Integration operators in average radial integrability spaces of analytic functions

Conditions for stochastic integrability in UMD Banach spaces

A detailed theory of stochastic integration in UMD Banach spaces has been developed recently by the authors. The present paper is aimed at giving various sufficient conditions for stochastic integrability.

Riemann integrability under weaker forms of continuity in infinite dimensional spaces

In classical analysis, the relationship between continuity and Riemann integrability is an intimate one: a continuous function on a closed and bounded interval is always Riemann integrable whereas a Riemann integrable function is continuous almost everywhere. In the setting of functions taking values in infinite dimensional spaces that include quasi Banach spaces, one encounters certain curious situations involving the breakdown of the above stated phenomena, besides the failure of the fundamental theorem of calculus and the non-existence of primitives for continuous functions! While some of t…

Improving integrability via absolute summability: a general version of Diestel's Theorem

A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the $(p,σ)$-absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banac…

Stochastic integration in quasi-Banach spaces

In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent $p\in (0,1]$. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic parti…

Mixed norm spaces and RM(p,q) spaces

In this paper we present the containment relationship between the spaces of analytic functions with average radial integrability RM(p,q) and a family of mixed norm spaces.

Weak characterizations of stochastic integrability and Dudley's theorem in infinite dimensions

In this paper we consider stochastic integration with respect to cylindrical Brownian motion in infinite dimensional spaces. We study weak characterizations of stochastic integrability and present a natural continuation of results of van Neerven, Weis and the second named author. The limitation of weak characterizations will be demonstrated with a nontrivial counterexample. The second subject treated in the paper addresses representation theory for random variables in terms of stochastic integrals. In particular, we provide an infinite dimensional version of Dudley's representation theorem for…

Differentiation and integration operators on weighted Banach spaces of holomorphic functions

We show that some previous results concerning the boundedness of differentiation and integration operators on weighted spaces given by radial weights in the unit disk or the complex plane might fail without some natural additional conditions. In view of this we develop a new elementary approach which is essentially different from the previous one and can be applied for weights and domains of general types. We also establish a new characterization of some popular classes of radial weights.

帐篷空间上的不等式与平均径向可积空间上的闭域积分算子

Inequalities on Tent Spaces and Closed Range Integration Operators on spaces of Average Radial Integrability

本文研究单位圆盘解析函数帐篷空间的反向 Carleson 测度不等式，给出关于可测集的充要条件，并将其扩展至解析帐篷空间，用于刻画 Pommerenke 积分算子在平均径向可积空间上的闭域性质。

Integration on locally compact noncommutative spaces

We present an ab initio approach to integration theory for nonunital spectral triples. This is done without reference to local units and in the full generality of semifinite noncommutative geometry. The main result is an equality between the Dixmier trace and generalised residue of the zeta function and heat kernel of suitable operators. We also examine definitions for integrable bounded elements of a spectral triple based on zeta function, heat kernel and Dixmier trace techniques. We show that zeta functions and heat kernels yield equivalent notions of integrability, which imply Dixmier trace…

Fock空间上具有非径向符号的自伴Toeplitz算子的范数界

Norm bounds for self-adjoint Toeplitz operators with non-radial symbols on the Fock space

本研究将Galbis的Fock空间自伴Toeplitz算子范数界推广至非径向有界可积符号，通过移植Nicola-Tilli等周不等式与适配Galbis的二维积分逼近论证完成拓展。

On trace Theorems for weighted mixed norm Sobolev spaces and applications

We prove trace theorems for weighted mixed norm Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the power in the weight function and the integrability power for the integration with respect to the vertical variable but not on the integrability powers for the integration with respect to the horizontal ones. They are new even in the un-weighted case and they recover classical results in the case of un-mixed norm spaces. The work is motivated by the study of re…

Function-measure kernels, self-integrability and uniquely-defined stochastic integrals

In this work we study the self-integral of a function-measure kernel and its importance on stochastic integration. A continuous-function measure kernel $K$ over $D \subset \mathbb{R}^{d}$ is a function of two variables which acts as a continuous function in the first variable and as a real Radon measure in the second. Some analytical properties of such kernels are studied, particularly in the case of cross-positive-definite type kernels. The self-integral of $K$ over a bounded set $D$ is the "integral of $K$ with respect to itself". It is defined using Riemann sums and denoted $\int_{D}K(x,dx)…

Integration and differentiation operators between growth spaces

For arbitrary radial weights $w$ and $u$, we study the integration operator between the growth spaces $H_w^\infty$ and $H_u^\infty$ on the complex plane. Also, we investigate the differentiation operator on the Hardy growth spaces $H_w^p$, $0<p<\infty$, defined on the unit disk or on the complex plane. As in the case $p=\infty$, the log-convex weights $w$ play a special role in the problems under consideration.

Volterra type integration operators between weighted Bergman spaces and Hardy spaces

Let $\mathcal{D}$ be the class of radial weights on the unit disk which satisfy both forward and reverse doubling conditions. Let $g$ be an analytic function on the unit disk $\mathbb{D}$. We characterize bounded and compact Volterra type integration operators \[ J_{g}(f)(z)=\int_{0}^{z}f(λ)g'(λ)dλ\] between weighted Bergman spaces $L_{a}^{p}(ω)$ induced by $\mathcal{D}$ weights and Hardy spaces $H^{q}$ for $0<p,q<\infty$.

Bourgain--Brezis--Mironescu 公式在任意开集上的 BV 函数及其应用

Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets and applications

本文在任意开集上为 BV 函数建立 Bourgain--Brezis--Mironescu 公式，确定非局部能量 Γ-极限为全变差常数倍，并给出 BV 函数刻画及紧性结果。