We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

In this paper we introduce Hardy-Lorentz spaces with variable exponents associated with dilations in ${{\mathbb{R}}^{n}}$. We establish maximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.

It is well known that functions in the analytic Besov space $B_{1}$ on the unit disk $\mathbb{D}$ admit an integral representation

$$\begin{eqnarray}f(z)=\int _{\mathbb{D}}\frac{z-w}{1-z\overline{w}}\,d{\it\mu}(w),\end{eqnarray}$$

${\it\mu}$

$|{\it\mu}|(\mathbb{D})<\infty$

$B_{p}$

$0<p\leq 1$

${\rm\Lambda}_{t}$

$t>1$

Tent spaces of vector-valued functions were recently studied by Hytönen, van Neerven and Portal with an eye on applications to $H^{\infty }$-functional calculi. This paper extends their results to the endpoint cases $p=1$ and $p=\infty $ along the lines of earlier work by Harboure, Torrea and Viviani in the scalar-valued case. The main result of the paper is an atomic decomposition in the case $p=1$, which relies on a new geometric argument for cones. A result on the duality of these spaces is also given.

We show that the modulus of the Bergman kernel $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}B(z, \zeta )$ of a general homogeneous Siegel domain of type II is ‘almost constant’ uniformly with respect to $z$ when $\zeta $ varies inside a Bergman ball. The control is expressed in terms of the Bergman distance. This result was proved by A. Korányi for symmetric Siegel domains of type II. Subsequently, R. R. Coifman and R. Rochberg used it to establish an atomic decomposition theorem and an interpolation theorem by functions in Bergman spaces $A^p$ on these domains. The atomic decomposition theorem and the interpolation theorem are extended here to the general homogeneous case using the same tools. We further extend the range of exponents $p$ via functional analysis using recent estimates.

In this article, we establish a new atomic decomposition for $f\,\in \,L_{w}^{2}\,\bigcap \,H_{w}^{p}$, where the decomposition converges in $L_{w}^{2}$-norm rather than in the distribution sense. As applications of this decomposition, assuming that $T$ is a linear operator bounded on $L_{w}^{2}$ and $0\,<\,p\,\le \,1$, we obtain (i) if $T$ is uniformly bounded in $L_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$; (ii) if $T$ is uniformly bounded in $H_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded on $H_{w}^{p}$; (iii) if $T$ is bounded on $H_{w}^{p}$, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$.

We obtain an atomic decomposition for weighted Triebel–Lizorkin spaces on spaces of homogeneous type, using the area function, the discrete Calderón reproducing formula and discrete sequence spaces.

In this paper, we shall give some characterizations of the Hardy space associated with twisted convolution, including Lusin area integral, Littlewood–Paley g-function and heat maximal function.

We study the coordinate rings of $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$ scheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here ${\mu }'$ gives the Jordan block structure of the nilpotent matrix. de Concini and Procesi [5] proved a conjecture of Kraft [12] that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer [22, 23]. The famous $q$-Kostka polynomial ${{\tilde{K}}_{\lambda \mu }}(q)$ is the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by $\lambda $ in the ring $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$. Lascoux and Schützenberger [15, 13] gave combinatorially a decomposition of ${{\tilde{K}}_{\lambda \mu }}(q)$ as a sum of “atomic” polynomials with non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model.

Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen [19] imply a direct-sum decomposition of the ideals of nilpotent orbit closures, arising from the inclusions of the corresponding sets. We carry out the restriction to the diagonal using a recent theorem of Broer [3]. This gives a direct-sum decomposition of the ideals yielding the $k\left[ \overline{{{\text{C}}_{\mu }}}\,\bigcap \,\text{t} \right]$, and a new proof of the atomic decomposition of the $q$-Kostka polynomials.