Commutators of Multilinear Calderón-Zygmund Operator on Weighted Herz-Morrey Spaces with Variable Exponents

In this paper, we acquire the boundedness of commutators generated by multilinear Calderón-Zygmund operator and BMO functions on products of weighted Herz-Morrey spaces with variable exponents.

The Uniformly Parabolic Equations of Higher Order with Discontinuous Data in Generalized Morrey Spaces and Elliptic Equations in Unbounded Domains

We study the regularity of the solutions of the Cauchy-Dirichlet problem for linear uniformly parabolic equations of higher order with vanishing mean oscillation (VMO) coefficients. We prove continuity in generalized parabolic Morrey spaces Mp,φ of sublinear operators generated by the parabolic Calderon-Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. We obtain strong solution belongs to the generalized Sobolev-Morrey space Wp,φm,1∘Q. Also we consider elliptic equation in unbounded domains.

BMO Functions Generated by A X ℝ n Weights on Ball Banach Function Spaces

Let X be a ball Banach function space on ℝ n . We introduce the class of weights A X ℝ n . Assuming that the Hardy-Littlewood maximal function M is bounded on X and X ′ , we obtain that BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A X ℝ n . As a consequence, we have BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A L p · ℝ n ℝ n , where L p · ℝ n is the variable exponent Lebesgue space. As an application, if a linear operator T is bounded on the weighted ball Banach function space X ω for any ω ∈ A X ℝ n , then the commutator b , T is bounded on X with b ∈ BMO ℝ n .

On BMO and Carleson Measures on Riemannian Manifolds

Let $\mathcal{M}$ be a Riemannian $n$-manifold with a metric such that the manifold is Ahlfors regular. We also assume either non-negative Ricci curvature or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions $u: \mathcal{M} \to \mathbb{R}$ by a Carleson measure condition of their $\sigma $-harmonic extension $U: \mathcal{M} \times (0,\infty ) \to \mathbb{R}$. We make crucial use of a $T(b)$ theorem proved by Hofmann, Mitrea, Mitrea, and Morris. As an application, we show that the famous theorem of Coifman–Lions–Meyer–Semmes holds in this class of manifolds: Jacobians of $W^{1,n}$-maps from $\mathcal{M}$ to $\mathbb{R}^n$ can be estimated against BMO-functions, which now follows from the arguments for commutators recently proposed by Lenzmann and the 2nd-named author using only harmonic extensions, integration by parts, and trace space characterizations.

Commutators of the Bilinear Hardy Operator on Herz Type Spaces with Variable Exponents

In this paper, we obtain the boundedness of bilinear commutators generated by the bilinear Hardy operator and BMO functions on products of Herz spaces and Herz-Morrey spaces with variable exponents.

Sharp maximal estimates for multilinear commutators of multilinear strongly singular Calderón–Zygmund operators and applications

In this paper, we aim to establish the sharp maximal pointwise estimates for the multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions or Lipschitz functions, respectively. As applications, the boundedness of these multilinear commutators on product of weighted Lebesgue spaces are obtained. It is interesting to note that there is no size condition assumption for the kernel of the multilinear strongly singular Calderón–Zygmund operator. Due to the stronger singularity for the kernel of the multilinear strongly singular Calderón–Zygmund operator, we need to be more careful in estimating the mean oscillation over the small balls to get the sharp maximal function estimates.

Higher-Order Commutators of Parametric Marcinkiewicz Integrals on Herz Spaces with Variable Exponent

LetΩ∈Ls(Sn-1)fors⩾1be a homogeneous function of degree zero andbbe BMO functions. In this paper, we obtain some boundedness of the parametric Marcinkiewicz integral operatorμΩρand its higher-order commutator[bm,μΩρ]on Herz spaces with variable exponent.