The Hardy Operator and Boyd Indices

Limiting interpolation spaces via extrapolation

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…

Hardy Space Estimates for Littlewood-Paley-Stein Square Functions and Calderón-Zygmund Operators

In this work, we give new sufficient conditions for a Littlewood-Paley-Stein square function and necessary and sufficient conditions for a Calderón-Zygmund operator to be bounded on Hardy spaces $H^p$ with indices smaller than $1$. New Carleson measure type conditions are defined for Littlewood-Paley-Stein operators, and we show that they are sufficient for the associated square function to be bounded from $H^p$ into $L^p$. New polynomial growth $BMO$ conditions are also introduced for Calderón-Zygmund operators. These results are applied to prove that Bony paraproducts can be constructed such…

与算子相关的局部Hardy空间及加倍度量测度空间上的球拟Banach函数空间及其应用

Local Hardy Spaces Associated with Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

本文在加倍度量测度空间框架下，研究与球拟Banach函数空间及满足高斯上界热核的非负自伴算子相关的局部Hardy空间，建立其多种刻画并应用于几类函数空间。

On a theorem of Livsic

The theory of symmetric, non-selfadjoint operators has several deep applications to the complex function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators such as Schrodinger operators in mathematical physics. Examples of simple symmetric operators include multiplication operators on various spaces of analytic functions such as model subspaces of Hardy spaces, deBranges-Rovnyak spaces and Herglotz spaces, ordinary differential operators (including Schrodinger operators from quantum mechanics), Toeplitz operator…

重排不变空间上分段乘子有界性的刻画

Characterization of the boundedness of the segment multiplier on rearrangement-invariant spaces

研究重排不变空间上分段乘子等变换的有界性，通过一对博伊德型指标刻画其有界性，还展示了分段乘子有界但非截断希尔伯特变换无界的重排不变空间。

Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces

We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\cdot)}(\varrho)$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result of the boundedness of the Hardy-Littlewood maximal operator in the space $L^{p(\cdot)}(X,\varrho)$ over a metric measure space $X$ satisfying the doubling condition. In the case where $X$ is bounded, the weight function satisfies a certain version of a general Muckenhoupt-type condition For a bounded or unbounded $X$ we also consider a class of weights of t…

Interpolation of Fredholm operators

We prove novel results on interpolation of Fredholm operators including an abstract factorization theorem. The main result of this paper provides sufficient conditions on the parameters $θ\in (0,1)$ and $q\in \lbrack 1,\infty ]$ under which an operator $A$ is a Fredholm operator from the real interpolation space $(X_{0},X_{1})_{θ,q}$ to $(Y_{0},Y_{1})_{θ,q} $ for a given operator $A\colon (X_{0},X_{1})\rightarrow (Y_{0},Y_{1})$ between compatible pairs of Banach spaces such that its restrictions to the endpoint spaces are Fredholm operators. These conditions are expressed in terms of the corre…

重排不变空间中有限Hilbert变换的延拓与积分表示

Extension and Integral Representation of the finite Hilbert Transform In Rearrangement Invariant Spaces

本文研究重排不变空间中有限Hilbert变换的延拓与反演，通过引入由该算子诱导的向量测度及其积分算子，给出了该算子的积分表示。