论文标题
作用于混合 - norm lebesgue空间的积分内核的Schur-type Banach模块
Schur-type Banach modules of integral kernels acting on mixed-norm Lebesgue spaces
论文作者
论文摘要
Schur's test states that if $K:X\times Y\to\mathbb{C}$ satisfies $\int_Y |K(x,y)|dν(y)\leq C$ and $\int_X |K(x,y)|dμ(x)\leq C$, then the associated integral operator acts boundedly on $L^p$ for all $p\in [1,\infty]$.我们得出了此结果的一个变体，以确保（加权）混合 - - - - - - - - - - - - - - - - - - norm lebesgue spaces $ l_w^{p，q} $的所有$ p，q \ in [1，\ infty] $。对于非阴性整体内核，我们的标准很清晰。即，仅当整体操作员在所有混合符号Lebesgue空间上都有有条体作用时，就会满足。 在此标准的动机上，我们引入了固体Banach模块$ \ MATHCAL {b} _m（x，y）的整体核的$，以使所有中的所有内核中的所有内核中的所有内核{b} _m（x，x，y）$ l_w^{p，q}（p，q}（p，n [1，\ infty] $，前提是权重$ v，w $是$ m $ - 中等。相反，如果$ \ MATHBF {A} $和$ \ MATHBF {B} $是坚实的Banach空间与混合 - lebesgue空间有关； I.e.，$ \ left（l^1 \ cap l^\ infty \ cap l^{1，\ infty} \ cap l^{\ infty，1} \ right）_v \ hookrightArrow \ mathbf {b} l^{1，\ infty} + l^{\ infty，1} \ right）_ {1/w} $对于某些权重$ v，w $，具体取决于权重$ m $。 内核代数$ \ MATHCAL {B} _M（X，X）$特别适合（广义）COORBIT理论应用：通常，需要验证许多技术条件以确保Coorbit Space理论适用于给定的连续框架$ψ$和Banach Space $ \ Mathbf $ \ Mathbf {a a a a} $ {a} $。我们表明，检查是否属于$ \ Mathcal {B} _M（X，X）$相关的某些与$ψ$关联的积分内核;这样可以确保coorbit Spaces $ \ operatorName {co}_ψ（l_κ^{p，q}）$对于所有$ p，q \ in [1，\ infty] $ in [1，\ infty] $，并且所有权重$κ$与$ m $兼容。
Schur's test states that if $K:X\times Y\to\mathbb{C}$ satisfies $\int_Y |K(x,y)|dν(y)\leq C$ and $\int_X |K(x,y)|dμ(x)\leq C$, then the associated integral operator acts boundedly on $L^p$ for all $p\in [1,\infty]$. We derive a variant of this result ensuring boundedness on the (weighted) mixed-norm Lebesgue spaces $L_w^{p,q}$ for all $p,q\in [1,\infty]$. For non-negative integral kernels our criterion is sharp; i.e., it is satisfied if and only if the integral operator acts boundedly on all of the mixed-norm Lebesgue spaces. Motivated by this criterion, we introduce solid Banach modules $\mathcal{B}_m(X,Y)$ of integral kernels such that all kernels in $\mathcal{B}_m(X,Y)$ map $L_w^{p,q}(ν)$ boundedly into $L_v^{p,q}(μ)$ for all $p,q \in [1,\infty]$, provided that the weights $v,w$ are $m$-moderate. Conversely, if $\mathbf{A}$ and $\mathbf{B}$ are solid Banach spaces for which all kernels $K\in\mathcal{B}_m(X,Y)$ map $\mathbf{A}$ into $\mathbf{B}$, then $\mathbf{A}$ and $\mathbf{B}$ are related to mixed-norm Lebesgue-spaces; i.e., $\left(L^1\cap L^\infty\cap L^{1,\infty}\cap L^{\infty,1}\right)_v\hookrightarrow\mathbf{B}$ and $\mathbf{A}\hookrightarrow\left(L^1 + L^\infty + L^{1,\infty} + L^{\infty,1}\right)_{1/w}$ for certain weights $v,w$ depending on the weight $m$. The kernel algebra $\mathcal{B}_m(X,X)$ is particularly suited for applications in (generalized) coorbit theory: Usually, a host of technical conditions need to be verified to guarantee that coorbit space theory is applicable for a given continuous frame $Ψ$ and a Banach space $\mathbf{A}$. We show that it is enough to check that certain integral kernels associated to $Ψ$ belong to $\mathcal{B}_m(X,X)$; this ensures that the coorbit spaces $\operatorname{Co}_Ψ(L_κ^{p,q})$ are well-defined for all $p,q\in [1,\infty]$ and all weights $κ$ compatible with $m$.
