BMO Functions Research Articles

Weighted estimates for commutators of strongly singular Calderón–Zygmund operators

In this paper, the authors establish the boundedness of commutators generated by strongly singular Calderon–Zygmund operators and weighted BMO functions on weighted Herz-type Hardy spaces. Moreover, the corresponding results for commutators generated by strongly singular Calderon–Zygmund operators and weighted Lipschitz functions can also be obtained.

Multilinear dyadic operators and their commutators

We obtain a generalized paraproduct decomposition of the pointwise product of two or more functions that naturally gives rise to multilinear dyadic paraproducts and Haar multipliers. We then study the boundedness properties of these multilinear operators and their commutators with dyadic BMO functions. We also characterize the dyadic BMO functions via the boundedness of (a) certain paraproducts, and (b) the commutators of multilinear Haar multipliers and paraproduct operators.

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.

Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants

For [Formula: see text] and [Formula: see text], let [Formula: see text] be the space of harmonic functions [Formula: see text] on the upper half-space [Formula: see text] satisfying [Formula: see text] and [Formula: see text] be the Campanato space on [Formula: see text]. We show that [Formula: see text] coincides with [Formula: see text] for all [Formula: see text], where the case [Formula: see text] was originally discovered by Fabes, Johnson and Neri [E. B. Fabes, R. L. Johnson and U. Neri, Spaces of harmonic functions representable by Poisson integrals of functions in BMO and [Formula: see text], Indiana Univ. Math. J. 25 (1976) 159–170] and yet the case [Formula: see text] was left open. Moreover, for the scaling invariant version of [Formula: see text], [Formula: see text], which comprises all harmonic functions [Formula: see text] on [Formula: see text] satisfying [Formula: see text] we show that [Formula: see text], where [Formula: see text] is the collection of all functions [Formula: see text] such that [Formula: see text] are in [Formula: see text]. Analogues for solutions to the heat equation are also established. As an application, we show that the spaces [Formula: see text] unify naturally [Formula: see text], [Formula: see text] and [Formula: see text] which can be effectively adapted and applicable to suit handling the well/ill-posedness of the incompressible Navier–Stokes system on [Formula: see text].

Orthogonal Polynomials on the Circle for the Weight w Satisfying Conditions $$w,w^{-1}\in \mathrm{BMO}$$ w , w - 1 ∈ BMO

For the weight w satisfying \(w,w^{-1}\in \mathrm{BMO}({\mathbb {T}})\), we prove the asymptotics of \(\{\Phi _n(e^{i\theta },w)\}\) in \(L^p[-\pi ,\pi ], 2\leqslant p<p_0\), where \(\{\Phi _n(z,w)\}\) are monic polynomials orthogonal with respect to w on the unit circle \({\mathbb {T}}\). Immediate applications include the estimates on the uniform norm and asymptotics of the polynomial entropies. The estimates on higher-order commutators between the Calderon–Zygmund operators and BMO functions play the key role in the proofs of main results.

Vector-Valued Singular Integrals with Non-smooth Kernels on Spaces of Homogeneous Type

In this article, we obtain weighted norm inequalities for vector-valued singular integrals with non-smooth kernels on spaces of homogeneous type. Vector-valued commutators of singular integrals and BMO functions are also studied.

Parameterized Littlewood--Paley operators and their commutators on Herz spaces with variable exponents

The aim of this paper is to deal with Littlewood--Paley operators with real parameters, including the parameterized Lusin area integrals and the parameterized Littlewood--Paley $g_{\lambda}^{\ast}$-functions, and their commutators on Herz spaces with two variable exponents $p(\cdot),~q(\cdot)$. By using the properties of the Lebesgue spaces with variable exponents, the boundedness of the parameterized Littlewood--Paley operators and their commutators generated respectively by BMO function and Lipschitz function is obtained on those Herz spaces.

On Hardy and BMO Spaces for Grushin Operator

We study Hardy and BMO spaces associated with the Grushin operator. We first prove atomic and maximal functions characterizations of the Hardy space. Further we establish a version of Fefferman–Stein decomposition of BMO functions associated with the Grushin operator and then obtain a Riesz transforms characterization of the Hardy space.

Weighted estimates for vector-valued multilinear square function

Let T be the multilinear square operator, respectively with certain smooth kernels and non-smooth kernels defined in (Xue and Yan in J. Math. Anal. Appl. 422:1342-1362, 2015) and (Hormozi et al. in arXiv preprint, 2015), and let $T^{*}$ be its corresponding maximal operator. In this paper, we prove the vector-valued weighted norm boundedness for T and $T^{*}$ and also establish multiple weighted inequalities for their corresponding iterated commutator generated by the vector-valued multilinear operator and BMO function.

Rough fractional integrals and its commutators on variable Morrey spaces

In this paper, the authors obtain the boundedness of fractional integrals with rough kernel on variable Morrey spaces. The corresponding boundedness for commutators generalized by the fractional integral and BMO function is also considered.

Weighted Norm Inequalities for Spectral Multipliers on Graphs

Let Γ be a graph endowed with a reversible Markov kernel p, whose associated operator P is defined by \(Pf(x) = {\sum }_{y} p(x, y)f(y)\). We assume that the kernels p n (x, y) associated to P n satisfy Gaussian upper bounds but do not assume they satisfy the Holder continuity property and the temporal regularity. Denote by L = I − P the discrete Laplacian on Γ. This article shows the weighted weak type (1, 1) estimates and the weighted L p norm inequalities for the spectral multipliers of L. We also obtain the weighted L p norm inequalities for the commutators of the spectral multipliers of L with BMO functions which are new even for the unweighted case.

Bellman function for extremal problems in BMO

In this paper we develop the method of nding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John{Nirenberg inequality and L p estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge{Amp ere equation in a parabolic strip for suciently smooth boundary conditions. In such a way, we have obtained an algorithm for constructing an exact Bellman function for a large class of integral functionals on the BMO space.

Oscillation and variation inequalities for singular integrals and commutators on weighted Morrey spaces

This paper is devoted to investigating the bounded behaviors of the oscillation and variation operators for Calderon-Zygmund singular integrals and the corresponding commutators on the weighted Morrey spaces. We establish several criterions of boundedness, which are applied to obtain the corresponding bounds for the oscillation and variation operators of Hilbert transform, Hermitian Riesz transform and their commutators with BMO functions, or Lipschitz functions on weighted Morrey spaces.

The Dirichlet problem with BMO boundary data and almost-real coefficients

It is known that a function, harmonic in a Lipschitz domain, is the Poisson extension of a BMO function if and only if its gradient satisfies a Carleson-measure condition. We show that the same is true of functions that satisfy elliptic equations in two-dimensional Lipschitz domains, provided the coefficients are independent of one coordinate and have small imaginary part.

Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions

We consider the Weinstein type equation Lλu=0 on (0,∞)×(0,∞), where Lλ=∂t2+∂x2−λ(λ−1)x2, with λ>1. In this paper we characterize the solutions of Lλu=0 on (0,∞)×(0,∞) representable by Bessel–Poisson integrals of BMO-functions as the ones satisfying certain Carleson properties.

POINTWISE ESTIMATES AND BOUNDEDNESS OF GENERALIZED LITTLEWOOD-PALEY OPERATORS IN BMO(ℝn)

In this paper, we study the generalized Littlewood-Paley operators. It is shown that the generalized g-function, Lusin area function and -function on any BMO function are either infinite everywhere, or finite almost everywhere, respectively; and in the latter case, such operators are bounded from BMO() to BLO(), which improve and generalize some previous results.

New Weighted Norm Inequalities for Certain Classes of Multilinear Operators and their Iterated Commutators

Applying the new class of multiple weights functions and new sharp maximal functions, we obtain the pointwise estimates, strong type and weak end-point estmates for certain classes of multilinear operators and their iterated commutators with new BMO functions.

Finitely randomized dyadic systems and BMO on metric measure spaces

We study the connection between BMO and dyadic BMO in metric measure spaces using finitely randomized dyadic systems, and give a Garnett–Jones type proof for a theorem of Uchiyama on a construction of certain BMO functions. We obtain a relation between the BMO norm of a suitable expectation over dyadic systems and the dyadic BMO norms of the original functions in different systems. The expectation is taken over only finitely randomized dyadic systems to overcome certain measurability questions. Applying our result, we derive Uchiyama’s theorem from its dyadic counterpart, which we also prove.

Marcinkiewicz integrals with non-smooth kernels on spaces of homogeneous type

In this article, we obtain weak type (1, 1) and strong type (p, p) (1 < p < ∞) inequalities for Marcinkiewicz integrals with non-smooth kernels on spaces of homogeneous type. Moreover, we also prove the Lp-boundedness of commutators of BMO functions and Marcinkiewicz integrals on spaces of homogeneous type.

Endpoint estimates for Marcinkiewicz integrals on weighted weak hardy spaces

In this paper, we obtain the (WH ω 1 , WH ω 1 ) type estimate for the Marcinkiewicz integral and the (WH , ω 1 , WH ω 1 ) type estimate for the commutator generated by a BMO function and the Marcinkiewicz integral, where the kernel satisfies a certain logarithmic type Lipschitz condition.