Tent Spaces Research Articles

A Class of $\alpha$-Carleson Measures

In the present paper, we introduce a class of $\alpha$-Carleson measures $\mathcal{C}_{\alpha,v}(\mathbb{R}^{n+1}_+)$, which is called by the vanishing $\alpha$-Carleson measures. We prove that $\mathcal{C}_{1/p,v}(\mathbb{R}^{n+1}_+)$ is just a predual of the tent space $\widetilde{T}_{\infty}^p$ ($0 \lt p \lt 1$). Furthermore, we construct the $\alpha$-Carleson measures and the vanishing $\alpha$-Carleson measures by the Campanato functions and its a subclass, respectively. Moreover, a characterization of the vanishing $\alpha$-Carleson measure by the compactness of Poisson integral is given in this paper. Finally, as some applications, we give the $(L^{2/\alpha},L^2)$ boundedness and compactness for some paraproduct operators.

Tent spaces at endpoints

In 1985, Coifman, Meyer, and Stein gave the duality of the tent spaces; that is, (Tqp(R+n+1))*=Tq'p'(R+n+1) for 1<p,q<∞, and (T∞1(R+n+1))*=C(R+n+1), (Tq1(R+n+1))*=Tq'∞(R+n+1) for 1<q<∞, where C(R+n+1) denotes the Carleson measure space on R+n+1. We prove that (Cv(R+n+1))*=T∞1(R+n+1), which we introduced recently, where Cv(R+n+1) is the vanishing Carleson measure space on R+n+1. We also give the characterizations of Tq∞(R+n+1) by the boundedness of the Poisson integral. As application, we give the boundedness and compactness on Lq(Rn) of the paraproduct πF associated with the tent space Tq∞(R+n+1), and we extend partially an interesting result given by Coifman, Meyer, and Stein, which establishes a close connection between the tent spaces T2p(R+n+1) (1≤p≤∞) and Lp(Rn), Hp(Rn) and BMO(Rn) spaces.

Tent spaces and Littlewood-Paley g-functions associated with Bergman spaces in the unit ball of ℂn

ABSTRACTIn this paper, a family of holomorphic spaces of tent type in the unit ball of is introduced, which is closely related to maximal and area integral functions in terms of the Bergman metric. It is shown that these spaces coincide with Bergman spaces. Furthermore, Littlewood-Paley type g-functions for the Bergman spaces are introduced in terms of the radial derivative, the complex gradient and the invariant gradient. The corresponding characterizations for Bergman spaces are obtained as well. As an application, we obtain new maximal and area integral characterizations for Besov spaces.

Natural Regeneration of Oak and Hornbeam under the Canopy and Log Cabins in the State Enterprise "Korsun-Shevchenko Forestry"

The reproduction of hornbeam both natural and cultural is an important issue nowadays. Seed propagation for this species is considered to be the best way to reproduce. The regularities of the process of regeneration under the canopy of oak and hornbeam forests and log cabins in fresh oak forests in Korsun-Shevchenkivskyi forestry are the key problems of the research presented. After having conducted the research we have obtained the results stating that in mature stands of the Korsun-Shevchenko forestry fresh oak dominated for fullness of the cover. Furthermore, all log cabins show intense natural seed recovery. However, the composition of the natural seed recovery is not always correlated with the composition of the felled tree stand. The feature of beech seed recovery is that it undergrowth kept under a tent spaces, self-seeding and appears in log cabins after logging parent stand. Thus we shoul summarize that forest recovery process in fresh oak and hornbeam forests of state enterprise "Korsun-Shevchenko forestry" is satisfactory. In normal conditions oak and hornbeam of forestry bloom and bear fruit every year, but abundant crops depend on weather conditions once in 3-5 years. Under a canopy of plantations of populations of the common oak in blocks of Korsun-Shevchenko forestry through intensive expansion of grass cover and shrubs, and with not enough seats of oak and lack of proper care resumes is unsatisfactory. The most successful is a natural renewal of hornbeam populations.

Interpolation and Embeddings of Weighted Tent Spaces

Given a metric measure space X, we consider a scale of function spaces T^{p,q}_s(X), called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on X we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call Z-spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy–Littlewood–Sobolev-type embeddings between weighted tent spaces.

Calderón–Zygmund decompositions in tent spaces and weak-type endpoint bounds for two quadratic functionals of Stein and Fefferman–Stein

We prove some Calderón–Zygmund type decompositions for functions in the tent spaces introduced by Coifman, Meyer and Stein. These decompositions will be useful in the operator theory on tent spaces developed in the author’s 2015 thesis. As an application

Tent space boundedness via extrapolation

We study the action of operators on tent spaces such as maximal operators, Calderon–Zygmund operators, Riesz potentials. We also consider singular non-integral operators. We obtain boundedness as an application of extrapolation methods in the Banach range. In the non Banach range, boundedness results for Calderon–Zygmund operators follow by using an appropriate atomic theory. We end with some consequences on amalgam spaces.

Hardy-Hausdorff spaces on the Heisenberg group

In this paper, by using the tent spaces on the Siegel upper half space, which are defined in terms of Choquet integrals with respect to Hausdorff capacity on the Heisenberg group, the Hardy-Hausdorff spaces on the Heisenberg group are introduced. Then, by applying the properties of the tent spaces on the Siegel upper half space and the Sobolev type spaces on the Heisenberg group, the atomic decomposition of the Hardy-Hausdorff spaces is obtained. Finally, we prove that the predual spaces of Q spaces on the Heisenberg group are the Hardy-Hausdorff spaces.

Embedding of Möbius Invariant Function Spaces into Tent Spaces

On the unit disk we introduce a new class of tent spaces \(T^q_{s,t}(\mu )\) for any positive Borel measure \(\mu \), consider a class of Mobius invariant spaces \(Z_p\) of analytic functions, and show that \(Z_p\) is contained in \(T^1_{p,1}\) if and only if \(\mu \) is a p-Carleson measure. This could be considered a continuation or variation of the work initiated by Xiao (Adv Math 217:2075–2088, 2008).

On existence and uniqueness for non-autonomous parabolic Cauchy problems with rough coefficients

We consider existence and uniqueness issues for the initial value problem of parabolic equations ∂ t u = divA∇u on the upper half space, with initial data in L p spaces. The coefficient matrix A is assumed to be uniformly elliptic, but merely bounded measurable in space and time. For real coefficients and a single equation, this is an old topic for which a comprehensive theory is available, culminating in the work of Aronson. Much less is understood for complex coefficients or systems of equations except for the work of Lions, mainly because of the failure of maximum principles. In this paper, we come back to this topic with new methods that do not rely on maximum principles. This allows us to treat systems in this generality when p ≥ 2, or under certain assumptions such as bounded variation in the time variable (a much weaker assumption that the usual Holder continuity assumption) when p < 2. We reobtain results for real coefficients, and also complement them. For instance, we obtain uniqueness for arbitrary L p data, 1 ≤ p ≤ ∞, in the class L ∞ (0, T ; L p (R n)). Our approach to the existence problem relies on a careful construction of propagators for an appropriate energy space, encompassing previous constructions. Our approach to the uniqueness problem, the most novel aspect here, relies on a parabolic version of the Kenig-Pipher maximal function, used in the context of elliptic equations on non-smooth domains. We also prove comparison estimates involving conical square functions of Lusin type and prove some Fatou type results about non-tangential convergence of solutions. Recent results on maximal regularity operators in tent spaces that do not require pointwise heat kernel bounds are key tools in this study.

Characterization of two-weighted inequalities for multilinear fractional maximal operator

In this paper, we restudied the two-weight problem of multilinear fractional maximal operator Mα. First, we gave a characterization of two-weight inequalities for Mα related to a multilinear analogue of Sawyer’s two-weight condition S(p→,q), which essentially improved and extended some known results before. This was done mainly by using the technique of the well known atomic decomposition of tent space. Then we obtained the strong boundedness of Mα associated with multiple weight A(p→,q) class, which removed the power bump condition assumed in the known results before. Finally, a new two-weight B(p→,q) condition was introduced and the two-weight inequality of Mα with B(p→,q) condition was established.

Weighted tent spaces with Whitney averages: factorization, interpolation and duality

In this paper, we introduce a new scale of tent spaces which covers, the (weighted) tent spaces of Coifman-Meyer-Stein and of Hofmann-Mayboroda-McIntosh, and some other tent spaces considered by Dahlberg, Kenig-Pipher and Auscher-Axelsson in elliptic equations. The strong factorizations within our tent spaces, with applications to quasi-Banach complex interpolation and to multiplier-duality theory, are established. This way, we unify and extend the corresponding results obtained by Coifman-Meyer-Stein, Cohn-Verbitsky and Hyt\"onen-Ros\'en.

Real interpolation of weighted tent spaces

Let and be a Muckenhoupt weight. In this article, the authors study the real interpolation of the weighted tent space . For all , , and , the authors show that , where and denotes the weighted Lorentz-tent space, which is introduced in this article. As an application, the authors prove a real interpolation result on the weighted Hardy spaces for all and , which, when , seals a gap existing in the original proof of a corresponding result of Fefferman et al. [Trans. Amer. Math. Soc. 1974;191:75–81].

Embedding Bergman spaces into tent spaces

Let \(A^p_\omega \) denote the Bergman space in the unit disc \(\mathbb {D}\) of the complex plane induced by a radial weight \(\omega \) with the doubling property \(\int _{r}^1\omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds\). The tent space \(T^q_s(\nu ,\omega )\) consists of functions such that $$\begin{aligned} \begin{aligned} \Vert f\Vert _{T^q_s(\nu ,\omega )}^q =\int _\mathbb {D}\left( \int _{\varGamma (\zeta )}|f(z)|^s\,d\nu (z)\right) ^\frac{q}{s}\omega (\zeta )\,dA(\zeta ) <\infty ,\quad 0<q, \; s<\infty . \end{aligned} \end{aligned}$$ Here \(\varGamma (\zeta )\) is a non-tangential approach region with vertex \(\zeta \) in the punctured unit disc \(\mathbb {D}{\setminus }\{0\}\). We characterize the positive Borel measures \(\nu \) such that \(A^p_\omega \) is embedded into the tent space \(T^q_s(\nu ,\omega )\), where \(1+\frac{s}{p}-\frac{s}{q}>0\), by considering a generalized area operator. The results are provided in terms of Carleson measures for \(A^p_\omega \).

ON VECTOR-VALUED TENT SPACES AND HARDY SPACES ASSOCIATED WITH NON-NEGATIVE SELF-ADJOINT OPERATORS

AbstractIn this paper, we study Hardy spaces associated with non-negative self-adjoint operators and develop their vector-valued theory. The complex interpolation scales of vector-valued tent spaces and Hardy spaces are extended to the endpoint p=1. The holomorphic functional calculus of L is also shown to be bounded on the associated Hardy space H1L(X). These results, along with the atomic decomposition for the aforementioned space, rely on boundedness of certain integral operators on the tent space T1(X).

ON THE WELL-POSEDNESS OF PARABOLIC EQUATIONS OF NAVIER–STOKES TYPE WITH DATA

We develop a strategy making extensive use of tent spaces to study parabolic equations with quadratic nonlinearities as for the Navier–Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier–Stokes equations in$\mathbb{R}^{n}$with small initial data in$\mathit{BMO}^{-1}(\mathbb{R}^{n})$. We then study another model where neither pointwise kernel bounds nor self-adjointness are available.

Embedding theorems for Bergman spaces via harmonic analysis

Let \(A^p_\omega \) denote the Bergman space in the unit disc induced by a radial weight \(\omega \) with the doubling property \(\int _{r}^1\omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds\). The positive Borel measures such that the differentiation operator of order \(n\in \mathbb {N}\cup \{0\}\) is bounded from \(A^p_\omega \) into \(L^q(\mu )\) are characterized in terms of geometric conditions when \(0<p,q<\infty \). En route to the proof a theory of tent spaces for weighted Bergman spaces is built.

Non-uniformly local tent spaces

We develop a theory of `non-uniformly local' tent spaces on metric measure spaces. As our main result, we give a remarkably simple proof of the atomic decomposition.

Well-posedness of quasi-geostrophic equations with data in Besov-[formula omitted] spaces

By wavelets and multi-resolution analysis, we convert the bilinear estimate on tent spaces associated with Besov-Q spaces Ḃp,pγ1,γ2(Rn) to the estimates of the numerical quantities. As an application, we get the well-posedness of the quasi-geostrophic equations with initial data in Ḃp,pγ1,γ2(R2).

Wavelets and the well-posedness of incompressible magneto-hydrodynamic equations in Besov type [formula omitted]-space

In this paper, we introduce a class of Besov type Q-spaces Ḃp,pγ1,γ2(Rn) to study the well-posedness of the fractional magneto-hydrodynamic (FMHD) equations. Applying wavelets and multi-resolution analysis, we obtain the boundedness of a semigroup operator from Ḃp,pγ1,γ2(Rn) to some tent spaces Bp,m,m′γ1,γ2. As an application, we prove the global well-posedness of equations (FMHD) with data in Ḃp,pγ1,γ2(Rn). Compared with the method of Fourier transform, the advantage of our method can be applied to the well-posedness with initial data in Ḃp,pγ1,γ2(Rn), where p≠2.