Characterization Of BMO Research Articles

Gradient estimates for Schrödinger operators with characterizations of 𝐵𝑀𝑂_{ℒ} on Heisenberg groups

Let L = − Δ H n + V \mathcal {L}=-\Delta _{\mathbb {H}^n}+V be a Schrödinger operator with the nonnegative potential V V belonging to the reverse Hölder class B Q B_{Q} , where Q Q is the homogeneous dimension of the Heisenberg group H n \mathbb {H}^n . In this paper, we obtain pointwise bounds for the spatial derivatives of the heat and Poisson kernels related to L \mathcal {L} . As an application, we characterize the space B M O L ( H n ) BMO_{\mathcal {L}}(\mathbb {H}^n) , associated to the Schrödinger operator L \mathcal {L} , in terms of two Carleson type measures involving the spatial derivatives of the heat kernel of the semigroup { e − s L } s > 0 \{e^{-s\mathcal {L}}\}_{s>0} and the Poisson kernel of the semigroup { e − s L } s > 0 \{e^{-s\sqrt {\mathcal {L}}}\}_{s>0} , respectively. At last, we pose a conjecture about the converse characterization of B M O L ( H n ) BMO_{\mathcal {L}}(\mathbb {H}^n) .

The John\u2013Nirenberg inequality in ball Banach function spaces and application to characterization of BMO

Our goal is to obtain the John–Nirenberg inequality for ball Banach function spaces X, provided that the Hardy–Littlewood maximal operator M is bounded on the associate space X' by using the extrapolation. As an application we characterize BMO, the bounded mean oscillation, via the norm of X.

Extrapolation to Herz spaces with variable exponents and applications

This paper gives an extension of the extrapolation theory to Herz spaces with variable exponents. By using this extrapolation theory, we establish the Fefferman–Stein inequalities, the Rubio de Francia inequalities, the John–Nirenberg inequalities, the characterizations of BMO and the boundedness of the geometrical maximal operator on Herz spaces with variable exponents.

A characterization of BMO in terms of endpoint bounds for commutators of singular integrals

We provide a characterization of BMO in terms of endpoint boundedness of commutators of singular integrals. In particular, in one dimension, we show that ∥b∥BMO ≃ B, where B is the best constant in the endpoint L log L modular estimate for the commutator [H, b]. We provide a similar characterization of the space BMO in terms of endpoint boundedness of higher order commutators of the Hilbert transform. In higher dimension we give the corresponding characterization of BMO in terms of the first order commutators of the Riesz transforms. We also show that these characterizations can be given in terms of commutators of more general singular integral operators of convolution type.

On an Equivalent Norm on the Space BMO

We extend the inequality proved by S. V. Bochkarev to a larger class of convolution operators assuming that the Fourier transforms of the kernels of these operators satisfy certain conditions in the spirit of the Hormander–Mikhlin multiplier theorem. Therefore, we give a new characterization of BMO.

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.

Vector-Valued John–Nirenberg Inequalities and Vector-Valued Mean Oscillations Characterization of BMO

We establish a vector-valued John–Nirenberg inequality for oscillations measured in a Banach function space (B.f.s.) norm. This inequality generalizes several existing results on John–Nirenberg inequalities on function spaces such as rearrangement-invariant B.f.s. and Lebesgue spaces with variable exponents. Moreover, this inequality also offers a new characterization of BMO in terms of the weighted vector-valued mean oscillation.

Synthesis of 1,4-butanediol Monooleate and 1,4-butanediol, 9-hydroxy-10-methoxy-monostearate from Palm Oil as Modifier of Epoxy Resin for Coating

1,4-butanediol moonooleate (BMO) and 1,4-butanediol,9-hydroxy-10-methoxy-monostearate (BHMMS) were synthesized. BMO was synthesized via Fisher Esterification between oleic acid from palm oil and 1,4-butanediol by using sulfuric acid as catalyst. Furthermore, BMO product was used as raw material for synthesis of BHMMS. BHMMS was synthesis via continuous two-step reaction i.e. epoxidation and hydroxylation. Characterizations of BMO and BHMMS products were conducted by analysis of ester number, acid number, iodine number and FTIR. In this research, obtained product of BMO and BHMMS were used as modifier of epoxy by reacting with epoxy and tolonate to produce polyurethane modified epoxy (PME) based on BMO (PMEBMO) and BHMMS (PMEBHMMS) from palm oil. Characterization of PME was conducted by determine residual isocyanate to calculate the isocyanate conversion and FTIR analysis. Curing process of PME products to obtain film sheets were done by adding versamid 140 as hardener. Characterizations of film sheets were conducted by determine of mechanical properties such as tensile strength and elongation at break. The result showed that mechanical properties of PME were higher than the mechanical properties of unmodified epoxy.

Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces

An atomic decomposition of Hardy spaces by atoms associated with Banach function space is developed. Inspired by these decompositions, a criterion on a general Banach function space is introduced so that the characterization of BMO by using that Banach function space is valid.

Characterizations of BMO by $A_{p}$ Weights and $p$-Convexity

We show that the Lebesgue spaces for defining BMO can be replaced by $p$-convex rearrangement-invariant quasi-Banach function spaces associated with $A_{p}$-weighted measures.

Some new characterizations on spaces of functions with bounded mean oscillation

AbstractLet X be a space of homogeneous type. The authors introduce some generalized approximations to the identity (for short, GAI) with optimal decay conditions in the sense that these conditions are the sufficient and necessary conditions for these GAI's to characterize BMO(X), the space of functions with bounded mean oscillation on X. The authors also obtain a new John‐Nirenberg‐type inequality associated with GAI's, which leads some new characterizations of BMO(X) in terms of rearrangement functions, and certain maximal functions related to GAI's. Some variants of these characterizations for BMO‐type spaces on X of Duong and Yan are also established. Moreover, the equivalence between BMO and the space of BMO type on Ahlfors n ‐regular quasimetric measure spaces is obtained, which confirms an assertion of Duong and Yan (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Localized Hardy spaces associated with operators

Let L be a linear operator in L 2(ℝ n ) and generate an analytic semigroup {e −tL } t≥0 with kernels satisfying an upper bound of Poisson type. In this article, the authors introduce the localized Hardy space via molecules and show that , where and are Hardy spaces associated with L and L + I, respectively. Characterizations of via the localized Lusin area function and are established. Then, the authors introduce the localized BMO space bmo L (ℝ n ) and prove that the dual of is bmo L*(ℝ n ), where L* denotes the adjoint operator of L in L 2(ℝ n ). The John–Nirenberg inequality for elements in bmo L (ℝ n ) and a characterization of bmo L (ℝ n ) via BMO L (ℝ n ) are also established, where BMO L (ℝ n ) is the BMO space associated with L. As applications, the authors obtain the characterizations of the localized Hardy space associated to the Schrödinger operator L = −Δ + V, where is a nonnegative potential, in terms of the localized Lusin-area functions and the localized radial maximal functions.

Factorization theorem for product Hardy spaces

We extend the well known factorization theorems on the unit disk to prod- uct Hardy spaces, which generalizes the previous results obtained by Coifman, Rochberg and Weiss. The basic tools are the boundedness of a certain bilinear form on R2 + × R2 + and the characterization of BMO(R2 + × R2 +) recently obtained by Ferguson, Lacey and Sadosky.

A representation theorem of harmonic functions and its application to BMO on manifolds

In this paper a representation theorem of harmonic functions on manifolds is set up. As an application, a characterization of BMO in terms of Carleson measures is obtained.

A characterization of BMO on manifolds

A characterization of BMO on manifolds

𝐵𝑀𝑂(𝜌) and Carleson measures

This paper concerns certain generalizations of BMO {\text {BMO}} , the space of functions of bounded mean oscillation. Let ρ \rho be a positive nondecreasing function on ( 0 , ∞ ) (0,\infty ) with ρ ( 0 + ) = 0 \rho (0 + ) = 0 . A locally integrable function on R m {{\mathbf {R}}^m} is said to belong to BMO ( ρ ) {\text {BMO}}(\rho ) if its mean oscillation over any cube Q Q is O ( ρ ( l ( Q ) ) ) O(\rho (l(Q))) , where l ( Q ) l(Q) is the edge length of Q Q . Carleson measures are known to be closely related to BMO {\text {BMO}} . Generalizations of these measures are shown to be similarly related to the spaces BMO ( ρ ) {\text {BMO}}(\rho ) . For a cube Q Q in R m , | Q | {{\mathbf {R}}^m},\;|Q| denotes its volume and R ( Q ) R(Q) is the set { ( x , y ) ∈ R + m + 1 : x ∈ Q , 0 > y > l ( Q ) } \{ (x,y) \in {\mathbf {R}}_ + ^{m + 1}:x \in Q,\;0 > y > l(Q)\} . A measure μ \mu on R + m + 1 {\mathbf {R}}_ + ^{m + 1} is called a ρ \rho -Carleson measure if | μ | ( R ( Q ) ) = O ( ρ ( l ( Q ) ) | Q | ) |\mu |(R(Q)) = O(\rho (l(Q))|Q|) , for all cubes Q Q . L. Carleson proved that a compactly supported function in BMO {\text {BMO}} can be represented as the sum of a bounded function and the balyage, or sweep, of some Carleson measure. A generalization of this theorem involving BMO ( ρ ) {\text {BMO}}(\rho ) and ρ \rho -Carleson measures is proved for a broad class of growth functions, and this is used to represent BMO ( ρ ) {\text {BMO}}(\rho ) as a dual space. The proof of the theorem is based on a proof of J. Garnett and P. Jones of Carleson’s theorem. Another characterization of BMO ( ρ ) {\text {BMO}}(\rho ) using ρ \rho -Carleson measures is a corollary. This result generalizes a characterization of BMO {\text {BMO}} due to C. Fefferman. Finally, an atomic decomposition of the predual of BMO ( ρ ) {\text {BMO}}(\rho ) is given.

A characterization of BMO and $BMO_{ϱ}$

A characterization of BMO and $BMO_{ϱ}$

A Remark on Carleson's Characterization of BMO

A Remark on Carleson's Characterization of BMO

Another characterization of BMO

The following characterization of functions of bounded mean oscillation (BMO) is proved. f is in BMO if and only if \[ f = α log ⁡ g ∗ − β log ⁡ h ∗ + b f = \alpha \log {g^ \ast } - \beta \log {h^ \ast } + b \] where g ∗ , ( h ∗ ) {g^ \ast },({h^ \ast }) is the Hardy-Littlewood maximal function of g, (h), respectively, b is bounded and ‖ f ‖ BMO ⩽ c ( α + β + ‖ b ‖ ∞ ) {\left \| f \right \|_{{\text {BMO}}}} \leqslant c(\alpha + \beta + {\left \| b \right \|_\infty }) .

A remark on Carleson’s characterization of BMO

L. Carleson showed that if φ ∈ BMO ( R n ) \varphi \in \text {BMO}(R ^n) and supp φ \varphi is compact, then φ \varphi can be written in the form φ ( x ) = Σ k = 1 ∞ ∫ P t k ( y ) ( x − y ) b k ( y ) d y + b 0 ( x ) \varphi (x) = \Sigma _{k = 1}^\infty \smallint {P_{{t_k}(y)}}(x - y){b_k}(y)dy + {b_0}(x) where Σ k = 0 ∞ ‖ b k ‖ ∞ ⩽ C ‖ φ ‖ BMO , t k ( y ) > 0 \Sigma _{k = 0}^\infty {\left \| {{b_k}} \right \|_\infty } \leqslant C{\left \| \varphi \right \|_{{\text {BMO}}}},{t_k}(y) > 0 and P t ( x ) = C n t ( | x | 2 + t 2 ) − ( n + 1 ) / 2 {P_t}(x) = {C_n}t{(|x{|^2} + {t^2})^{ - (n + 1)/2}} is the Poisson kernel. We show that we can take b 2 = b 3 = ⋯ = 0 {b_2} = {b_3} = \cdots = 0 . This can be generalized on the space of homogeneous type with certain assumptions.