Lixin Yan

Sun Yat-Sen University | SYSU · mathema

Let \(G\cong {\mathbb {R}}^{d} \ltimes {\mathbb {R}}\) be a finite-dimensional two-step nilpotent group with the group multiplication \((x,u)\cdot (y,v)\rightarrow (x+y,u+v+x^{T}Jy)\) where J is a skew-symmetric matrix satisfying a degeneracy condition with \(2\le \textrm{rank}\, J <d\). Consider the maximal function defined by $$\begin{aligned} {\...

Let $X$ be a metric space with doubling measure, and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. Let $f$ be in the space $ {\rm BMO}_L(X)$ associated with the operator $L$ and we define its distance from the subspace $L^{\infty}(X)$ under the $ {\rm BMO}_L(X)$ norm as follows: $$ {\rm...

Let ${\frak M}^\alpha$ be the spherical maximal operators of complex order $\alpha$ on ${\mathbb R^n} $ for all $ n\geq 2$. In this article we show that when $n\geq 2$, suppose \begin{eqnarray*} \|{\frak M}^{\alpha}f\|_{L^p({\mathbb R^n})} \leq C\|f \|_{L^p({\mathbb R^n})} \end{eqnarray*} holds for some $\alpha$ and $p\geq 2$, then we must have ${\...

Let $\Delta_{\mathbb{T}^m\times \mathbb{R}^n}$ denote the Laplace-Beltrami operator on the product spaces $\mathbb{T}^m\times \mathbb{R}^n$. In this article we show that $$ \left\|e^{it\Delta_{\mathbb{T}^m\times \mathbb{R}^n}}f\right\|_{L^p(\mathbb{T}^m\times \mathbb{R}^n\times [0,1])} \leq C \|f\|_{W^{\alpha,p}(\mathbb{T}^m\times\mathbb{R}^n)} $$...

We introduce the Hardy spaces for Fourier integral operators on Riemannian manifolds with bounded geometry. We then use these spaces to obtain improved local smoothing estimates for Fourier integral operators satisfying the cinematic curvature condition, and for wave equations on compact manifolds. The estimates are essentially sharp, for all $2<p<...

A large part of the theory of Hardy spaces on products of Euclidean spaces has been extended to the setting of products of stratified Lie groups. This includes characterisation of Hardy spaces by square functions and by atomic decompositions, proof of the duality of Hardy spaces with BMO, and description of many interpolation spaces. Until now, how...

Let $G\cong\mathbb{R}^{d} \ltimes \mathbb{R}$ be a finite-dimensional two-step nilpotent group with the group multiplication $(x,u)\cdot(y,v)\rightarrow(x+y,u+v+x^{T}Jy)$ where $J$ is a skew-symmetric matrix satisfying a degeneracy condition with $2\leq {\rm rank}\, J <d$. Consider the maximal function defined by $$ {\frak M}f(x, u)=\sup_{t>0}\big|...

Let L L be a non-negative self-adjoint operator acting on the space L 2 ( X ) L^2(X) , where X X is a positive measure space. Let L = ∫ 0 ∞ λ d E L ( λ ) { L}=\int _0^{\infty } \lambda dE_{ L}({\lambda }) be the spectral resolution of L L and S R ( L ) f = ∫ 0 R d E L ( λ ) f S_R({ L})f=\int _0^R dE_{ L}(\lambda ) f denote the spherical partial sum...

Let ΔSn denote the Laplace-Beltrami operator on the n-dimensional unit sphere Sn, n≥2. In this paper we show that‖eitΔSnf‖L4([0,2π)×Sn)≤C‖f‖Wα,4(Sn) holds if α>(n−2)/4. The range of α is sharp in the sense that the estimate fails for α<(n−2)/4. As a consequence, we obtain space-time Lp-estimates for eitΔSn for 2≤p≤∞. We also prove that the maximal...

We obtain weak-type $(p, p)$ endpoint bounds for Bochner–Riesz means for the Hermite operator $H = -\Delta + |x|^2$ in ${\mathbb{R}}^n, n\ge 2$ and for other related operators, for $1\le p\le 2n/(n+2)$, extending earlier results of Thangavelu and of Karadzhov.

Let H=−Δ+|x|2 be the Hermite operator in Rn. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with H which is defined by SRλ(H)f(x)=∑k=0∞(1−2k+nR2)+λPkf(x). Here Pkf is the k-th Hermite spectral projection operator. For 2≤p<∞, we prove thatlimR→∞⁡SRλ(H)f=fa.e. for all f∈Lp(Rn) provided that λ>λ(p)/2 and λ(p...

Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$, where $X$ is a metric measure space. Let ${ L}=\int_0^{\infty} \lambda dE_{ L}({\lambda})$ be the spectral resolution of ${ L}$ and $S_R({ L})f=\int_0^R dE_{ L}(\lambda) f$ denote the spherical partial sums in terms of the resolution of ${ L}$. In this article we give a s...

Consider a non-doubling manifold with ends M=Rn♯Rm where Rn=Rn×Sm−n for m>n≥3. We say that an operator L has a generalised Poisson kernel if L generates a semigroup e−tL whose kernel pt(x,y) has an upper bound similar to the kernel of e−tΔ where Δ is the Laplace-Beltrami operator on M. An example for operators with generalised Gaussian bounds is th...

Let $\Delta_{\mathbb S^n}$ denote the Laplace-Beltrami operator on the $n$-dimensional unit sphere $\mathbb S^n$. In this paper we show that $$ \| e^{it \Delta_{\mathbb S^n}}f \|_{L^4([0, 2\pi) \times \mathbb S^n)} \leq C \| f\|_{W^{\alpha, 4} (\mathbb S^n)} $$ holds provided that $n\geq 2$, $\alpha> {(n-2)/4}.$ The range of $\alpha$ is sharp up to...

In this paper we prove spectral multiplier theorems for abstract self-adjoint operators on spaces of homogeneous type. We have two main objectives. The first one is to work outside the semigroup context. In contrast to previous works on this subject, we do not make any assumption on the semigroup. The second objective is to consider polynomial off-...

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. In this paper, we study sharp endpoint $L^p$-Sobolev estimates for the solution of the initial value problem for the Schr\"odinger equation, $i \partial_t u + L u=0 $ and show that for all $f\in L^p(X), 1<p<\infty,$ \beg...

Let $H = -\Delta + |x|^2$ be the Hermite operator in ${\mathbb R}^n$. In this paper we study almost everywhere convergence of Bochner-Riesz means for the Hermite operator $H$. We prove that $$ \lim\limits_{R\to \infty} S_R^{\lambda}(H) f(x)=f(x) \ \ \text{a.e.} $$ for $f\in L^p(\mathbb R^n)$ provided that $p\geq 2$ and $\lambda> 2^{-1}\max\big\{ n\...

Let L be a non-negative self-adjoint operator acting on \(L^2(X)\) where X is a space of homogeneous type with a dimension n. Suppose that the heat operator \(e^{-tL}\) satisfies the generalized Gaussian \((p_0, p'_0)\)-estimates of order m for some \(1\le p_0 < 2\). In this paper we prove sharp endpoint \(L^p\)-Sobolev bound for the Schrödinger gr...

We show that for an entire function φ belonging to the Fock space F2(Cn) on the complex Euclidean space Cn, the integral operatorSφF(z)=∫CnF(w)ez⋅w¯φ(z−w¯)dλ(w),z∈Cn, is bounded on F2(Cn) if and only if there exists a function m∈L∞(Rn) such thatφ(z)=∫Rnm(x)e−2(x−i2z)2dx,z∈Cn. Here dλ(w)=π−ne−|w|2dw is the Gaussian measure on Cn. With this character...

We describe a simple but surprisingly effective technique of obtaining spectral multiplier results for abstract operators which satisfy the finite propagation speed property for the corresponding wave equation propagator. We show that, in this setting, spectral multipliers follow from resolvent type estimates. The most notable point of the paper is...

Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian upper bounds and H\"older's continuity in $x$ but we do not require the semigroup to satisfy the preservation cond...

We investigate the Hardy space H L 1 H^1_L associated with a self-adjoint operator L L defined in a general setting by Hofmann, Lu, Mitrea, Mitrea, and Yan [Mem. Amer. Math. Soc. 214 (2011), pp. vi+78]. We assume that there exists an L L -harmonic non-negative function h h such that the semigroup exp ⁡ ( − t L ) \exp (-tL) , after applying the Doob...

Consider a non-doubling manifold with ends $M = \mathfrak{R}^{n}\sharp\, {\mathbb R}^{m}$ where $\mathfrak{R}^n=\mathbb{R}^n\times \mathbb{S}^{m-n}$ for $m> n \ge 3$. We say that an operator $L$ has a generalised Poisson kernel if $\sqrt{ L}$ generates a semigroup $e^{-t\sqrt{L}}$ whose kernel $p_t(x,y)$ has an upper bound similar to the kernel of...

We show that for an entire function $\varphi$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, that the integral operator \begin{eqnarray*} S_{\varphi}F(z)= \frac{1}{\pi^n}\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} \varphi(z- \bar{w}) e^{-\left\vert w\right\vert^2}dw, \ \ \ \ \ z\in \mathbb{...

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat kernel of $L$ satisfies a Gaussian upper bound. It is known that the operator $(I+L)^{-s } e^{itL}$ is bounded on $L^p(X)$ for $s> n|{1/ 2}-{1/p}| $ and $ p\in (1, \infty)$ (see for example, \cite{C...

We obtain new multilinear multiplier theorems for symbols of restricted smoothness which lie locally in certain Sobolev spaces. We provide applications concerning the boundedness of the commutators of Calderón and Calderón–Coifman–Journé.

Let $L$ be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and H\"older's continuity. Also assume that $L$ has a bounded holomorphic functional calculus on $L^2(\mathbb{R}^n)$. In this paper, we construct a frame decomposition for the functions belonging to the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ associated to $...

Let L be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and Hölder's continuity. Also assume that L has a bounded holomorphic functional calculus on L ² (R ⁿ ). In this paper, we construct a frame decomposition for the functions belonging to the Hardy space H L¹ (R ⁿ ) associated to L, and for functions in the Le...

In this paper we prove spectral multiplier theorems for abstract self-adjoint operators on spaces of homogeneous type. We have two main objectives. The first one is to work outside the semigroup context. In contrast to previous works on this subject, we do not make any assumption on the semigroup. The second objective is to consider polynomial off-...

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat kernel of $L$ satisfies a Gaussian upper bound. In this paper we show sharp endpoint estimate for the Schr\"odinger group $e^{itL}$ on the Hardy space $H^1_L(X)$ associated with $L$ such that \begin...

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with the upper dimension $n$. Suppose that the heat operator $e^{-tL}$ satisfies the generalized Gaussian $(p_0, p'_0)$-estimates of order $m$ for some $1\leq p_0 < 2$. In this article we prove sharp endpoint $L^p$-Sobolev bounds for the Schr...

Let $(X,d,\mu)$ be a space of homogeneous type. We establish three equivalent characterizations of the exponential square class associated to the classical dyadic square function on $(X,d,\mu)$: exponential square integrability, improved good-$\lambda$ inequalities, and the sharp $L^p$ lower bound for the classical dyadic square function. This resu...

We obtain optimal weak-type $(p, p)$ bounds for Bochner-Riesz means for the Hermite operator $H=-\Delta +|x|^2$ in ${\mathbb R}^n, n\geq 2$ and for other related operators for $1\leq p\leq 2n/(n+2)$, extending earlier results of Thangavelu and of Karadzhov.

We consider the Lt²Lxr estimates for the solutions to the wave and Schrödinger equations in high dimensions. For the homogeneous estimates, we show Lt²Lx∞ estimates fail at the critical regularity in high dimensions by using stable Lévy process in Rd. Moreover, we show that some spherically averaged Lt²Lx∞ estimate holds at the critical regularity....

We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schr\"odinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^\infty$ estimates fail at the critical regularity in high dimensions by using stable L\'evy process in $\R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^\infty$ estima...

Let L be a Schrödinger operator of the form L = −Δ + V acting on L2(ℝn) where the nonnegative potential V belongs to the reverse Hölder class Bq for some q ≥ n. In this article we will show that a function f ∈ L2,λ(ℝn), 0 < λ < n, is the trace of the solution of Lu = −utt + Lu = 0, u(x, 0) = f(x), where u satisfies a Carleson type condition $$\math...

We investigate $L^p$ boundedness of the maximal Bochner-Riesz means for self-adjoint operators of elliptic type. Assuming the finite speed of propagation for the associated wave operator, from the restriction type estimates we establish the sharp $L^p$ boundedness of the maximal Bochner-Riesz means for the elliptic operators. As applications, we ob...

Let $X$ be a metric measure space with a doubling measure and $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$. Assume that $L$ generates an analytic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables $x$ and $y$. In this article we continue a stud...

In this paper, we consider the stochastic heat equation of the form (Formula Presented), where 1 < β < α < 2, W (t, x) is a fractional Brownian sheet, ∆θ:= −(−∆)θ/2 denotes the fractional Lapalacian operator and f: [0, T ] × R × R → R is a nonlinear measurable function. We introduce the existence, uniqueness and Hölder regularity of the solution. A...

This paper comprises two parts. In the first, we study $L^p$ to $L^q$ bounds for spectral multipliers and Bochner-Riesz means with negative index in the general setting of abstract self-adjoint operators. In the second we obtain the uniform Sobolev estimates for constant coefficients higher order elliptic operators $P(D)-z$ and all $z\in {\mathbb C...

Let $\L$ be a Schr\"odinger operator of the form $\L=-\Delta+V$ acting on $L^2(\mathbb R^n)$, $n\geq3$, where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ Let ${\rm BMO}_{{\mathcal{L}}}(\RR)$ denote the BMO space associated to the Schr\"odinger operator $\L$ on $\RR$. In this article we show that for...

Let $X_1$ and $X_2$ be metric spaces equipped with doubling measures and let $L_1$ and $L_2$ be nonnegative self-adjoint second-order operators acting on $L^2(X_1)$ and $L^2(X_2)$ respectively. We study multivariable spectral multipliers $F(L_1, L_2)$ acting on the Cartesian product of $X_1$ and $X_2$. Under the assumptions of the finite propagatio...

The aim of this article is to develop the theory of product Hardy spaces associated with operators which possess the weak assumption of Davies--Gaffney heat kernel estimates, in the setting of spaces of homogeneous type. We also establish a Calder\'on--Zygmund decomposition on product spaces, which is of independent interest, and use it to study th...

Let L=-δ+μ be the generalized Schrödinger operator on Rn, n≥3, where μ≢0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Based on Shen's work for the fundamental solution of L in [23], we establish the following upper bound for semigroup kernels Kt(x,y), associated to e-tL,0≤Kt(x,y)≤Cht(x-y...

For all $k,n\in \mathbb N$, we obtain the boundedness of the $n$th dimensional Calder\'on-Coifman-Journ\'e multicommutator \[ (f,a_1,\dots , a_k)\mapsto \mathrm{p.v.} \int_{\mathbb R^n} f(y) \left( \prod_{l=1}^n \dfrac{1}{(y_l-x_l)^{k+1}} \right ) \left[\prod_{j=1}^k\int_{x_1}^{y_1} \cdots \int_{x_n}^{y_n} a_j(u_1, \dots, u_n) \, du_1\! \cdots \! d...

The main aim of this article is to establish boundedness of singular integrals with non-smooth kernels on product spaces. Let $L_1$ and $L_2$ be non-negative self-adjoint operators on $L^2(\mathbb{R}^{n_1})$ and $L^2(\mathbb{R}^{n_2})$, respectively, whose heat kernels satisfy Gaussian upper bounds. First, we obtain an atomic decomposition for func...

Let $L$ be the infinitesimal generator of an analytic semigroup $\{e^{-tL}\}_{t\ge0}$ on $L^2({\mathbb R}^n)$ with suitable upper bounds on its heat kernels. Given $1\leq p<\infty$ and $\lambda \in (0, n)$, a function $f$ (with appropriate bound on its size $|f|$) belongs to Campanato space ${\mathscr{L}_L^{p,\lambda}({\mathbb{R}^n})}$ associated t...

Let \((X, d, \mu )\) be a metric measure space endowed with a distance \(d\) and a nonnegative Borel doubling measure \(\mu \) . Let \(L\) be a second-order non-negative self-adjoint operator on \(L^2(X)\) . Assume that the semigroup \(e^{-tL}\) generated by \(L\) satisfies Gaussian upper bounds. In this article we establish a discrete characteriza...

Let $L$ be a nonnegative, self-adjoint operator satisfying Gaussian estimates on $L^2(\RR^n)$. In this article we give an atomic decomposition for the Hardy spaces $ H^p_{L,max}(\R)$ in terms of the nontangential maximal functions associated with the heat semigroup of $L$, and this leads eventually to characterizations of Hardy spaces associated to...

Let L be a Schrodinger operator of the form L=-\Delta+V acting on L^2(Rn) where the nonnegative potential V belongs to the reverse Holder class Bq for some q>= n. Let BMO_L(Rn) denote the BMO space on Rn associated to the Schrodinger operator L. In this article we will show that a function f in BMO_L(Rn) is the trace of the solution of L'u=-u_tt+Lu...

Let (X, d, µ) be a metric measure space endowed with a metric d and a nonnegative Borel doubling measure µ. Let L be a non-negative self-adjoint operator of order m on X. Assume that L generates a holomorphic semigroup e −tL whose kernels p t (x, y) satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables x and...

Let (X, d, µ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure µ. Let L be a second order self-adjoint positive operator on L 2(X). Assume that the semigroup e−tL generated by −L satisfies the Gaussian upper bounds on L 2(X). In this article we study a local version of Hardy space h L 1 (X) associated wi...

Motivated by the problem of spherical summability of products of Fourier series, we study the boundedness of the bilinear Bochner-Riesz multipliers $(1-|\xi|^2-|\eta|^2)^\delta_+$ and we make some advances in this investigation. We obtain an optimal result concerning the boundedness of these means from $L^2\times L^2 $ into $L^1$ with minimal smoot...

Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy generalized $m$-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjo...

Let L be a non-negative self-adjoint operator acting on L 2(X), where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e −tL whose kernel p t (x,y) has a Gaussian upper bound but there is no assumption on the regularity in variables x and y. In this article we study weighted L p -norm inequalities for spectral multi...

Let L be the generator of an analytic semigroup whose heat kernel satisfies an upper bound of Poisson type acting on L2(X) where X is a (possibly non-doubling) space of polynomial upper bound on volume growth. The aim of this paper is to introduce a new class of Besov spaces associated with the operator L so that when L is the Laplace operator −Δ o...

We consider abstract non-negative self-adjoint operators on $L^2(X)$ which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition which in the case of the standard Laplace operator is equivalent to $(p,2)$ restriction estimate of Stein and Tomas. Next we show t...

Let X be a metric space with doubling measure, and L be a non-negative, self-adjoint operator satisfying Davies-Gafiney bounds on L2(X). In this article we develop a theory of Hardy and BMO spaces as- sociated to L, including an atomic (or molecular) decomposition, square function characterization, duality of Hardy and BMO spaces. Further specializ...

This article is concerned with some weighted norm inequalities for the so-called horizontal (i.e. involving time derivatives) area integrals associated to a non-negative self-adjoint operator satisfying a pointwise Gaussian estimate for its heat kernel, as well as the corresponding vertical (i.e. involving space derivatives) area integrals associat...

Let L=−Δ+V be a Schrödinger operator on ℝn where V is a nonnegative function in the space L1loc(ℝn) of locally integrable functions on ℝn. In this paper we provide an atomic decomposition for the Hardy space H1L(ℝn) associated to L in terms of the maximal function characterization. We then adapt our argument to give an atomic decomposition for the...

Let (X, d, μ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ. Let L be a non-negative self-adjoint operator on L2(X). Assume that the semigroup e-tL generated by L satisfies the Davies-Gaffney estimates. Let HpL(X) be the Hardy space associated with L. We prove a Hörmander-type spectral multiplier the...

Let w be some ApAp weight and enjoy reverse Hölder inequality, and let L=−Δ+VL=−Δ+V be a Schrödinger operator on RnRn, where V∈Lloc1(Rn) is a non-negative function on RnRn. In this article we introduce weighted Hardy spaces HL,w1(Rn) associated to L in terms of the area function characterization, and prove their atomic characters. We show that the...

We prove a H 1 -coercive estimate for differential forms of arbitrary degrees in semi-convex domains of the Euclidean space. The key result is an integral identity involving a boundary term in which the Weingarten matrix of the boundary intervenes, established for any Lipschitz domain Ω⊆ℝ n whose outward unit normal ν belongs to L 1 n-1 (∂Ω), the L...

We study the fully inhomogeneous Dirichlet problem for the Laplacian in bounded convex domains in RnRn, when the size/smoothness of both the data and the solution are measured on scales of Besov and Triebel–Lizorkin spaces. As a preamble, we deal with the Dirichlet and Regularity problems for harmonic functions in convex domains, with optimal nonta...

Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ have Gaussian upper bounds but possess no regularity in variables $x$ and $y$. In this article, we study weighted $L^p$-norm inequalities for spectral multi...

The aim of this article is threefold. Firstly, we study Hardy spaces, h p L (Ω), associated with an operator L which is either the Dirichlet Laplacian ∆ D or the Neumann Laplacian ∆ N on a bounded Lipschitz domain Ω in R n , for 0 < p ≤ 1. We obtain equivalent characterizations of these function spaces in terms of maximal functions and atomic decom...

We obtain endpoint estimates for multilinear singular integrals operators whose kernels satisfy regularity conditions signicantly,weaker than those of the stan- dard Calder on-Zygmund kernels. As a consequence, we deduce endpoint L,=m estimates for the mth order commutator of Calder,on. Our results reproduce known estimates for m = 1; 2 but are new...

In this article we prove Cotlar's inequality for the maximal singular integrals associated with operators whose ker-nels satisfy regularity conditions weaker than those of the stan-dard m-linear Calderón-Zygmund kernels. The present study is motivated by the fundamental example of the maximal mth or-der Calderón commutators whose kernels are not re...

We obtain a local Tb theorem for singular integral operators on spaces of homogeneous type by using tree selection algorithm of the dyadic model and the BCR algorithm, which extends an earlier result of M. Christ [Colloq. Math. 60/61, No. 2, 601–628 (1990; Zbl 0758.42009)].

Let A=-(Ñ-i[(a)\vec])(Ñ-i[(a)\vec]) +V{A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V} be a magnetic Schrödinger operator acting on L2(\mathbb Rn){L^2({\mathbb R}^n)}, n≥ 1, where [(a)\vec]=(a1, ¼, an) Î L2loc(\mathbb Rn, \mathbb Rn){{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)} and 0 £ V Î L1loc(\mathbb Rn){0\...

Let L be the infinitesimal generator of an analytic semigroup on L²(ℝⁿ) with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space $H_{L}^{1}(ℝ^{n})$ and a $\text{BMO}_{L}(ℝ^{n})$ space associated with the operator L were introduced and studied. In this paper we define a class of $...

We characterize in terms of Fourier spectrum the boundary values of functions in the complex Hardy spaces H p(C±), 1 ≤ p ≤∞ . As an applica- tion we extend the Bedrosian identity, originally stated for square-integrable functions, to the Lp(R )c ases.

Let L be the infinitesimal generator of an analytic semigroup on L2 (ℝ) with suitable upper bounds on its heat kernels, and L has a bounded holomorphic functional calculus on L2 (ℝ). In this article, we introduce new function spaces H L1 (ℝ × ℝ) and BMOL(ℝ × ℝ) (dual to the space H L*1(ℝ × ℝ) in which L* is the adjoint operator of L) associated wit...

Given p ∈ [1,∞) and λ ∈ (0, n), we study Morrey space \(L^{p,\lambda}({\Bbb R}^n)\) of all locally integrable complex-valued functions f on \({\Bbb R}^n\) such that for every open Euclidean ball B ⊂ \({\Bbb R}^n\) with radius rB there are numbers C = C(f ) (depending on f ) and c = c(f,B) (relying upon f and B) satisfying $$r^{-\lambda}_B\sum_B \ve...

In this paper, we introduce some new function spaces of Sobolev type on metric measure spaces. These new function spaces are defined by variants of Poincaré inequalities associated with generalized approximations of the identity, and they generalize the classical Sobolev spaces on Euclidean spaces. We then obtain two characterizations of these new...

Given $p\in [1,\infty)$ and $\lambda\in (0,n)$, we study Morrey space ${\rm L}^{p,\lambda}({\mathbb R}^n)$ of all locally integrable complex-valued functions $f$ on ${\mathbb R}^n$ such that for every open Euclidean ball $B\subset{\mathbb R}^n$ with radius $r_B$ there are numbers $C=C(f)$ (depending on $f$) and $c=c(f,B)$ (relying upon $f$ and $B$)...

Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMO_L space associated with L which was introduced recently by Duong and Yan. We discuss applications of the new BMO_L spaces in the theory of singular integration such as BMO_L estimates and interpolation results for fractional powers, p...

Let A=-(Ñ-i[(a)\vec])2+VA=-(\nabla-i\vec{a})^2+V be a magnetic Schrödinger operator acting on L 2(R n ), n≥1, where [(a)\vec]=(a1,¼,an) Î L2loc\vec{a}=(a_1,\cdots,a_n)\in L^2_{\rm loc} and 0≤V∈L 1 loc. Following [1], we define, by means of the area integral function, a Hardy space H 1 A associated with A. We show that Riesz transforms (∂/∂x k -i...

Let X be a space of homogeneous type with finite measure. Let T be a singular integral operator which is bounded on Lp(X), 1 < p < ∞. We give a sufficient condition on the kernel k(x, y) of T so that when a function b ε BMO (X), the commutator [b, T] (f) = T (bf) − bT (f) is bounded on spaces Lp for all p, 1< p < ∞. Key wordsCommutator–homogeneous...

In this paper, we introduce and develop some new function spaces of BMO (bounded mean oscillation) type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type. The new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity, and they generalize the classical...

Let L L be the infinitesimal generator of an analytic semigroup on L 2 ( R n ) L^2({\mathbb R}^n) with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space H L 1 H_L^1 by means of an area integral function associated with the operator L L . By using a variant of the maximal function associated with the semig...

Let $\mathscr{X}$ be a space of homogeneous type. Assume that $L$ has a bounded holomorphic functional calculus on $L^2(\Omega)$ and $L$ generates a semigroup with suitable upper bounds on its heat kernels where $\Omega$ is a measurable subset of $\mathscr{X}$ . For appropriate bounded holomorphic functions $b$ , we can define the operators $b(L)$...

In this paper, we give a new characterization of the Morrey–Campanato spaces by using the convolution φ t B *f(x) to replace the minimizing polynomial P B f of a function f in the Morrey-Campanato norm, where is an appropriate Schwartz function.

Let T be a singular integral operator bounded on L-P(R-n) for some p, 1 < p < infinity. The authors give a sufficient condition on the kernel of T so that when b is an element ofBMO, the commutator [b, T](f) = T(bf) - bT(f) is bounded on the space L-P for all p, 1 < p < infinity). The condition of this paper is weaker than the usual pointwise Horma...

Let Ω be a special Lipschitz domain on Rn, and L be a second-order elliptic self-adjoint operator in divergence form L=−div(A∇) on Lipschitz domain Ω subject to Neumann boundary condition. In this paper, we give a simple proof of the atomic decomposition for Hardy spaces HNp(Ω) of Ω for a range of p, by means of nontangential maximal function assoc...

We make use of the Beylkin-Coifman-Rokhlin wavelet decomposition algorithm on the Calderón-Zygmund kernel to obtain some fine estimates on the operator and prove the T(l) theorem on Besov and Triebel-Lizorkin spaces. This extends previous results of Frazier et al., and Han and Hofmann.

We prove L p -estimates for the Littlewood-Paley function associated with a second order divergence form operator L=−div A∇ with bounded measurable complex coefficients in ℝn .

Let X be a space of homogeneous type, and L be the generator of a semigroup with Gaussian kernel bounds on L²(X). We define the Hardy spaces H_s^p(X) of X for a range of p, by means of area integral function associated with the Poisson semigroup of L, which is proved to coincide with the usual atomic Hardy spaces H<su...

The classical Hardy-Littlewood-Sobolev theorems for Riesz potentials (−Δ)−α/2 are extended to the generalised fractional integrals L –α/2 for 0 < α < n, where L=−div A∇ is a uniformly complex elliptic operator with bounded measurable coefficients in ℝn.

The aim of this paper is to establish a sufficient condition for certain weighted norm inequalities for singular integral operators with non-smooth kernels and for the commutators of these singular integrals with BMO functions. Our condition is applicable to various singular integral operators, such as the second derivatives of Green operators asso...

In this paper we use wavelets to characterize weighted Triebel-Lizorkin spaces. Our weights belong to the Muckenhoupt class Aq and our weighted Triebel-Lizorkin spaces. are weighted atomic Triebel-Lizorkin spaces.

We prove Lp -estimates for the Littlewood–Paley g-function associated with a complex elliptic operator L = − div A with bounded measurable coefficients in n.