link topological structure. Our work demonstrates that complex energies can be experimentally measured in quantum simulators via non-Hermitian absorption spectroscopy, thereby opening the door for exploring various complex-energy properties in non-Hermitian quantum systems, such as trapped ions, cold atoms, superconducting circuits or solid-state spin systems.

quant-ph↗

arXiv2023

A compact extension of Journ\'{e}'s $T1$ theorem on product spaces

We prove a compact version of the $T1$ theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral operator $T$ admits the compact full and partial kernel representations, and satisfies the weak compactness property, the diagonal $\mathrm{CMO}$ condition, and the product $\mathrm{CMO}$ condition, then $T$ can be extended to a compact operator on $L^p(w)$ for all $1<p<\infty$ and $w \in A_p(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})$. Even in the unweighted setting, it is the first time to give a compact extension of Journ\'{e}'s $T1$ theorem on product spaces.

math.CA↗

arXiv2022

A class of multilinear bounded oscillation operators on measure spaces and applications

In this paper, we develop a comprehensive weighted theory for a class of Banach-valued multilinear bounded oscillation operators on measure spaces, which merges multilinear Calder\'{o}n-Zygmund operators with a quantity of operators beyond the multilinear Calder\'{o}n-Zygmund theory. We prove that such multilinear operators and corresponding commutators are locally pointwise dominated by two sparse dyadic operators, respectively. We also establish three kinds of typical estimates: local exponential decay estimates, mixed weak type estimates, and sharp weighted norm inequalities. Beyond that, based on Rubio de Francia extrapolation for abstract multilinear compact operators, we obtain weighted compactness for commutators of specific multilinear operators on spaces of homogeneous type. A compact extrapolation allows us to get full range of exponents, while weighted interpolation for multilinear compact operators is crucial to the compact extrapolation. These are due to a weighted Fr\'{e}chet-Kolmogorov theorem in the quasi-Banach range, which gives a characterization of relative compactness of subsets in weighted Lebesgue spaces. As applications, we illustrate multilinear bounded oscillation operators with examples including multilinear Hardy-Littlewood maximal operators on measure spaces, multilinear $\omega$-Calder\'{o}n-Zygmund operators on spaces of homogeneous type, multilinear Littlewood-Paley square operators, multilinear Fourier integral operators, higher order Calder\'{o}n commutators, maximally modulated multilinear singular integrals, and $q$-variation of $\omega$-Calder\'{o}n-Zygmund operators.

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 Hyt\"{o}nen. Advances have greatly improved conceptual understanding of classical objects such as Calder\'{o}n-Zygmund operators. However, plenty of operators do not fit into the class of Calder\'{o}n-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) Calder\'{o}n-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↗

arXiv2022

The $\mathrm{CMO}$-Dirichlet problem for elliptic systems in the upper half-space

We prove that for any second-order, homogeneous, $N \times N$ elliptic system $L$ with constant complex coefficients in $\mathbb{R}^n$, the Dirichlet problem in $\mathbb{R}^n_+$ with boundary data in $\mathrm{CMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ is well-posed under the assumption that $d\mu(x', t) := |\nabla u(x)|^2\, t \, dx' dt$ is a strong vanishing Carleson measure in $\mathbb{R}^n_+$ in some sense. This solves an open question posed by Martell et al. The proof relies on a quantitative Fatou-type theorem, which not only guarantees the existence of the pointwise nontangential boundary trace for smooth null-solutions satisfying a strong vanishing Carleson measure condition, but also includes a Poisson integral representation formula of solutions along with a characterization of $\mathrm{CMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ in terms of the traces of solutions of elliptic systems. Moreover, we are able to establish the well-posedness of the Dirichlet problem in $\mathbb{R}^n_+$ for a system $L$ as above in the case when the boundary data belongs to $\mathrm{XMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$, which lines in between $\mathrm{CMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ and $\mathrm{VMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$. Analogously, we formulate a new brand of strong Carleson measure conditions and a characterization of $\mathrm{XMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ in terms of the traces of solutions of elliptic systems.

math.CA↗

arXiv2022

Carleson measure estimates, corona decompositions, and perturbation of elliptic operators without connectivity

Let $Ω$ be an open set with Ahlfors-David regular boundary satisfying the corkscrew condition. When $Ω$ is connected in some quantitative form one can establish that for any real elliptic operator with bounded coefficients, the quantitative absolute continuity of elliptic measures is equivalent to the fact that all bounded null solutions satisfy Carleson measure estimates. In turn, in the same setting these equivalent properties are stable under Fefferman-Kenig-Pipher perturbations. However, without connectivity, there is no Fefferman-Kenig-Pipher perturbation result available. In this paper, we work with a corona decomposition associated with the elliptic measure and show that it is equivalent to the fact that bounded null solutions satisfy partial/weak Carleson measure estimates, or to the fact that the Green function is comparable to the distance to the boundary in the corona sense. This characterization has profound consequences. We extend Fefferman-Kenig-Pipher's perturbation to non-connected settings. For the Laplacian, these corona decompositions or, equivalently, the partial/weak Carleson measure estimates are meaningful enough to characterize the uniform rectifiability of the boundary. As a consequence, we obtain that the boundary of the set is uniformly rectifiable if bounded null solutions for any Fefferman-Kenig-Pipher perturbation of the Laplacian satisfy Carleson measure estimates. For Kenig-Pipher operators any of the properties of the characterization is stable under transposition or symmetrization of the matrices of coefficients. As a result, we obtain that Carleson measure estimates for bounded null-solutions of non-symmetric variable operators satisfying an $L^1$-Kenig-Pipher condition occur if and only if the boundary of the open set is uniformly rectifiable. Our results generalize previous work in settings where quantitative connectivity.

math.CA↗

arXiv2022

Extrapolation for multilinear compact operators and applications

This paper is devoted to studying the Rubio de Francia extrapolation for multilinear compact operators. It allows one to extrapolate the compactness of $T$ from just one space to the full range of weighted spaces, whenever an $m$-linear operator $T$ is bounded on weighted Lebesgue spaces. This result is indeed established in terms of the multilinear Muckenhoupt weights $A_{\vec{p}, \vec{r}}$, and the limited range of the $L^p$ scale. To show extrapolation theorems above, by means of a new weighted Fréchet-Kolmogorov theorem, we present the weighted interpolation for multilinear compact operators. To prove the latter, we also need to bulid a weighted interpolation theorem in mixed-norm Lebesgue spaces. As applications, we obtain the weighted compactness of commutators of many multilinear operators, including multilinear $ω$-Calderón-Zygmund operators, multilinear Fourier multipliers, bilinear rough singular integrals and bilinear Bochner-Riesz means. Beyond that, we establish the weighted compactness of higher order Calderón commutators, and commutators of Riesz transforms related to Schrödinger operators.

math.CA↗

arXiv2021

Absolute continuity of degenerate elliptic measure

Let $\Omega \subset \mathbb{R}^{n+1}$ be an open set whose boundary may be composed of pieces of different dimensions. Assume that $\Omega$ satisfies the quantitative openness and connectedness, and there exist doubling measures $m$ on $\Omega$ and $\mu$ on $\partial \Omega$ with appropriate size conditions. Let $Lu=-\mathrm{div}(A\nabla u)$ be a real (not necessarily symmetric) degenerate elliptic operator in $\Omega$. Write $\omega_L$ for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) $\omega_L \in A_{\infty}(\mu)$, (ii) the Dirichlet problem for $L$ is solvable in $L^p(\mu)$ for some $p \in (1, \infty)$, (iii) every bounded null solution of $L$ satisfies Carleson measure estimates with respect to $\mu$, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q(\mu)$ for all $q \in (0, \infty)$ for any null solution of $L$, and (v) the Dirichlet problem for $L$ is solvable in $\mathrm{BMO}(\mu)$. On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of $\omega_L$ with respect to $\mu$ in terms of local $L^2(\mu)$ estimates of the truncated conical square function for any bounded null solution of $L$. This is also equivalent to the finiteness $\mu$-almost everywhere of the truncated conical square function for any bounded null solution of $L$.

math.AP↗

arXiv2021

Two-weight extrapolation on function spaces and applications

This paper is devoted to studying the extrapolation theory of Rubio de Francia on general function spaces. We present endpoint extrapolation results including $A_1$, $A_p$, and $A_\infty$ extrapolation in the context of Banach function spaces, and also on modular spaces. We also include several applications that can be easily obtained using extrapolation: local decay estimates for various operators, Coifman--Fefferman inequalities that can be used to show some known sharp $A_1$ inequalities, Muckenhoupt--Wheeden and Sawyer's conjectures are also presented for many operators, which go beyond Calder\'{o}n--Zygmund operators. Finally, we obtain two-weight inequalities for Littlewood--Paley operators and Fourier integral operators on weighted Banach function spaces.

math.CA↗

arXiv2021

On the $A_\infty$ condition for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Let $Ω\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (i.e., quantitatively open and path-connected) satisfiying the capacity density condition. Let $L_0 u=-\mathrm{div}(A_0 \nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$ be two real uniformly elliptic operators in $Ω$, with $ω_{L_0}, ω_L$ the associated elliptic measures. We establish the equivalence between the following properties: (i) $ω_L \in A_{\infty}(ω_{L_0})$, (ii) $L$ is $L^p(ω_{L_0})$-solvable for some $p\in (1,\infty)$, (iii) bounded null solutions of $L$ satisfy Carleson measure estimates with respect to $ω_{L_0}$, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q(ω_{L_0})$ for some (or for all) $q\in (0,\infty)$ for any null solution of $L$, and (v) $L$ is $\mathrm{BMO}(ω_{L_0})$-solvable. Moreover, in each of the properties (ii)-(v) it is enough to consider the class of solutions $u(X)=ω_L^X(S)$ with arbitrary Borel sets $S\subset\partialΩ$. Also, we characterize the absolute continuity of $ω_{L_0}$ with respect to $ω_L$ in terms of some qualitative local $L^2(ω_{L_0})$ estimates for the truncated conical square function for any bounded null solution of $L$. This is also equivalent to the finiteness $ω_{L_0}$-a.e. of the truncated conical square function for any bounded null solution of $L$. As applications, we show that $ω_{L_0}\llω_L$ if the disagreement of the coefficients satisfies some qualitative quadratic estimate in truncated cones for $ω_{L_0}$-a.e. vertex. Finally, when $L_0$ is either the transpose of $L$ or its symmetric part, we obtain the corresponding absolute continuity when the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for $ω_{L_0}$-a.e. vertex.

math.CA↗

arXiv2021

Extrapolation on function and modular spaces, and applications

We generalize the extrapolation theory of Rubio de Francia to the context of Banach function spaces and modular spaces. Our results are formulated in terms of some natural weighted estimates for the Hardy-Littlewood maximal function and are stated in measure spaces and for general Muckenhoupt bases. Finally, we give several applications in analysis and partial differential equations.

math.CA↗