BMO Spaces Research Articles

Duality, BMO and Hankel operators on Bernstein spaces

In this paper we deal with the problem of describing the dual space (Bκ1)⁎ of the Bernstein space Bκ1, that is the space of entire functions of exponential type (at most) κ>0 whose restriction to the real line is Lebesgue integrable. We provide several characterizations, showing that such dual space can be described as a quotient of the space of entire functions of exponential type κ whose restriction to the real line are in a suitable BMO-type space, or as the space of symbols b for which the Hankel operator Hb is bounded on the Paley–Wiener space Bκ/22. We also provide a characterization of (Bκ1)⁎ as the BMO space w.r.t. the Clark measures of the inner function ei2κz on the upper half-plane, in analogy with the known description of the dual of backward-shift invariant 1-spaces on the torus. Furthermore, we show that the orthogonal projection Pκ:L2(R)→Bκ2 induces a bounded operator from L∞(R) onto (Bκ1)⁎.Finally, we show that Bκ1 is the dual space of a suitable VMO-type space or as the space of symbols b for which the Hankel operator Hb on the Paley–Wiener space Bκ/22 is compact.

Sharp analytic version of Fefferman's inequality

Let T be a unit circle. Assume further that f is an element of the Hardy space H1(T) and g belongs to the analytic BMO space on T. The paper contains the identification of the optimal universal constant C in the estimate|12π∫Tf(ζ)‾g(ζ)dζ|≤C‖f‖H1(T)‖g‖BMO(T). Actually, the inequality is studied in the stronger form, involving the Littlewood-Paley function on the left and the sharp maximal function of g on the right. The proof rests on the construction of an appropriate plurisuperharmonic function on a parabolic domain and the application of probabilistic techniques.

Weighted Hardy factorizations in terms of Multilinear Calderón-Zygmund and fractional integral operators

The foundational paper of Coifman-Rochberg-Weiss provided a constructive proof of the weak factorizations of the classical Hardy space in terms of Riesz transforms. In this paper, we establish the factorization theorems for the weighted Hardy spaces. The results are also extended to the multilinear setting. Furthermore, we show that the weighted BMO space and weighted Lipschitz spaces are necessary for the weighted boundedness of commutators of the multilinear Calderón-Zygmund and fractional integral operators, respectively.

Complex interpolation between noncommutative martingale BMO spaces and Hardy–Orlicz spaces

Complex interpolation between noncommutative martingale BMO spaces and Hardy–Orlicz spaces

Some variable exponent boundedness and commutators estimates for fractional Rough Hardy operators on central Morrey space

In this article, we study the boundedness of the fractional Rough Hardy operator and its adjoint operators on the central Morrey space with a variable exponent. We also establish the same boundedness for their commutators when the symbol functions are on the λ-central BMO space with a variable exponent.

Blow up criteria for three-dimensional incompressible Navier-Stokes-Landau-Lifshitz system in the whole space

In this paper, we investigate the blow up criteria for the local smooth solutions to the three-dimensional incompressible Navier-Stokes-Landau-Lifshitz system via the components of the velocity field u or velocity gradient ∇u and the gradient orientation field ∇d. More precisely, the first result is the Serrin-type blow up criterion with the components of u and ∇d. The remaining is relevant to ∇u and ∇d corresponding to the Besov space with negative index. As corollaries, the blow up criteria in both of the Besov space with negative index and BMO space with reference to u and ∇d are obtained.

Weighted estimates for fractional bilinear Hardy operators on variable exponent Morrey–Herz space

In this article, we analyze the boundedness for the fractional bilinear Hardy operators on variable exponent weighted Morrey–Herz spaces MK˙q,p(⋅)α(⋅),λ(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${M\\dot{K}^{\\alpha (\\cdot ),\\lambda}_{q,p(\\cdot)}(w)}$\\end{document}. Similar estimates are obtained for their commutators when the symbol functions belong to BMO space with variable exponents.

Characterizations of a class of weighted 𝐵𝑀𝑂 spaces via commutators of intrinsic square functions

The intrinsic square functions including the Lusin area function, Littlewood-Paley g g -function and g λ ∗ g^\ast _\lambda -function dominate pointwisely the classical Littlewood-Paley functions and can be used to characterize the weighted Hardy spaces and more general Musielak-Orlicz Hardy spaces etc. This paper shows that for b ∈ B M O ( R n ) b\in \mathrm {BMO(\mathbb {R}^n)} , the commutators generated by these intrinsic square functions with b b are bounded from H ω p ( R n ) H^p_\omega (\mathbb {R}^n) to L ω p ( R n ) L^p_\omega (\mathbb {R}^n) for some 0 > p ≤ 1 0>p\le 1 and ω ∈ A ∞ \omega \in A_\infty if and only if b ∈ B M O ω , p ( R n ) b\in \mathcal {BMO}_{\omega ,p}(\mathbb {R}^n) , which are a class of non-trivial subspaces of B M O ( R n ) \mathrm {BMO(\mathbb {R}^n)} .

The commutator of the Bergman projection on strongly pseudoconvex domains with minimal smoothness

Consider a bounded, strongly pseudoconvex domain D⊂Cn with minimal smoothness (namely, the class C2) and let b be a locally integrable function on D. We characterize boundedness (resp., compactness) in Lp(D),p>1, of the commutator [b,P] of the Bergman projection P in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on b. We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N. Kerzman & E. M. Stein, and E. Ligocka.

Riesz transform associated with the fractional Fourier transform and applications in image edge detection

The fractional Hilbert transform was introduced by Zayed [30, Zayed, 1998] and has been widely used in signal processing. In view of its connection with the fractional Fourier transform, Chen, the first, second and fourth authors of this paper in [6, Chen et al., 2021] studied the fractional Hilbert transform and other fractional multiplier operators on the real line. The present paper is concerned with a natural extension of the fractional Hilbert transform to higher dimensions: this extension is the fractional Riesz transform and is given by multiplication which a suitable chirp function on the fractional Fourier transform side. In addition to a thorough study of the fractional Riesz transform, in this work we also investigate the boundedness of singular integral operators with chirp functions on rotation invariant spaces, chirp Hardy spaces and their relation to chirp BMO spaces, as well as applications of the theory of fractional multipliers in partial differential equations. Through numerical simulation, we provide physical and geometric interpretations of high-dimensional fractional multipliers. Finally, we present an application of the fractional Riesz transforms in edge detection which verifies a hypothesis insinuated in [26, Xu et al., 2016]. In fact our numerical implementation confirms that amplitude, phase, and direction information can be simultaneously extracted by controlling the order of the fractional Riesz transform.

Hardy and BMO spaces on Weyl chambers

Abstract Let W be a finite reflection group associated with a root system R in ℝ d {\mathbb{R}^{d}} . Let C + {C_{+}} denote a positive Weyl chamber distinguished by a choice of R + {R_{+}} , a set of positive roots. We define and investigate Hardy and BMO spaces on C + {C_{+}} in the framework of boundary conditions given by a homomorphism η ∈ Hom ⁡ ( W , ℤ ^ 2 ) {\eta\in\operatorname{Hom}(W,\widehat{\mathbb{Z}}_{2})} which attaches the ± {\pm} signs to the facets of C + {C_{+}} . Specialized to orthogonal root systems, atomic decompositions in H η 1 {H^{1}_{\eta}} and h η 1 {h^{1}_{\eta}} are obtained and the duality problem is also treated.

A note on commutators of singular integrals with BMO and VMO functions in the Dunkl setting

AbstractOn equipped with a root system R, multiplicity function , and the associated measure , we consider a (nonradial) kernel , which has properties similar to those from the classical theory of singular integrals and the Dunkl convolution operator associated with K. Assuming that b belongs to the BMO space on the space of homogeneous type , we prove that the commutator is a bounded operator on for all . Moreover, is compact on , provided . The paper extends results of Han, Lee, Li, and Wick.

Characterizations of the BMO and Lipschitz Spaces via Commutators on Weak Lebesgue and Morrey Spaces

Characterizations of the BMO and Lipschitz Spaces via Commutators on Weak Lebesgue and Morrey Spaces

Duality of generalized Hardy and BMO spaces associated with singular partial differential operator

Duality of generalized Hardy and BMO spaces associated with singular partial differential operator

Pointwise multipliers on Orlicz-Campanato spaces

Let ψ:(0,∞)→(0,∞) be a non-decreasing function with critical lower and upper type indices σ−⁎ and σ+⁎. Given a Campanato type space Cψ(Rn) that was defined associated to the function ψ, under σ−⁎>⌊σ+⁎⌋−1 when n≥2 or σ−⁎>σ+⁎−1 when n=1, the authors prove that any function is a pointwise multiplier on Cψ(Rn) if and only if it is bounded and belongs to a Musielak-Orlicz-Campanato space CΨ(Rn) that was associated toΨ(x,r):=|∫r1ψ(t)dtt|+∫1max⁡{r,e+|x|}(max⁡{r,e+|x|}t)⌊σ+⁎⌋ψ(t)dtt, where x∈Rn and r∈(0,∞). This generalizes the known characterizations of pointwise multipliers on the BMO(Rn) space and the Campanato space.

Operator-valued anisotropic Hardy and BMO spaces

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ, and A be an expansive dilation on Rn. In this article, the author introduces the operator-valued anisotropic BMO space BMOA(Rn,M) and obtains a predual space of BMOA(Rn,M), which is the operator-valued anisotropic Hardy space HA1(Rn,M). Moreover, as the Lq-spaces analogues of the BMO spaces, the author introduces the operator-valued anisotropic BMO-type space LqMOA(Rn,M)(2<q<∞), and establishes the duality between LqMOA(Rn,M) and the operator-valued anisotropic Hardy space HAp(Rn,M) with p=(q−1)/q. In addition, the author also obtains some interpolation results and the equivalence of HAp(Rn,M) and Lp(Rn,M) with p∈(1,∞). As an application, the boundedness of anisotropic Calderón-Zygmund operators on BMOA(Rn,M) is established. When A:=2In×n, part of results coincides with those of T. Mei [33], where In×n denotes the n×n unit matrix.

Calderón–Zygmund operators on RBMO

Let μ \mu be an n n -dimensional finite positive measure on R m \mathbb {R}^m . We obtain a T 1 T1 condition sufficient for the boundedness of Calderón–Zygmund operators on R B M O ( μ ) \mathrm {RBMO}(\mu ) , the regular BMO space of Tolsa [Math. Ann. 319 (2001), pp. 89–149].

Parametric Marcinkiewicz Integral and Its Higher-Order Commutators on Variable Exponents Morrey-Herz Spaces

In this article, we prove the boundedness of the parametric Marcinkiewicz integral and its higher-order commutators generated by BMO spaces on the variable Morrey-Herz space. All the results are new even when α · is a constant.

Quantitative Hölder estimates for even singular integral operators on patches

In this paper we show a constructive method to obtain C˙σ estimates of even singular integral operators on characteristic functions of domains with C1+σ regularity, 0<σ<1. This kind of functions were shown in first place to be bounded (classically only in the BMO space) to obtain global regularity for the vortex patch problem [5,2]. This property has then been applied to solve different type of problems in harmonic analysis and PDEs. Going beyond in regularity, the functions are discontinuous on the boundary of the domains, but C˙σ in each side. This C˙σ regularity has been bounded by the C1+σ norm of the domain [8,14,16]. Here we provide a quantitative bound linear in terms of the C1+σ regularity of the domain. This estimate shows explicitly the dependence of the lower order norm and the non-self-intersecting property of the boundary of the domain. As an application, this quantitative estimate is used in a crucial manner to the free boundary incompressible Navier-Stokes equations providing new global-in-time regularity results in the case of viscosity contrast [12].

Hardy spaces associated to generalized Hardy operators and applications

In this paper, we will study the Hardy and BMO spaces associated to the generalized Hardy operator L_{alpha }= (-Delta )^{alpha /2}+a|x|^{-alpha }. Similarly to the classical Hardy and BMO spaces, we will prove that our new function spaces will enjoy some important results such as molecular decomposition and duality. As applications, we show the boundedness of the spectral multiplier of Laplace transform type and the Sobolev norm inequalities involving the generalized Hardy operator.