BMO Functions Research Articles

Carleson Measures on Locally Finite Trees

We provide a characterization of Carleson measures on locally finite trees. This characterization establishes the connection between Carleson measures and the boundedness of a suitable Poisson integral between Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-spaces. Additionally, when the tree has bounded degree, we investigate the relationship between Carleson measures and BMO functions defined on the boundary of the tree.

Arbitrary finite intersections of doubling measures and applications

We make major progress on a folkloric conjecture in analysis by constructing a measure on the real line which is doubling on all n-adic intervals for any finite list of n∈N, yet not doubling overall. In particular, we extend previous results in the area, including those of Boylan-Mills-Ward and Anderson-Hu, by using a wide array of substantially new ideas. In addition, we provide several nontrivial applications to reverse Hölder weights, Ap weights, Hardy spaces, BMO and VMO function classes, and connect our results with key principles and conjectures across number theory.

Commutators on Spaces of Homogeneous Type in Generalized Block Spaces

In this paper, we are interested in studying a generalized block space (denoted as Bφp,r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{B}^{p,r}_\\varphi $$\\end{document}) on a space of homogeneous type. We show that this space is the predual of certain generalized Morrey–Lorentz space. By duality, we obtain the Bφp,r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{B}^{p,r}_\\varphi $$\\end{document}-bound of operators of Calderón–Zygmund type. In addition, we prove a weak Hardy factorization in terms of commutators of integral operator of Calderón–Zygmund type in block spaces. Thanks to the Hardy factorization result, we obtain a characterization of functions in BMO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{BMO}$$\\end{document} via the boundedness of commutators of homogeneous linear Calderón–Zygmund operators in the generalized block space (resp. the generalized Morrey–Lorentz space). Finally, we study a compactness characterization of commutators of Calderón–Zygmund type in generalized Morrey–Lorentz spaces.

Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces

Abstract In this article, we show continuity of commutators of Calderón-Zygmund operators [ b , T ] \left[b,T] with BMO functions in generalized Orlicz-Morrey spaces M Φ , φ ( R n ) {M}^{\Phi ,\varphi }\left({{\mathbb{R}}}^{n}) . We give necessary and sufficient conditions for the boundedness of the genuine Calderón-Zygmund operators T T and for their commutators [ b , T ] \left[b,T] on generalized Orlicz-Morrey spaces, respectively.

Locally uniform domains and extension of bmo functions

We prove that for a domain \(\Omega\subset\mathbb{R}^n\), being \((\epsilon,\delta)\) in the sense of Jones is equivalent to being an extension domain for bmo, the nonhonomogeneous version of the space of functions of bounded mean oscillation on \(\Omega\). Such domains, which can be identified as local versions of uniform domains (defined by requiring the presence of length cigars between nearby points), allow a definition of bmo\((\Omega)\) in terms of "small" and "large" cubes contained in \(\Omega\), where the scale is closely tied to the geometry of the domain.

Weighted norm inequalities for multilinear strongly singular Calderón–Zygmund operators on RD‐spaces

AbstractLet be an RD‐space, namely, a space of homogeneous type in the sense of Coifman and Weiss with the Borel measure μ satisfying the reverse doubling condition on . Based on this space, the authors define a multilinear strongly singular Calderón–Zygmund operator whose kernel does not need any size condition and has more singularities near the diagonal than that of a standard multilinear Calderón–Zygmund operator. For such an operator, we establish its boundedness on product of weighted Lebesgue spaces by means of the pointwise estimate for the sharp maximal function. In addition, the endpoint estimates of the type are also obtained. Moreover, we prove weighted boundedness results for multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions. These results contribute to the extension of multilinear strongly singular Calderón–Zygmund operator theory in the Euclidean case to the context of space of homogeneous type.

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.

Estimate for some integral operators and their commutators on generalized fractional mixed Morrey spaces

In this paper, we first establish the definition of a generalized fractional mixed Morrey space Lp→,η,φ(Rn), where p→=(p1,p2,⋯,pn), 1<p→≤∞ and φ:(0,∞)→(0,∞) is an increasing function satisfying certain doubling conditions. Second, we prove that the Calderón-Zygmund integral operator T and its commutator [b,T] which is generated by T and b∈BMO(Rn) are bounded from spaces Lp→,η,φ(Rn) into spaces Lp→,η,φ(Rn); furthermore, we prove that the fractional integral operator Iα and its commutator [b,Iα] formed by b∈BMO(Rn) and Iα are bounded from the spaces Lp→,η,φ(Rn) into spaces Lq→,η−α,φ(Rn), where 1<p→<q→<∞, ∑j=1n1qj=∑j=1n1pj−α and ∑j=1n1pj≥η>α>0. Finally, the boundedness for the fractional maximal operator Mα, the commutator [b,Mα] associated with BMO functions and the commutator [b,T] formed by T and b∈Lipβ(Rn) on spaces Lp→,η,φ(Rn) is established, respectively.

A revisit to “On BMO and Carleson measures on Riemannian manifolds”

Let $\mathcal {M}$ be an Ahlfors $n$-regular Riemannian manifold such that either the Ricci curvature is non-negative or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. In the paper [IMRN, 2022, no. 2, 1245-1269] of Brazke–Schikorra–Sire, the authors characterised the BMO function $u : \mathcal {M} \to \mathbb {R}$ by a Carleson measure condition of its $\sigma$-harmonic extension $U:\mathcal {M}\times \mathbb {R}_+ \to \mathbb {R}$. This paper is concerned with the similar problem under a more general Dirichlet metric measure space setting, and the limiting behaviours of BMO &amp; Carleson measure, where the heat kernel admits only the so-called diagonal upper estimate. More significantly, without the Ricci curvature condition, we relax the Ahlfors regularity to a doubling property, and remove the pointwise bound on the gradient of the heat kernel. Some similar results for the Lipschitz function are also given, and two open problems related to our main result are considered.

BMO embeddings, chord-arc curves, and Riemann mapping parametrization

We consider the space of chord-arc curves on the plane passing through infinity with their parametrization γ defined on the real line, and embed this space into the product of the BMO Teichmüller spaces. The fundamental theorem we prove about this representation is that γ↦log⁡γ′ is a biholomorphic homeomorphism into the complex Banach space of BMO functions. Using these two equivalent complex structures, we develop a clear exposition on the analytic dependence of involved mappings between certain subspaces. Especially, we examine the parametrization of a chord-arc curve by using the Riemann mapping and its dependence on the arc-length parametrization. As a consequence, we can solve the conjecture of Katznelson, Nag, and Sullivan by showing that this dependence is not continuous.

Boundedness of intrinsic square functions and commutators on generalized central Morrey spaces

In this paper, the authors establish the boundedness for a large class of intrinsic square functions G α , g α , g λ ~ , α ∗ and their commutators [ b , G α ] , [ b , g α ] and [ b , g λ ~ , α ∗ ] generated with λ-central BMO functions b ∈ CBM O p , λ ( R n ) on generalized central Morrey spaces B q , φ ( R n ) for 1 < q < ∞ , 0 < α ≤ 1 , respectively. All of the results are new even on the central Morrey spaces B q , λ ( R n ) .

Weighted estimates for commutators associated to singular integral operator satisfying a variant of Hörmander's condition

&lt;abstract&gt;&lt;p&gt;In this paper, we establish some boundedness for commutators generated by the singular integral operator satisfying a variant of Hörmander's condition and a weighted BMO function on weighted Hardy spaces and weighted Herz spaces. As an application, we obtain some classical results.&lt;/p&gt;&lt;/abstract&gt;

Paraproducts, Bloom BMO and sparse BMO functions

We address L^p(\mu)\to L^p(\lambda) bounds for paraproducts in the Bloom setting. We introduce certain “sparse BMO” functions associated with sparse collections with no infinitely increasing chains, and use these to express sparse operators as sums of paraproducts and martingale transforms – essentially, as Haar multipliers – as well as to obtain an equivalence of norms between sparse operators \mathcal{A}_{\mathcal{S}} and compositions of paraproducts \Pi^*_a\Pi_b .

On weighted norm inequalities for oscillatory integral operators

We prove weighted norm inequalities with Muckenhoupt’s A_p-weights, for a wide class of oscillatory integral operators. As a consequence, one also obtains the boundedness of commutators of the aforementioned operators with functions in BMO.

A real-variable construction with applications to BMO–Teichmüller theory

With the use of real-variable techniques, we construct a weight function ω on the interval [0,2π) which is doubling and satisfies logω is a BMO function, but which is not a Muckenhoupt weight (A∞). Applications to the BMO–Teichmüller space and the space of chord-arc curves are considered.

Optimal boundary regularity for viscosity solutions of fully nonlinear elliptic equations

We establish optimal boundary regularity results for viscosity solutions to second order fully nonlinear uniformly elliptic equations in the form F(D2u(x),Du(x),x)=f(x) in Ω. In particular, we obtain sharp estimates for borderline cases f∈Ln(Ω) and f∈BMO(Ω). For source functions in BMO(Ω), we obtain C1,Log−Lip interior regularity for flat solutions of non-convex elliptic equations. As a consequence, we obtain C1,Log−Lip estimates near the boundary; which again is an optimal estimate.

Quantitative weighted bounds for composite operators on spaces of homogeneous type

Let T1 and T2 be Calderón–Zygmund operators, and let T1,b denote the commutator of T1 and the BMO function b. By establishing bisublinear sparse dominations, we obtain the quantitative weighted bounds for composite operators T1T2 and T1,bT2 on Lp(X,ω), with ω∈Ap(X).

Boundedness of Littlewood-Paley Operators in Variable Morrey Spaces

In this paper, the authors prove norm inequalities for the intrinsic square functions and commutators generated by this class operator and BMO function in variable Morrey spaces. This implies that the same norm inequalities for the Lusin area integrals, the Littlewood-Paley operators and the continuous square functions. As application, we get the boundedness for convolution Calderón-Zygmund operators in generalized Morrey spaces.

Commutators of Multilinear Calderón-Zygmund Operator on Weighted Herz-Morrey Spaces with Variable Exponents

In this paper, we acquire the boundedness of commutators generated by multilinear Calderón-Zygmund operator and BMO functions on products of weighted Herz-Morrey spaces with variable exponents.

Boundedness of Integral Operators in Generalized Weighted Grand Lebesgue Spaces with Non-doubling Measures

The goal of this paper is to prove the boundedness of fundamental integral operators of Harmonic Analysis in generalized weighted grand Lebesgue spaces $$L^{p),\varphi }_w$$ defined on domains in $${\mathbb {R}}^n$$ without assuming that the underlying measure $$\mu $$ is doubling. Operators under consideration involve maximal and Calderon–Zygmund operators, also commutators of singular integrals with BMO functions. Essential part of this paper is devoted to the weighted Sobolev-type inequalities in these spaces for fractional integrals with non-doubling measures satisfying the growth condition. We discuss the case when a weight function defines the absolute continuous measure of integration, or plays the role of multiplier in the definition of the norm. All these results are new even for the classical weighted grand Lebesgue spaces $$L^{p),\theta }_w$$ defined with respect to non-doubling measures. The results are obtained under the Muckenhoupt condition on weights. The results are obtained for weight functions satisfying Muckenhoupt $$A_p$$ conditions defined with respect to non-homogeneous spaces.