Tent Spaces Research Articles

Harmonic Analysis on Chord Arc Domains

In the present paper we study the solvability of the Dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients which are small perturbations of given operators in rough domains beyond the Lipschitz category. In our approach, the development of the theory of tent spaces on these domains is essential.

New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets

In this paper, the authors establish new characterizations of the recently introduced Besov-type spaces $\dot{B}^{s,\tau}_{p,q}({\mathbb R}^n)$ and Triebel-Lizorkin-type spaces $\dot{F}^{s,\tau}_{p,q}({\mathbb R}^n)$ with $p\in (0,\infty]$, $s\in{\mathbb R}$, $\tau\in [0,\infty)$, and $q\in (0,\infty]$, as well as their preduals, the Besov-Hausdorff spaces $B\dot{H}^{s,\tau}_{p,q}(\R^n)$ and Triebel-Lizorkin-Hausdorff spaces $F\dot{H}^{s,\tau}_{p,q}(\R^n)$, in terms of the local means, the Peetre maximal function of local means, and the tent space (the Lusin area function) in both discrete and continuous types. As applications, the authors then obtain interpretations as coorbits in the sense of H. Rauhut in [Studia Math. 180 (2007), 237-253] and discretizations via the biorthogonal wavelet bases for the full range of parameters of these function spaces. Even for some special cases of this setting such as $\dot F^s_{\infty,q}({\mathbb R}^n)$ for $s\in{\mathbb R}$, $q\in (0,\infty]$ (including $\mathop\mathrm{BMO} ({\mathbb R}^n)$ when $s=0$, $q=2$), the $Q$ space $Q_\alpha ({\mathbb R}^n)$, the Hardy-Hausdorff space $HH_{-\alpha}({\mathbb R}^n)$ for $\alpha\in (0,\min\{\frac n2,1\})$, the Morrey space ${\mathcal M}^u_p({\mathbb R}^n)$ for $1<p\le u<\infty$, and the Triebel-Lizorkin-Morrey space $\dot{\mathcal{E}}^s_{upq}({\mathbb R}^n)$ for $0<p\le u<\infty$, $s\in{\mathbb R}$ and $q\in(0,\infty]$, some of these results are new.

Verifications of Pointwise Multipliers of Relations between Weighted Bergman Spaces and Hardy Spaces

The justification of Pointwise multipliers from weighted Bergman spaces and Hardy spaces to weighted Bergman spaces which are Hilbert are characterized by using Bloch type spaces, BMOA type spaces, weighted Bergman spaces and tent spaces. Keywords: Pointwise multipliers; Hardy spaces; Bergman spaces; Bloch type spaces; BMOA type spaces; tent spaces. 2010 Mathematics Subject Classification: 47B38

Singular integral operators on tent spaces

We extend the recent results concerning boundedness of the maximal regularity operator on tent spaces. This leads us to develop a singular integral operator theory on tent spaces. Such operators have operator-valued kernels. A seemingly appropriate condition on the kernel is time–space decay measured by off-diagonal estimates with various exponents.

Abstract capacitary estimates and the completeness and separability of certain classes of non-locally convex topological vector spaces

We are concerned with establishing completeness and separability criteria for large classes of topological vector spaces which are typically non-locally convex, including Lebesgue-like spaces, Lorentz spaces, Orlicz spaces, mixed-normed spaces, tent spaces, and discrete Triebel–Lizorkin and Besov spaces. For vector spaces of measurable functions we also derive pointwise convergence results. Our approach relies on abstract capacitary estimates and works in certain cases of interest even in the absence of a background measure space and/or of a vector space structure.

Carleson Measure and Tent Spaces on the Siegel Upper Half Space

The Hausdorff capacity on the Heisenberg group is introduced. The Choquet integrals with respect to the Hausdorff capacity on the Heisenberg group are defined. Then the fractional Carleson measures on the Siegel upper half space are discussed. Some characterized results and the dual of the fractional Carleson measures on the Siegel upper half space are studied. Therefore, the tent spaces on the Siegel upper half space in terms of the Choquet integrals are introduced and investigated. The atomic decomposition and the dual spaces of the tent spaces are obtained at the last.

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 $H_{\Phi,L^*}({\mathcal X})$.

Continuous Characterizations of Besov-Lizorkin-Triebel Spaces and New Interpretations as Coorbits

We give characterizations for homogeneous and inhomogeneous Besov-Lizorkin-Triebel spaces (H. Triebel 1983, 1992, and 2006) in terms of continuous local means for the full range of parameters. In particular, we prove characterizations in terms of Lusin functions (tent spaces) and spaces involving the Peetre maximal function to apply the classical coorbit space theory according to Feichtinger and Gröchenig (H. G Feichtinger and K. Gröchenig 1988, 1989, and 1991). This results in atomic decompositions and wavelet bases for homogeneous spaces. In particular we give sufficient conditions for suitable wavelets in terms of moment, decay and smoothness conditions.

The wavelet characterization of Herz-type Hardy spaces with variable exponent

‎In this paper‎, ‎using the atomic theory of the Herz-type Hardy spaces‎ ‎with variable exponent‎, ‎we give their wavelet characterization by‎ ‎means of some discrete tent spaces with variable exponent at the‎ ‎origin‎.

Whitney coverings and the tent spaces T1, q(γ) for the Gaussian measure

We introduce a technique for handling Whitney decompositions in Gaussian harmonic analysis and apply it to the study of Gaussian analogues of the classical tent spaces $T^{1,q}$ of Coifman, Meyer and Stein.

Change of angle in tent spaces

We prove sharp bounds for the equivalence of norms in tent spaces with respect to changes of angles. Some applications are given.

The maximal regularity operator on tent spaces

Recently, Auscher and Axelsson gave a new approach to non-smooth boundary value problems with $L^2$ data, that relies on some appropriate weighted maximal regularity estimates. As part of the development of the corresponding $L^p$ theory, we prove here the relevant weighted maximal estimates in tent spaces $T^{p, 2}$ for $p$ in a certain open range. We also study the case $p=\infty$.

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.

Well-posedness and regularity of generalized Navier–Stokes equations in some critical Q-spaces

We study the well-posedness and regularity of the generalized Navier–Stokes equations with initial data in a new critical space Q α ; ∞ β , − 1 ( R n ) = ∇ ⋅ ( Q α β ( R n ) ) n , β ∈ ( 1 2 , 1 ) , which is larger than some known critical homogeneous Besov spaces. Here Q α β ( R n ) is a space defined as the set of all measurable functions with sup ( l ( I ) ) 2 ( α + β − 1 ) − n ∫ I ∫ I | f ( x ) − f ( y ) | 2 | x − y | n + 2 ( α − β + 1 ) d x d y < ∞ where the supremum is taken over all cubes I with edge length l ( I ) and edges parallel to the coordinate axes in R n . In order to study the well-posedness and regularity, we give a Carleson measure characterization of Q α β ( R n ) by investigating a new type of tent spaces and an atomic decomposition of the predual for Q α β ( R n ) . In addition, our regularity results apply to the incompressible Navier–Stokes equations with initial data in Q α ; ∞ 1 , − 1 ( R n ) .

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.

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let $L$ be a linear operator in $L^2({{\mathbb R}^n})$ and generate an analytic semigroup $\{e^{-tL}\}_{t\ge 0}$ with kernels satisfying an upper bound of Poisson type, whose decay is measured by $\theta(L)\in (0,\infty].$ Let $\omega$ on $(0,\infty)$ be of upper type 1 and of critical lower type $\widetilde p_0(\omega)\in (n/(n+\theta(L)), 1]$ and $\rho(t)={t^{-1}}/\omega^{-1}(t^{-1})$ for $t\in (0,\infty)$. In this paper, the authors first introduce the VMO-type space $\mathrm{VMO}_{\rho,L}({\mathbb R}^n)$ and the tent space $T^{\infty}_{\omega,\mathrm v}({\mathbb R}^{n+1}_+)$ and characterize the space $\mathrm{VMO}_{\rho,L}({\mathbb R}^n)$ via the space $T^{\infty}_{\omega,\mathrm v}({{\mathbb R}}^{n+1}_+)$. Let $\widetilde{T}_{\omega} ({{\mathbb R}}^{n+1}_+)$ be the Banach completion of the tent space $T_{\omega}({\mathbb R}^{n+1}_+)$. The authors then prove that $\widetilde{T}_{\omega}({\mathbb R}^{n+1}_+)$ is the dual space of $T^{\infty}_{\omega,\mathrm v}({\mathbb R}^{n+1}_+)$. As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^\ast}({\mathbb R}^n)$ is the space $B_{\omega,L}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$ and $B_{\omega,L}({\mathbb R}^n)$ the Banach completion of the Orlicz-Hardy space $H_{\omega,L}({\mathbb R}^n)$. These results generalize the known recent results by particularly taking $\omega(t)=t$ for $t\in (0,\infty)$.

Some remarks on strong factorization of tent spaces

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Tent spaces associated with semigroups of operators

We study tent spaces on general measure spaces ( Ω , μ ) . We assume that there exists a semigroup of positive operators on L p ( Ω , μ ) satisfying a monotone property but do not assume any geometric/metric structure on Ω. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman's H 1 –BMO duality theory. We also get a H 1 –BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative L p spaces.

A new class of function spaces connecting Triebel–Lizorkin spaces and Q spaces

Let s ∈ R , τ ∈ [ 0 , ∞ ) , p ∈ ( 1 , ∞ ) and q ∈ ( 1 , ∞ ] . In this paper, we introduce a new class of function spaces F ˙ p , q s , τ ( R n ) which unify and generalize the Triebel–Lizorkin spaces with both p ∈ ( 1 , ∞ ) and p = ∞ and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the relationship between Triebel–Lizorkin spaces and Q spaces, which answers a question posed by Dafni and Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures and Q α ( R n ) , J. Funct. Anal. 208 (2004) 377–422]. Moreover, via the Hausdorff capacity, we introduce a new class of tent spaces F T ˙ p , q s , τ ( R + n + 1 ) and determine their dual spaces F W ˙ p ′ , q ′ − s , τ / q ( R n ) , where s ∈ R , p , q ∈ [ 1 , ∞ ) , max { p , q } > 1 , τ ∈ [ 0 , q ( max { p , q } ) ′ ] , and t ′ denotes the conjugate index of t ∈ ( 1 , ∞ ) ; as an application of this, we further introduce certain Hardy–Hausdorff spaces F H ˙ p , q s , τ ( R n ) and prove that the dual space of F H ˙ p , q s , τ ( R n ) is just F ˙ p ′ , q ′ − s , τ / q ( R n ) when p , q ∈ ( 1 , ∞ ) .

Conical square function estimates in UMD Banach spaces and applications to H ∞-functional calculi

We study conical square function estimates for Banach-valued functions and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-McIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator A with certain off-diagonal bounds such that A always has a bounded H ∞-functional calculus on these spaces. This provides a new way of proving functional calculus of A on the Bochner spaces L p (ℝ n ; X) by checking appropriate conical square function estimates and also a conical analogue ofBourgain’s extension of the Littlewood-Paley theory to the UMD-valued context. Even when X = ℂ, our approach gives refined p-dependent versions of known results.