BMO Spaces Research Articles

Products of Functions in $$H^1_{\rho }({{\mathcal {X}}})$$ H ρ 1 ( X ) and $${\mathrm {BMO}}_{\rho }({{\mathcal {X}}})$$ BMO ρ ( X ) over RD-Spaces and Applications to Schrödinger Operators

Let $$({{\mathcal {X}}},d,\mu )$$ be an RD-space, $$H^1_{\rho }({{\mathcal {X}}})$$ , and $${\mathrm {BMO}}_{\rho }({{\mathcal {X}}})$$ be, respectively, the local Hardy space and the local BMO space associated with an admissible function $$\rho $$ . Under an additional assumption that there exists a specific generalized approximation of the identity, the authors prove that the product $$f\times g$$ of $$f\in H^1_{\rho }({{\mathcal {X}}})$$ and $$g\in {\mathrm {BMO}}_{\rho }({{\mathcal {X}}})$$ , viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from $$H^1_{\rho }({{\mathcal {X}}})\times {\mathrm {BMO}}_{\rho } ({{\mathcal {X}}})$$ into $$L^1({{\mathcal {X}}})$$ and from $$H^1_{\rho }({{\mathcal {X}}}) \times {\mathrm {BMO}}_{\rho } ({{\mathcal {X}}})$$ into $$H^{\log }({{\mathcal {X}}})$$ , which is of wide generality. The authors also give out four applications of this result to Schrodinger operators, respectively, over different underlying spaces, where three of these applications are new.

Norm, essential norm and weak compactness of weighted composition operators between dual Banach spaces of analytic functions

In this paper we estimate the norm and the essential norm of weighted composition operators from a large class of – non-necessarily reflexive – Banach spaces of analytic functions on the open unit disk into weighted type Banach spaces of analytic functions and Bloch type spaces. We also show the equivalence of compactness and weak compactness of weighted composition operators from these weighted type spaces into a class of Banach spaces of analytic functions, that includes a large family of conformally invariant spaces like BMOA and analytic Besov spaces.

An equivalent characterization of BMO with Gauss measure

Let γ be the Gauss measure on ℝn: We establish a Calderon-Zygmund type decomposition and a John-Nirenberg type inequality in terms of the local sharp maximal function and the median value of function over cubes. As an application, we obtain an equivalent characterization of known BMO space with Gauss measure.

Weighted p-Adic Hardy Operators and Their Commutators on p-Adic Central Morrey Spaces

In this paper, we establish necessary and sufficient conditions for boundedness of weighted p-adic Hardy operators on p-adic Morrey spaces, p-adic central Morrey spaces and p-adic \(\lambda \)-central BMO spaces, respectively, and obtain their sharp bounds. We also give the characterization of weight functions for which the commutators generated by weighted p-adic Hardy operators and \(\lambda \)-central BMO functions are bounded on the p-adic central Morrey spaces. This result is different from that on Euclidean spaces due to the special structure of p-adic integers.

Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting

Fix λ > 0. Consider the Hardy space H 1 (R+,dm�) in the sense of Coifman and Weiss, where R+ := (0,∞) and dm� := x 2� dx with dx the Lebesgue measure. Also consider the Bessel operators △� := − d 2 dx2 −

Endpoint estimates for multilinear singular integral operators

Abstract In this paper, we establish three kinds of endpoint estimates for a class of multilinear singular integral operators and obtain the boundedness of this kind of multilinear singular integral operators on the product of BMO spaces, product of LMO spaces, and product of λ-central BMO spaces, respectively. Moreover, as special cases, the corresponding results of multilinear Calderón–Zygmund operators can be deduced.

Parabolic BMO and global integrability of supersolutions to doubly nonlinear parabolic equations

We prove that local and global parabolic BMO spaces are equal thus extending the classical result of Reimann and Rychener. Moreover, we show that functions in parabolic BMO are exponentially integrable in a general class of space-time cylinders. As a corollary, we establish global integrability for positive supersolutions to a wide class of doubly nonlinear parabolic equations.

The Holmes–Wick theorem on two-weight bounds for higher order commutators revisited

A sufficient condition for the two-weight boundedness of higher order commutators was recently obtained by Holmes and Wick in terms of an intersection of two BMO spaces. We provide an alternative proof, showing that the higher order case can be deduced by a classical Cauchy integral argument from the corresponding first order result of Holmes, Lacey and Wick.

A note on the essential norm of weighted composition operators on BMOA

We give some new estimates for the essential norm of weighted composition operators on the space BMOA. As a corollary, we obtain a new characterization for the compactness of weighted composition operators on the space BMOA.

Essential norm of a Volterra-type integral operator from Hardy spaces to some analytic function spaces

In this paper, we obtain some estimates of essential norm of the Volterra-type integral operator $T_g$, where \[ T_gf(z)=\int ^z_0f(\zeta )g'(\zeta )\,d\zeta , \] from Hardy spaces to the BMOA space, Besov spaces, Berg\-man spaces and Bloch-type spaces.

Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type

We give an explicit construction of Haar functions associated to a system of dyadic cubes in a geometrically doubling quasi-metric space equipped with a positive Borel measure, and show that these Haar functions form a basis for Lp. Next we focus on spaces X of homogeneous type in the sense of Coifman and Weiss, where we use these Haar functions to define a discrete square function, and hence to define dyadic versions of the function spaces H1(X) and BMO(X). In the setting of product spaces X˜=X1×⋯×Xn of homogeneous type, we show that the space BMO(X˜) of functions of bounded mean oscillation on X˜ can be written as the intersection of finitely many dyadic BMO spaces on X˜, and similarly for Ap(X˜), reverse-Hölder weights on X˜, and doubling weights on X˜. We also establish that the Hardy space H1(X˜) is a sum of finitely many dyadic Hardy spaces on X˜, and that the strong maximal function on X˜ is pointwise comparable to the sum of finitely many dyadic strong maximal functions. These dyadic structure theorems generalize, to product spaces of homogeneous type, the earlier Euclidean analogues for BMO and H1 due to Mei, and Li, Pipher and Ward.

Boundedness of certain commutators over non-homogeneous metric measure spaces

Let \((\mathcal {X},d,\mu )\) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let T be a Calderon-Zygmund operator with kernel satisfying only the size condition and some Hormander-type condition, and \(b\in \widetilde{\mathrm{RBMO}}(\mu )\) (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator \(T_b:=bT-Tb\) generated by T and b from the atomic Hardy space \(\widetilde{H}^1(\mu )\) with the discrete coefficient into the weak Lebesgue space \(L^{1,\,\infty }(\mu )\). From this and an interpolation theorem for sublinear operators which is also proved in this paper, the authors further show that the commutator \(T_b\) is bounded on \(L^p(\mu )\) for all \(p\in (1,\infty )\). Moreover, the boundedness of the commutator generated by the generalized fractional integral \(T_\alpha \,(\alpha \in (0,1))\) and the \(\widetilde{\mathrm{RBMO}}(\mu )\) function from \(\widetilde{H}^1(\mu )\) into \(L^{1/{(1-\alpha )},\,\infty }(\mu )\) is also presented.

Variation operators for semigroups and Riesz transforms acting on weighted $$L^p$$ L p and BMO spaces in the Schrödinger setting

In this paper, we extend the results of Schrodinger type operators (Betancor et al., Rev Mat Complut 26:485–534, 2013) and (Betancor et al., Potential Anal 38:711–739, 2013) to the weighted \({L^p({{\mathbb {R}}^{n}})}\) and \(\mathrm{BMO}\) spaces respectively, with the new type weight class defined (Bongioanni et al., J Math Anal Appl 373:563–579, 2011) which includes the Muckenhoupt class.

Carleson measures, riemann-stieltjes operators and multipliers on F(p, q, s) spaces in the unit ball of ℂn

This paper is devoted to characterizing the Riemann-Stieltjes operators and pointwise multipliers on F(p, q, s) spaces in the unit ball of ℂn, which contain many classical function spaces, such as the Bloch space, BMOA and Qs spaces. The boundedness and compactness of these operators on F(p,q,s) spaces are characterized by means of an embedding theorem, i.e., F(p,q,s) spaces boundedly embedded into the tent-type spaces T∞p,s (μ).

Some estimates for commutators of Riesz transform associated with Schrödinger type operators

Let L 1 = −Δ + V be a Schr:dinger operator and let L 2 = (−Δ)2 + V 2 be a Schrodinger type operator on ℝ n (n ⩾ 5), where V≠ 0 is a nonnegative potential belonging to certain reverse Holder class B s for s ⩾ n/2. The Hardy type space $$H_{{L_2}}^1$$ is defined in terms of the maximal function with respect to the semigroup $$\left\{ {{e^{ - t{L_2}}}} \right\}$$ and it is identical to the Hardy space $$H_{{L_1}}^1$$ established by Dziubanski and Zienkiewicz. In this article, we prove the L p -boundedness of the commutator R b = bRf - R(bf) generated by the Riesz transform $$R = {\nabla ^2}L_2^{ - 1/2}$$ , where $$b \in BM{O_\theta }(\varrho )$$ , which is larger than the space BMO(ℝ n ). Moreover, we prove that R b is bounded from the Hardy space $$H_{\mathcal{L}_1 }^1 $$ into weak $$L_{weak}^1 (\mathbb{R}^n )$$ .

Existence and asymptotic behavior to the incompressible nematic liquid crystal flow in the whole space

In this article, first of all, the global existence and asymptotic stability of solutions to the incompressible nematic liquid crystal flow is investigated when initial data are a small perturbation near the constant steady state (0,δ0); here, δ0 is a constant vector with |δ0|=1. Precisely, we show the existence and asymptotic stability with small initial data for . The initial data class of us is not entirely included in the space BMO−1×BMO and contains strongly singular functions and measures. As an application, we obtain a class of asymptotic existence of a basin of attraction for each self‐similar solution with homogeneous initial data. We also study global existence of a large class of decaying solutions and construct an explicit asymptotic formula for ∣x∣→∞, relating the self‐similar profile (U(x),D(x)) to its corresponding initial data (u0,d0). In two dimensions, we obtain higher‐order asymptotics of (u(x),d(x)). Copyright © 2016 John Wiley & Sons, Ltd.

Commutators in the two-weight setting

Let $R$ be the vector of Riesz transforms on $\mathbb{R}^n$, and let $\mu,\lambda \in A_p$ be two weights on $\mathbb{R}^n$, $1 < p < \infty$. The two-weight norm inequality for the commutator $[b, R] : L^p(\mathbb{R}^n;\mu) \to L^p(\mathbb{R}^n;\lambda)$ is shown to be equivalent to the function $b$ being in a BMO space adapted to $\mu$ and $\lambda$. This is a common extension of a result of Coifman-Rochberg-Weiss in the case of both $\lambda$ and $\mu$ being Lebesgue measure, and Bloom in the case of dimension one.

Hardy and Bmo Spaces on Graphs, Application to Riesz Transform

Let $\Gamma$ be a graph with the doubling property for the volume of balls and $P$ a reversible random walk on $\Gamma$. We introduce $H^1$ Hardy spaces of functions and $1$-forms adapted to $P$ and prove various characterizations of these spaces. We also characterize the dual space of $H^1$ as a $BMO$-type space adapted to $P$. As an application, we establish $H^1$-$H^1$ and $H^1$-$L^1$ boundedness of the Riesz transform.

Upper bound for multi-parameter iterated commutators

We show that the product BMO space can be characterized by iterated commutators of a large class of Calder\'on-Zygmund operators. This result follows from a new proof of boundedness of iterated commutators in terms of the BMO norm of their symbol functions, using Hyt\"onen's representation theorem of Calder\'on-Zygmund operators as averages of dyadic shifts. The proof introduces some new paraproducts which have BMO estimates.

Weighted BMO and Hankel operators between Bergman spaces

We introduce a family of weighted BMO spaces in the Bergman metric on the unit ball of C n and use them to characterize complex functions f such that the big Hankel operators Hf and H ¯ f are both bounded or compact from a weighted Bergman space into a weighted Lesbegue space with possibly different exponents and different weights. As a consequence, when the symbol function f is holomorphic, we characterize bounded and compact Hankel operators H ¯ f between weighted Bergman spaces. In particular,