VMO Spaces Research Articles

The commutator of the Bergman projection on strongly pseudoconvex domains with minimal smoothness

Consider a bounded, strongly pseudoconvex domain D⊂Cn with minimal smoothness (namely, the class C2) and let b be a locally integrable function on D. We characterize boundedness (resp., compactness) in Lp(D),p>1, of the commutator [b,P] of the Bergman projection P in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on b. We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N. Kerzman & E. M. Stein, and E. Ligocka.

BOUNDEDNESS AND COMPACTNESS OF CAUCHY-TYPE INTEGRAL COMMUTATOR ON WEIGHTED MORREY SPACES

AbstractIn this paper we study boundedness and compactness characterizations of the commutators of Cauchy type integrals on bounded strongly pseudoconvex domains D in $\mathbb C^{n}$ with boundaries $bD$ satisfying the minimum regularity condition $C^{2}$ , based on the recent results of Lanzani–Stein and Duong et al. We point out that in this setting the Cauchy type integral is the sum of the essential part which is a Calderón–Zygmund operator and a remainder which is no longer a Calderón–Zygmund operator. We show that the commutator is bounded on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1<p<\infty $ ) if and only if b is in the BMO space on $bD$ . Moreover, the commutator is compact on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1<p<\infty $ ) if and only if b is in the VMO space on $bD$ .

The Two-Dimensional Liquid Crystal Droplet Problem with a Tangential Boundary Condition

This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition and a volume constraint. We establish in 2D the existence of an optimal shape that has two cusps on the boundary. We also prove that the boundary of the droplet is a chord–arc curve with its normal vector field in the VMO space, and its arc-length parameterization belongs to the Sobolev space $$H^{3/2}$$ . In fact, the boundary curves of such droplets closely resemble the so-called Weil–Petersson class of planar curves. In addition, the asymptotic behavior of the optimal shape when the volume becomes extremely large or small is studied.

Compactness of commutator of Riesz transforms in the two weight setting

We characterize the compactness of commutators in the Bloom setting. Namely, for a suitably non-degenerate Calderón–Zygmund operator T, and a pair of weights σ,ω∈Ap, the commutator [T,b] is compact from Lp(σ)→Lp(ω) if and only if b∈VMOν, where ν=(σ/ω)1/p. This extends the work of the first author, Holmes and Wick. The weighted VMO spaces are different from the classical VMO space. In dimension d=1, compactly supported and smooth functions are dense in VMOν, but this need not hold in dimensions d≥2. Moreover, the commutator in the product setting with respect to little VMO space is also investigated.

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.

Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type

Abstract In this paper, we study the boundedness and compactness of the commutator of Calderón– Zygmund operators T on spaces of homogeneous type (X, d, µ) in the sense of Coifman and Weiss. More precisely, we show that the commutator [b, T] is bounded on the weighted Morrey space L ω p , k ( X ) L_\omega ^{p,k}\left( X \right) with κ ∈ (0, 1) and ω ∈ Ap (X), 1 < p < ∞, if and only if b is in the BMO space. We also prove that the commutator [b, T] is compact on the same weighted Morrey space if and only if b belongs to the VMO space. We note that there is no extra assumptions on the quasimetric d and the doubling measure µ.

The Cauchy integral, bounded and compact commutators

We study the commutator of the well-known Cauchy integral operator with a locally integrable function $b$ on $\mathbb R$, and establish the characterisation of the BMO space on $\mathbb R$ via the $L^p$ boundedness of this commutator. Moreover, we also establish the characterisation of the VMO space on $\mathbb R$ via the compactness of this commutator.

VMO Spaces Associated with Operators with Gaussian Upper Bounds on Product Domains

In this paper, we extend the well-known result “the predual of Hardy space \(H^1\) is VMO” to the product setting, associated with differential operators. Let \(L_i\), \(i = 1, 2\), be the infinitesimal generators of the analytic semigroups \(\{e^{-tL_i}\}\) on \(L^2({\mathbb {R}})\). Assume that the kernels of the semigroups \(\{e^{-tL_i}\}\) satisfy the Gaussian upper bounds. We introduce the VMO spaces VMO\(_{L_1, L_2}(\mathbb {R}\times \mathbb {R})\) associated with operators \(L_1\) and \(L_2\) on the product domain \(\mathbb {R}\times \mathbb {R}\), then show that the dual space of VMO\(_{L_1, L_2}(\mathbb {R}\times \mathbb {R})\) is the Hardy space \(H^1_{L_1^*, L_2^*}(\mathbb {R}\times \mathbb {R})\) associated with the adjoint operators \(L^*_1\) and \(L^*_2\).

The Uniform Hopf Inequality for discontinuous coefficients and optimal regularity in BMO for singular problems

We consider some singular second order semilinear problems which include, among many other special cases, the boundary layer equations such as they were treated by O.A. Oleinik in her pioneering works. We consider diffusion linear operator with possible discontinuous coefficients and prove an optimal criterion to get a quantitative strong maximum principle that we call Uniform Hopf Inequality UHI. Since the solutions of the singular semilinear problems under consideration are not Lipschitz continuous we carry out a careful study of the regularity of solutions when the coefficients of the diffusion matrix are merely in the vmo space and bounded. We prove that the gradient of the solution is still p-integrable, in absence of any continuity assumption on the spatial potential coefficient in the singular term. To this end, the UHI property is used several times. We also apply and improve previous a priori estimates due to S. Campanato in 1965.

VMO Space Associated with Parabolic Sections and its Application

AbstractIn this paper we define a space VMO𝒫 associated with a family 𝒫 of parabolic sections and show that the dual of VMO𝒫 is the Hardy space . As an application, we prove that almost everywhere convergence of a bounded sequence in implies weak* convergence

Теоремы вложения для P-ичных пространств Харди и VMO

Теоремы вложения для P-ичных пространств Харди и VMO

Vanishing mean oscillation spaces associated with operators satisfying Davies–Gaffney estimates

Let $(\mathcal{X}, d, \mu)$ be a metric measure space, $L$ a linear operator which has a bounded $H_\infty$ functional calculus and satisfies the Davies-Gaffney estimate, $\Phi$ a concave function on $(0,\infty)$ of critical lower type $p_\Phi^-\in(0,1]$ and $\rho(t)\equiv t^{-1}/\Phi^{-1}(t^{-1})$ for all $t\in(0,\infty)$. In this paper, the authors introduce the generalized VMO space ${\mathrm {VMO}}_{\rho,L}({\mathcal X})$ associated with $L$, and establish its characterization via the tent space. As applications, the authors show that $({\mathrm {VMO}}_{\rho,L}({\mathcal X}))^*=B_{\Phi,L^*}({\mathcal X})$, where $L^*$ denotes the adjoint operator of $L$ in $L^2({\mathcal X})$ and $B_{\Phi,L^*}({\mathcal X})$ the Banach completion of the Orlicz-Hardy space $H_{\Phi,L^*}({\mathcal X})$.

On a generalized Stokes problem

Abstract We deal with a generalization of the Stokes system. Instead of the Laplace operator, we consider a general elliptic operator and a pressure gradient with small perturbations. We investigate the existence and uniqueness of a solution as well its regularity properties. Two types of regularity are provided. Aside from the classical Hilbert regularity, we also prove the Hölder regularity for coefficients in VMO space.

VMO spaces associated with divergence form elliptic operators

Consider the second order divergence form elliptic operator L with complex bounded coefficients. In this paper we introduce and develop a function space VMOL of vanishing mean oscillation associated with the operator L. Using the theory of tent spaces and the Littlewood-Paley theory, we prove that the Hardy space \({H_{L}^1}\) of Hofmann and Mayboroda introduced in Hofmann and Mayboroda (Math Ann 344:37–116, 2009) is the dual of our \({{\rm VMO}_{L^{\ast}}}\), in which L* is the adjoint operator of L. We also give an equivalent characterization of the space VMOL in the context of the theory of tent spaces.

Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t −1/ω −1(t −1) for \({t\in (0,\infty).}\) In this paper, the authors introduce the generalized VMO spaces \({{\mathop{\rm VMO}_ {\rho, L}({\mathbb R}^n)}}\) associated with L, and characterize them via tent spaces. As applications, the authors show that \({({\rm VMO}_{\rho,L} ({\mathbb R}^n))^\ast=B_{\omega,L^\ast}({\mathbb R}^n)}\), where L * denotes the adjoint operator of L in \({L^2({\mathbb R}^n)}\) and \({B_{\omega,L^\ast}({\mathbb R}^n)}\) the Banach completion of the Orlicz–Hardy space \({H_{\omega,L^\ast}({\mathbb R}^n)}\). Notice that ω(t) = t p for all \({t\in (0,\infty)}\) and \({p\in (0,1]}\) is a typical example of positive concave functions satisfying the assumptions. In particular, when p = 1, then ρ(t) ≡ 1 and \({({\mathop{\rm VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)}\), where \({H_{L^\ast}^1({\mathbb R}^n)}\) was the Hardy space introduced by Hofmann and Mayboroda.

BMO-scale of distribution on ℝ n

Let S′ be the class of tempered distributions. For ƒ ∈ S′ we denote by J−α ƒ the Bessel potential of ƒ of order α. We prove that if J−α ƒ ∈ BMO, then for any λ ∈ (0, 1), J−α(f)λ ∈ BMO, where (f)λ = λ−nf(φ(λ−1)), φ ∈ S. Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order α > 0 belongs to the VMO space.

BMO from dyadic BMO on the bidisc

AbstractWe generalize to the bidisc a theorem of Garnett and Jones relating the space BMO of functions of bounded mean oscillation to its martingale counterpart, dyadic BMO. Namely, translation‐averages of suitable families of dyadic BMO functions belong to BMO. As a corollary, we deduce a biparameter version of a theorem of Burgess Davis connecting the Hardy space H1 to martingale H1. We also prove the analogs of the theorem of Garnett and Jones in the one‐parameter and biparameter VMO spaces of functions of vanishing mean oscillation.

On the Boundedness and Compactness of Operators of Hankel Type

Boundedness and compactness results are obtained for commutators of multiplication operators with a general class of integral operators J of a critical homogeneity which are modeled on the Bergman projection. It is shown that the commutator [ M f , J ] is bounded or compact on L p whenever the function ƒ is in an appropriately defined BMO or VMO space, respectively. Special cases of the general results include commutators with the Bergman projection in strictly pseudoconvex domains in C n and finite type domains in C 2. In addition, it is shown that, in the case of the Bergman projection in a strictly pseudoconvex domain, the sufficient condition is also necessary.

Martingale Hardy spaces, BMO and VMO spaces with nonlinearly ordered stochastic basis

В работе обобщаются н екоторые теоремы о мартингальных прост ранствах Харди и пространсвах ВМО и VMO. Д оказывается, что двой ственным к атомному пространсв у Харди по необязатель но линейно упорядоче нному стохастическому баз ису является соответ ствующее пространство ВМО+, а д войственным к так наз ываемому «срезанному» двумер ному двоичному пространс тву VMO служит атомное п ространство Харди. Далее устанавл ивается обобщение классичес кого неравенства Хар ди для функций из «срезанно го» двумерного двоич ного пространства Харди.

Dyadic martingale Hardy and VMO spaces on the plane

Dyadic martingale Hardy and VMO spaces on the plane