
Mingming CaoSpanish National Research Council | CSIC · ICMAT
Mingming Cao
Doctor of Philosophy
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49
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Skills and Expertise
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April 2019 - December 2022
September 2018 - March 2019
Education
September 2013 - June 2018
September 2009 - June 2013
Publications
Publications (49)
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 multilinea...
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...
In recent years, dyadic analysis has attracted a lot of attention due to the A2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2$$\end{document} conjecture. It has be...
We prove a compact version of the T 1 T1 theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral operator T T admits the compact full and partial kernel representations, and satisfies the weak compactness property, the diagonal C M O CMO condition, and the product C M O CMO condition, then T T can be extended to a...
Let Ω⊂Rn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^{n+1}$$\end{document}, n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepa...
In this paper, 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 $\vec{w...
This paper is devoted to studying the extrapolation theory of Rubio de Francia on general function spaces. We present endpoint extrapolation results including A1$A_1$, Ap$A_p$, and A∞$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 e...
In recent years, sharp or quantitative weighted inequalities have attracted considerable attention on account of the $$A_2$$ 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 ope...
We prove that for any second-order, homogeneous, $$N \times N$$ N × N elliptic system L with constant complex coefficients in $$\mathbb {R}^n$$ R n , the Dirichlet problem in $$\mathbb {R}^n_+$$ R + n with boundary data in $$\textrm{CMO}(\mathbb {R}^{n-1}, \mathbb {C}^N)$$ CMO ( R n - 1 , C N ) is well-posed under the assumption that $$d\mu (x', t)...
We, for the first time, prove a compact version of $T1$ theorem for singular integrals of Zygmund type on $\mathbb{R}^3$. That is, if a singular integral 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...
We prove a compact version of $T1$ theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral $T$ admits compact full and partial kernel representations, and satisfies the weak compactness property, the diagonal $\mathrm{CMO}$ condition, and the product $\mathrm{BMO}$ condition, then $T$ can be extended to a compact o...
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 commut...
Let $\Omega \subset \mathbb {R}^{n+1}$ , $n\ge 2$ , be a $1$ -sided nontangentially accessible domain, that is, a set which is quantitatively open and path-connected. Assume also that $\Omega $ satisfies the capacity density condition. Let $L_0 u=-\mathop {\operatorname {div}}\nolimits (A_0 \nabla u)$ , $Lu=-\mathop {\operatorname {div}}\nolimits (...
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 o...
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 vani...
In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $$L^p$$ L p , for some finite p , is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $$A_\infty $$ A ∞ . In turn, any of these conditions occurs if and...
Let $\Omega$ be an open set with Ahlfors-David regular boundary satisfying the corkscrew condition. When $\Omega$ 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...
In this paper, we establish several different characterizations of the vanishing mean oscillation space associated with Neumann Laplacian \(\Delta _N\), written \({\text {VMO}}_{\Delta _N}({\mathbb {R}^n})\). We first describe it with the classical \({\text {VMO}}({\mathbb {R}^n})\) and certain \({\text {VMO}}\) on the half-spaces. Then we demonstr...
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 r...
Let Sα be the multilinear square function defined on the cone with aperture α≥1. In this paper, we investigate several kinds of weighted norm inequalities for Sα. We first obtain a sharp weighted estimate in terms of aperture α and w→∈Ap→. By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bum...
This paper is devoted to studying the two-weight extrapolation theory of Rubio de Francia. We start to establish endpoint extrapolation results including $A_1$, $A_p$ and $A_\infty$ extrapolation in the context of Banach function spaces and modular spaces. Furthermore, we present extrapolation for commutators in weighted Banach function spaces. Bey...
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...
Let $\Omega \subset \mathbb{R}^{n+1}$, $n\ge 2$, 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 $\Omega$, with $\omega_{L_0}, \omega_L$ the ass...
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 $S_{\alpha,L}$ associated with an abstract operator $L$. We first establish two-weight inequali...
In this paper, we establish several different characterizations of the vanishing mean oscillation space associated with Neumann Laplacian $\Delta_N$, written ${\rm VMO}_{\Delta_N}(\mathbb{R}^n)$. We first describe it with the classical ${\rm VMO}(\mathbb{R}^n)$ and certain ${\rm VMO}$ on the half-spaces. Then we demonstrate that ${\rm VMO}_{\Delta_...
Let $S_{\alpha}$ be the multilinear square function defined by means of the cone in $\mathbb{R}_+^{n+1}$ with aperture $\alpha \geq 1$. In this paper we prove that the estimate \begin{equation*} \begin{split} ||S_{\alpha}(\vec{f})||_{L^{p}(\nu_{\vec{w}})} \le C_{n,m,\psi,\vec{p}} \alpha^{mn} [\vec{w}]_{A_{\vec{p}}}^{\max\{\frac{1}{2},\frac{p_1'}{p}...
Let \(m\ge 2, \lambda > 1\) and define the multilinear Littlewood–Paley–Stein operators by \( g_{\lambda ,\mu }^*(\vec {f})(x) = (\iint _{{\mathbb {R}}^{n+1}_{+}} (\frac{t}{t + |x - y|})^{m \lambda } |\int _{({{\mathbb {R}}^n})^{\kappa }} s_t(y,\vec {z}) \prod _{i=1}^{\kappa } f_i(z_i) \ \mathrm{d}\mu (z_1) \cdots \mathrm{d}\mu (z_{\kappa })|^2 \fr...
Let $\kappa \ge 2, \lambda > 1$ and define the multilinear Littlewood-Paley-Stein operators by $$g_{\lambda,\mu}^*(\vec{f})(x) = \bigg(\iint_{\mathbb{R}^{n+1}_{+}} \vartheta_t(x, y) \bigg|\int_{\mathbb{R}^{n \kappa}} s_t(y,\vec{z}) \prod_{i=1}^{\kappa} f_i(z_i) \ d\mu(z_i)\bigg|^2 \frac{d\mu(y) dt}{t^{m+1}}\bigg)^{\frac12}, $$ where $\vartheta_t(x,...
In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $L^p$, for some finite $p$, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $A_\infty$. In turn, any of these conditions occurs if and only if the gr...
In this paper, we investigate the boundedness of the multilinear pseudo-differential operator Tσ. First, we establish the local exponential decay estimates for Tσ. In terms of the corresponding commutators Tσ,Σb, we obtain the local subexponential decay estimates. Secondly, we derive the weighted mixed weak type inequality for Tσ, which parallels S...
We investigate the quantitative weighted estimates for a large class of the multilinear Littlewood–Paley square operators. Our kernels satisfy the minimal regularity assumption, called (Formula presented.)-Hörmander condition. We respectively establish the pointwise sparse domination for the multilinear square functions and their iterated commutato...
In this note, the dyadic Carleson embedding theorem in the multi-parameter setting will be presented. In the case of one parameter, a new simple dyadic proof will be given by using the technique of principal cubes. © 2018 University of Debrecen Institute of Mathematics. All rights reserved.
Let Iα→ be the bi-parameter fractional integral operator on Rⁿ¹×Rⁿ², Iα→(f)(x)=∫Rn1 ×Rn2[Formula presented]dy,0<αi<ni,i=1,2.In this paper, we give a characterization of two-weight norm inequality for the commutator of Iα→. We show that for μ,λ∈Ap,q(Rn→), ‖[b,Iα→]‖Lp(μp)→Lq(λq)≃‖b‖bmo(ν), where ν=μλ⁻¹, and [Formula presented]−[Formula presented]=[Fo...
Let $n\ge 2$ and $g_{\lambda}^{*}$ be the well-known high dimensional
Littlewood-Paley function which was defined and studied by E. M. Stein,
$$g_{\lambda}^{*}(f)(x)=\bigg(\iint_{\R^{n+1}_{+}}
\Big(\frac{t}{t+|x-y|}\Big)^{n\lambda} |\nabla P_tf(y,t)|^2 \frac{dy
dt}{t^{n-1}}\bigg)^{1/2}, \ \quad \lambda > 1$$ where $P_tf(y,t)=p_t*f(x)$,
$p_t(y)=t^{-...
Let $I_{\alpha}$ be the linear and $\mathcal{I}_{\alpha}$ be the bilinear fractional integral operators. In the linear setting, it is known that the two-weight inequality holds for the first order commutators of $I_{\alpha}$. But the method can't be used to obtain the two weighted norm inequality for the higher order commutators of $I_{\alpha}$. In...
Let $\mu=\mu_{n_1} \times \mu_{n_2}$, where $\mu_{n_1}$ and $\mu_{n_2}$ are upper doubling measures on $\R^{n_1}$ and $\R^{n_2}$ respectively. Let the pseudo-accretive function $b=b_1 \otimes b_2$ satisfy a bi-parameter Carleson condition. In this paper, we established the $L^2(\mu)$ boundedness of non-homogeneous Littlewood-Paley $g_{\lambda}^*$-f...
In this paper, we present a local $Tb$ theorem for the non-homogeneous
Littlewood-Paley $g_{\lambda}^{*}$-function with non-convolution type kernels
and upper power bound measure $\mu$. We show that, under the assumptions $\supp
b_Q \subset Q$, $|\int_Q b_Q d\mu| \gtrsim \mu(Q)$ and $||b_Q||^p_{L^p(\mu)}
\lesssim \mu(Q)$, the norm inequality $\big\...
In this paper, the multilinear fractional strong maximal operator
$\mathcal{M}_{\mathcal{R},\alpha}$ associated with rectangles and corresponding
multiple weights $A_{(\vec{p},q),\mathcal{R}}$ are introduced. Under the dyadic
reverse doubling condition, a necessary and sufficient condition for two-weight
inequalities is given. As consequences, we f...
It is well-known that the $L^p$ boundedness and weak $(1,1)$ estiamte $(\lambda>2)$ of the classical Littlewood-Paley $g_{\lambda}^{*}$-function was first studied by Stein, and the weak $(p,p)$ $(p>1)$ estimate was later given by Fefferman for $\lambda=2/p$. In this paper, we investigated the $L^p(\mu)$ boundedness of the non-homogeneous Littlewood...
The main result of this paper is a bi-parameter $Tb$ theorem for Littlewood-Paley $g$-function, where $b$ is a tensor product of two pseudo-accretive functions. Instead of the doubling measure, we work with a product measure $\mu = \mu_n \times \mu_m$, where the measures $\mu_n$ and $\mu_m$ are only assumed to be upper doubling. The main techniques...
This paper will be devoted to study the two-weight norm inequalities of the multilinear fractional maximal operator $\mathcal{M}_{\alpha}$ and the multilinear fractional integral operator $\mathcal{I}_{\alpha}$. The entropy conditions in the multilinear setting will be introduced and the entropy bounds for $\mathcal{M}_\alpha$ and $\mathcal{I}_\alp...
In this paper, we restudied the two-weight problem of multilinear fractional maximal operator $\mathcal{M}_{\alpha}$. First, we gave a characterization of two-weight inequalities for $\mathcal{M}_{\alpha}$ related to a multilinear analogue of Sawyer's two-weight condition $S_{(\vec{p},q)}$, which essentially improved and extended some known results...
Let m,n >= 1 and let g(gimel 1) ,(lambda 2)be the biparameter Littlewood-Paley g(gimel 1)*-function defined by g(gimel 1) ,(lambda 2) (f)(x) = (integral integral(m+1)(R+)(t(2)/t(2) +vertical bar x(2) - y(2)vertical bar))(m lambda 2) integral integral(n+1)(R+)(t(1)/t(1) +vertical bar x(1) - y(1)vertical bar))(m lambda 1) x vertical bar theta(t1+t2)f...
In this paper, we investigated the boundedness of multilinear fractional
strong maximal operator $\mathcal{M}_{\mathcal{R},\alpha}$ associated with
rectangles or related to more general basis with multiple weights
$A_{(\vec{p},q),\mathcal{R}}$. In the rectangles setting, we first gave an
end-point estimate of $\mathcal{M}_{\mathcal{R},\alpha}$, whi...