Interpolation between noncommutative martingale Hardy and BMO spaces: the case

Let $\mathcal {M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal {M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal {M}$ . For $0<p <\infty $ , let $\mathsf {h}_p^c(\mathcal {M})$ denote the noncommutative column conditioned martingale Hardy space and $\mathsf {bmo}^c(\mathcal {M})$ denote the column “little” martingale BMO space associated with the filtration $(\mathcal {M}_n)_{n\geq 1}$ . We prove the following real interpolation identity: if $0<p <\infty $ and $0<\theta <1$ , then for $1/r=(1-\theta )/p$ , $$ \begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$ with equivalent quasi norms. For the case of complex interpolation, we obtain that if $0<p<q<\infty $ and $0<\theta <1$ , then for $1/r =(1-\theta )/p +\theta /q$ , $$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$ with equivalent quasi norms. These extend previously known results from $p\geq 1$ to the full range $0<p<\infty $ . Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned $L_p$ -spaces are also shown to form interpolation scale for the full range $0<p<\infty $ when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned $L_p$ -spaces. We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

Weighted Lebesgue and central Morrey estimates for -adic multilinear Hausdorff operators and its commutators

UDC 517.9 We establish the sharp boundedness of -adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights and Muckenhoupt weights. Moreover, the boundedness for the commutators of -adic multilinear Hausdorff operators on the such spaces with symbols in central BMO space is also obtained.

On the Space of Bounded Mean Oscillations

The space of bounded mean oscillations, abbreviated BMO, was first introduced by F. John and L. Nirenberg in 1961 in the context of partial differential equations. Later, C. Fefferman proved that the BMO is the dual space of well-known Hardy space, popularly known as H1 space and became the center of attraction for mathematicians. With the help of BMO space, many mathematical phenomenon can be characterized clearly. In this article, we discuss the connections of function of bounded mean oscillations with weight functions, sharp maximal functions and Carleson measure.

Two characterizations of central BMO space via the commutators of Hardy operators

This article addresses two characterizations of BMO ⁢ ( ℝ n ) {\mathrm{BMO}(\mathbb{R}^{n})} -type space via the commutators of Hardy operators with homogeneous kernels on Lebesgue spaces: (i) characterization of the central BMO ⁢ ( ℝ n ) {\mathrm{BMO}(\mathbb{R}^{n})} space by the boundedness of the commutators; (ii) characterization of the central BMO ⁢ ( ℝ n ) {\mathrm{BMO}(\mathbb{R}^{n})} -closure of C c ∞ ⁢ ( ℝ n ) {C_{c}^{\infty}(\mathbb{R}^{n})} space via the compactness of the commutators. This is done by exploiting the center symmetry of Hardy operator deeply and by a more explicit decomposition of the operator and the kernel function.

A Characterization of Central BMO Space via the Commutator of Fractional Hardy Operator

This paper is devoted in characterizing the central BMO ℝn space via the commutator of the fractional Hardy operator with rough kernel. Precisely, by a more explicit decomposition of the operator and the kernel function, we will show that if the symbol function belongs to the central BMO ℝn space, then the commutator are bounded on Lebesgue space. Conversely, the boundedness of the commutator implies that the symbol function belongs to the central BMO ℝn space by exploiting the center symmetry of the Hardy operator deeply.

Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type

In this paper, we study the boundedness and compactness of the commutator of Calderón– Zygmund operators T on spaces of homogeneous type (X, d, µ) in the sense of Coifman and Weiss. More precisely, we show that the commutator [b, T] is bounded on the weighted Morrey space L ω p , k ( X ) L_\omega ^{p,k}\left( X \right) with κ ∈ (0, 1) and ω ∈ Ap (X), 1 < p < ∞, if and only if b is in the BMO space. We also prove that the commutator [b, T] is compact on the same weighted Morrey space if and only if b belongs to the VMO space. We note that there is no extra assumptions on the quasimetric d and the doubling measure µ.

Well-posedness and blow-up criterion for the Chaplygin gas equations in ℝN

We establish a well-posedness theory and a blow-up criterion for the Chaplygin gas equations in [Formula: see text] for any dimension [Formula: see text]. First, given [Formula: see text], [Formula: see text], we prove the well-posedness property for solutions [Formula: see text] in the space [Formula: see text] for the Cauchy problem associated with the Chaplygin gas equations, provided the initial density [Formula: see text] is bounded below. We also prove that the solution of the Chaplygin gas equations depends continuously upon its initial data [Formula: see text] in [Formula: see text] if [Formula: see text], and we state a blow-up criterion for the solutions in the classical BMO space. Finally, using Osgood’s modulus of continuity, we establish a refined blow-up criterion of the solutions.

Lorentz Estimates for Weak Solutions of Quasi-linear Parabolic Equations with Singular Divergence-free Drifts

This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.

A note on transport equation in quasiconformally invariant spaces

In this note, we study the well-posedness of the Cauchy problem for the transport equation in the BMO space and certain Triebel–Lizorkin spaces.