We introduce connections between the Cuntz relations and the Hardy space H_2 of the open unit disk . We then use them to solve a new kind of multipoint interpolation problem in H_2, where for instance, only a linear combination of the values of a function at given points is preassigned, rather than the values at the points themselves. We also consider the case of de Branges-Rovnyak spaces.
We studied complex interpolation noncommutative Hardy space associated with semi-finite von Neumann algebra and extend Pisier's interpolation theorem for this case.
Let $(X, Y)$ be a couple of quasi-Banach lattices of measurable functions on $\mathbb T \times Ω$ satisfying some additional assumptions. The K-closedness of a couple of Hardy-type spaces $(X_A, Y_A)$ in $(X, Y)$ and the stability of the real interpolation $(X_A, Y_A)_{θ, p} = (X_A + Y_A) \cap (X, Y)_{θ, p}$ are shown to be equivalent to each other and to the BMO-regularity of the associated lattices $\left(L_1, \left(X^r\right)' Y^r\right)_{δ, q}$. The inclusion $\left(X^{1 - θ} Y^θ\right)_A \subset \left(X_A, Y_A \right)_{θ, \infty}$ is also characterized in these therms. New examples of cou…
We characterize the non-commutative Aleksandrov--Clark measures and the minimal realization formulas of contractive and, in particular, isometric non-commutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb{C} ^d$ is defined as the Hilbert space of square--summable power series in several non-commuting formal variables, and we interpret this space as the non-commutative and multi-variable analogue of the Hardy space of square--summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov--Clark measur…
We consider an infinite homogeneous tree V endowed with the usual metric d defined on graphs and a weighted measure μ. The metric measure space V,d,μ) is nondoubling and of exponential growth, hence the classical theory of Hardy spaces does not apply in this setting. We construct an atomic Hardy space H^1 on (V,d,μ) and investigate some of its properties, focusing in particular on real interpolation properties and on boundedness of singular integrals on H^1.
Let $\mathcal{M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal{M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\M$. For $0<p \leq\infty$, let $\h_p^c(\mathcal{M})$ denote the noncommutative column conditioned martingale Hardy space associated with the filtration $(\mathcal{M}_n)_{n\geq 1}$ and the index $p$. We prove that for $0<p<\infty$, the compatible couple $\big(\h_p^c(\mathcal{M}), \h_\infty^c(\mathcal{M})\big)$ is $K$-closed in the couple $\big(L_p(\mathcal{N}), L_\infty(\mathcal{N}) \big)$ for an appropriate amplified semifinite von …
This paper is devoted to the interpolation principle between spaces of weak type. We characterise interpolation spaces between two Marcinkiewicz spaces in terms of Hardy type operators involving suprema. We study general properties of such operators and their behavior on Lorentz gamma spaces. A particular emphasis is placed on elementary and comprehensive proofs.
Let $\mathcal{M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal{M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal{M}$. For $0<p <\infty$, let $\mathsf{h}_p^c(\mathcal{M})$ denote the noncommutative column conditioned martingale Hardy space and $\bmo^c(\M)$ denote the column \lq\lq little\rq\rq \ martingale BMO space associated with the filtration $(\mathcal{M}_n)_{n\geq 1}$. We prove the following real interpolation identity: if $0<p <\infty$ and $0<θ<1$, then for $1/r=(1-θ)/p$, \[ \big(\mathsf{h}_p^c(\mathcal{M}), \bmo^c(\mathcal{M})\…
This paper gives a systematic study of operator-valued local Hardy spaces. These spaces are localizations of the Hardy spaces defined by Tao Mei, and share many properties with Mei's Hardy spaces. We prove the ${\rm h}_1$-$\rm bmo$ duality, as well as the ${\rm h}_p$-${\rm h}_q$ duality for any conjugate pair $(p,q)$ when $1<p< \infty$. We show that ${\rm h}_1(\mathbb{R}^d, \mathcal M)$ and ${\rm bmo}(\mathbb{R}^d, \mathcal M)$ are also good endpoints of $L_p(L_\infty(\mathbb{R}^d) \overline{\otimes} \mathcal M)$ for interpolation. We obtain the local version of Calderón-Zygmund theory, and th…
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$. In this article, assuming that the powered Hardy--Littlewood maximal operator satisfies some Fefferman--Stein vector-valued maximal inequality on $X$ as well as it is bounded on both the weak ball quasi-Banach function space $WX$ and the associated space, the authors establish various Littlewood--Paley function characterizations of $WH_X({\mathbb R}^n)$ under some weak assumptions on the Littlewood--Paley functions. The authors also prove that the real interpolation intermediate space $(H_{X}({\mathbb R}^n),L^\infty({\mathbb R}^…
In this article, the authors introduce a class of mixed-norm Herz spaces, $\dot{E}^{\vecα,\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$, which is a natural generalization of mixed Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz-Thorin interpolation theorem on $\dot{E}^{\vecα,\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$. Applying these Riesz-Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the …
We show that de Branges-Rovnyak spaces include as special cases a number of spaces, such as the Hardy space, the Fock space, the Hardy-Sobolev space and the Dirichlet space. We present a general framework in which all these spaces can be obtained by specializing a sequence that appears in the construction. We show how to exploit this approach to solve interpolation problems in the Fock space.
In this survey article some classical results concerning real interpolation between Hardy spaces are briefly presented and then it is explained how those results can be used to establish Yano-type extrapolation theorems for Hardy spaces. Some new extensions and variants of certain classical endpoint theorems in harmonic analysis are obtained as applications of the extrapolation results presented here.
Let $(\mathbb{X},d,μ)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón--Zygmund operators in the critical case. The main novelties of this article exist in that the authors…
We give a complete characterization of limiting interpolation spa\-ces for the real method of interpolation using extrapolation theory. For this purpose the usual tools (e.g., Boyd indices or the boundedness of Hardy type operators) are not appropriate. Instead, our characterization hinges upon the boundedness of some simple operators (e.g. $f\mapsto f(t^{2})/t$, or $f\mapsto f(t^{1/2}% )$) acting on the underlying lattices that are used to control the $K$- and $J$-functionals. Reiteration formulae, extending Holmstedt's classical reiteration theorem to limiting spaces, are also proved and cha…
This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.
Introduced by Coifman, Meyer, and Stein, the tent spaces have seen wide applications in Harmonic Analysis. Their analytic cousins have seen some applications involving the derivatives of Hardy space functions. Moreover, the tent spaces have been a recent focus of research. We introduce the concept of interpolating and sampling sequences for analytic tent spaces analogously to the same concepts for Bergman spaces. We then characterize such sequences in terms of Seip's upper and lower uniform density. We accomplish this by exploiting a kind of Möbius invariance for the tent spaces.
