BMO Spaces Research Articles

Real interpolation of martingale Orlicz Hardy spaces and BMO spaces

In this article, the authors prove that the real interpolation spaces between martingale Orlicz Hardy spaces and martingale BMO spaces are martingale Orlicz–Lorentz Hardy spaces. Using sharp maximal functions, the authors also establish the characterizations of martingale Orlicz Hardy spaces.

Weighted Central BMO Spaces and Their Applications

In this paper, the central BMO spaces with Muckenhoupt A p weight is introduced. As an application, we characterize these spaces by the boundedness of commutators of Hardy operator and its dual operator on weighted Lebesgue spaces. The boundedness of vector-valued commutators on weighted Herz spaces is also considered.

Characterizations of Weighted BMO Space and Its Application

In this paper, we prove that the weighted BMO space $${\rm{BM}}{{\rm{O}}^p}(\omega ) = \left\{ {f \in L_{{\rm{loc}}}^1:\mathop {\sup }\limits_Q \left\| {{\chi _Q}} \right\|_{{L^p}(\omega )}^{ - 1}{{\left\| {(f - {f_Q}){\omega ^{ - 1}}{\chi _Q}} \right\|}_{{L^p}(\omega )}} < \infty } \right\}$$ is independent of the scale p ∈ (0, ∞) in sense of norm when ω ∈ A1. Moreover, we can replace Lp(ω) by Lp,∞(ω). As an application, we characterize this space by the boundedness of the bilinear commutators [b, T]j(j = 1, 2), generated by the bilinear convolution type Calderón-Zygmund operators and the symbol b, from \({L^{{p_1}}}(\omega ) \times {L^{{p_2}}}(\omega )\) to Lp(ω1−p) with 1 < p1,p2 < ∞ and 1/p =1/p1 + 1/p2. Thus we answer the open problem proposed by Chaffee affirmatively.

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.

Biparameter BMO under the action of a rotation

In this work, we aim to study the action of composing by a rotation on the biparameter BMO space in R2. This BMO space is not preserved by a rotation since it relies on the structure of axis-parallel rectangles. We will quantify this fact by interpolation inequalities. One straightforward application of the interpolation inequalities is a boundedness property of directional Hilbert transforms.

Functions of Bounded Mean Oscillation and Quasisymmetric Mappings on Spaces of Homogeneous Type

We establish a connection between the function space BMO and the theory of quasisymmetric mappings on \emph{spaces of homogeneous type} $\widetilde{X} :=(X,\rho,\mu)$. The connection is that the logarithm of the generalised Jacobian of an $\eta$-quasisymmetric mapping $f: \widetilde{X} \rightarrow \widetilde{X}$ is always in $\rm{BMO}(\widetilde{X})$. In the course of proving this result, we first show that on $\widetilde{X}$, the logarithm of a reverse-H\"{o}lder weight $w$ is in $\rm{BMO}(\widetilde{X})$, and that the above-mentioned connection holds on metric measure spaces $\widehat{X} :=(X,d,\mu)$. Furthermore, we construct a large class of spaces $(X,\rho,\mu)$ to which our results apply. Among the key ingredients of the proofs are suitable generalisations to $(X,\rho,\mu)$ from the Euclidean or metric measure space settings of the Calder\'{o}n--Zygmund decomposition, the Vitali Covering Theorem, the Radon--Nikodym Theorem, a lemma which controls the distortion of sets under an $\eta$-quasisymmetric mapping, and a result of Heinonen and Koskela which shows that the volume derivative of an $\eta$-quasisymmetric mapping is a reverse-H\"{o}lder weight.

Recent results for the Landau–Lifshitz equation

We give a survey on some recent results concerning the Landau–Lifshitz equation, a fundamental nonlinear PDE with a strong geometric content, describing the dynamics of the magnetization in ferromagnetic materials. We revisit the Cauchy problem for the anisotropic LL equation, without dissipation, for smooth solutions, and also in the energy space in dimension one. We also examine two approximations of the LL equation given by of the Sine–Gordon equation and cubic Schrödinger equations, arising in certain singular limits of strong easy-plane and easy-axis anisotropy, respectively. Concerning localized solutions, we review the orbital and asymptotic stability problems for a sum of solitons in dimension one, exploiting the variational nature of the solitons in the hydrodynamical frameworkFinally, we survey results concerning the existence, uniqueness and stability of self-similar solutions (expanders and shrinkers) for the isotropic LL equation with Gilbert term. Since expanders are associated with a singular initial condition with a jump discontinuity, we also review their well-posedness in spaces linked to the BMO space.

Bilinear Pseudo-Differential Operators with Exotic Class Symbols of Limited Smoothness

We consider the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class \(BS_{\rho , \rho }^m ({\mathbb {R}}^n) \) for \(m \in {\mathbb {R}}\) and \(0 \le \rho < 1\). The aim of this paper is to discuss smoothness conditions for symbols to assure the boundedness from \(h^p \times h^q\) to \(h^r\) for \(1 \le r \le 2 \le p,q \le \infty \) satisfying \(1/p+1/q=1/r\), where \(h^\infty \) is replaced by the space bmo.

Closed range integral operators on Hardy, BMOA and Besov spaces

ABSTRACT If , the integral operator on , BMOA and (Besov) spaces, is defined as . In this paper, we prove three necessary and sufficient conditions for the operator to have closed range on , BMOA and .

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.

A characterization of via the commutator of Hardy-type operators on mixed Herz spaces

In this paper, we introduce and study the mixed central bounded mean oscillation space , which is a new central version of the bounded mean oscillation spaces BMO. The pre-dual of is proved to be the mixed Herz–Hardy space . Moreover, we also give a characterization of via the boundedness of the commutators of n-dimensional Hardy operator and its dual operator on mixed Herz spaces .

Gagliardo–Nirenberg and Sobolev interpolation inequalities on Besov spaces

We establish the Gagliardo–Nirenberg and Sobolev interpolation inequalities on Besov spaces. In particular, we improve the recent results of Miyazaki where he proved the inequalities on BMO spaces (see Yoichi Miyazaki [Proc. Amer. Math. Soc. 148 (2020), pp. 4257–4261]).

Characterizations of Product Hardy Spaces in Bessel Setting

In this paper, we work in the setting of Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and the harmonic function theory in this setting introduced by Muckenhoupt–Stein, especially the generalised Cauchy–Riemann equations and the conjugate harmonic functions. We provide the equivalent characterizations of product Hardy spaces associated with Bessel operators in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions defined via Poisson and heat semigroups, based on the atomic decomposition, the generalised Cauchy–Riemann equations, the extension of Merryfield’s result which connects the product non-tangential maximal function and area function, and on the grand maximal function technique which connects the product non-tangential and radial maximal function. We then obtain directly the decomposition of the product BMO space associated with Bessel operators.

Weighted variable Lorentz regularity for degenerate elliptic equations in Reifenberg domains

Let w be a Muckenhoupt weight and Ω a Reifenberg flat domain in . Assume that q ∈ (0, ∞] and p(·) : Ω → (1, ∞) is a variable exponent satisfying the ‐Hölder continuous condition. In this article, the authors investigate the weighted variable Lorentz Lp(·), q(Ω, w) regularity of the weak solutions of second‐order degenerate elliptic equations in divergence form with Dirichlet boundary condition, under the assumption that the degenerate coefficients belong to weighted BMO spaces with small norms.

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

Abstract 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.

Some characterizations of the generalized BMO spaces via the boundedness of maximal function commutators

In this paper, we investigated the commutators of Hardy–Littlewood maximal function with symbol function b in generalized BMO spaces. Some characterizations of generalized BMO spaces are obtained via the strong and weak boundedness of the commutators on Orlicz–Morrey space.

Some endpoint estimates for bilinear Coifman-Meyer multipliers

In this paper we establish mapping properties of bilinear Coifman-Meyer multipliers acting on the product spaces H1(Rn)×bmo(Rn) and Lp(Rn)×bmo(Rn), with 1<p<∞. As application of these results, we obtain some related Kato-Ponce-type inequalities involving the endpoint space bmo(Rn), and we also study the pointwise product of a function in bmo(Rn) with functions in H1(Rn), h1(Rn) and Lp(Rn), with 1<p<∞.

Commutators of Hilbert transforms along monomial curves

The Hilbert transforms associated with monomial curves have a natural non-isotropic structure. We study the commutator of such Hilbert transforms and a symbol $b$ and prove the upper bound of this commutator when $b$ is in the corresponding non-isotropic BMO space by using the Cauchy integral trick. We also consider the lower bound of this commutator by introducing a new testing BMO space associated with the given monomial curve, which shows that the classical non-isotropic BMO space is contained in the testing BMO space. We also show that the non-zero curvature of such monomial curves are important, since when considering Hilbert transforms associated with lines, the parallel version of non-isotropic BMO space and testing BMO space have overlaps but do not have containment.

An explicit formula of Cauchy-Szego kernel for quaternionic Siegel upper half space and applications

In this paper we obtain an explicit formula of Cauchy--Szego kernel for quaternionic Siegel upper half space, and then based on this, we prove that the Cauchy--Szego projection on quaternionic Heisenberg group is a Calderon--Zygmund operator via verifying the size and regularity conditions for the kernel. Next, we also obtain a suitable version of pointwise lower bound for the kernel, which further implies the characterisations of the boundedness and compactness of commutator of the Cauchy--Szego operator via the BMO and VMO spaces on quaternionic Heisenberg group, respectively.

Essential norm of generalized Hilbert matrix from Bloch type spaces to BMOA and Bloch space

Let $ \mu $ be a positive Borel measure on the interval $ [0, 1) $. The Hankel matrix $ {\mathcal H}_\mu = (\mu_{n+k})_{n, k\geq 0} $ with entries $ \mu_{n, k} = \mu_{n+k} $ induces the operator $ {\mathcal H}_\mu(f)(z) = \sum\limits_{n = 0}^\infty\left(\sum\limits_{k = 0}^\infty\mu_{n,k}a_k\right)z^n $ on the space of all analytic functions $ f(z) = \sum^\infty_{n = 0}a_nz^n $ in the unit disk $ {\mathbb{D}} $. In this paper, we characterize the boundedness and compactness of $ {\mathcal H}_\mu $ from Bloch type spaces to the BMOA and the Bloch space. Moreover we obtain the essential norm of $ {\mathcal H}_\mu $ from $ {\mathcal{B}}^\alpha $ to $ {\mathcal{B}} $ and BMOA.