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Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first introduce the variable weak Hardy space on $mathbb R^n$, $W!H^{p(cdot)}(mathbb R^n)$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $W!H^{p(cdot)}(mathbb R^n)$, respectively, by means of atoms, molecules, the Lusin area function, the Littlewood-Paley $g$-function or $g_{lambda}^ast$-function. As an application, the authors establish the boundedness of convolutional $delta$-type and non-convolutional $gamma$-order Calderon-Zygmund operators from $H^{p(cdot)}(mathbb R^n)$ to $W!H^{p(cdot)}(mathbb R^n)$ including the critical case $p_-={n}/{(n+delta)}$, where $p_-:=mathopmathrm{ess,inf}_{xin rn}p(x).$
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying that there exists a constant $p_0in(0,p_-)$, where $p_-:=mathop{mathrm {ess,inf}}_{xin mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the
Let $L$ be a one-to-one operator of type $omega$ in $L^2(mathbb{R}^n)$, with $omegain[0,,pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(cdot): mathbb{R}^nto(0,,1]$ be a variable exponent
Let $L$ be a linear operator on $L^2(mathbb R^n)$ generating an analytic semigroup ${e^{-tL}}_{tge0}$ with kernels having pointwise upper bounds and $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally log-Holder conti
We present in this paper some embeddings of various dyadic martingale Hardy-amalgam spaces $H^S_{p,q},,, H^s_{p,q},,,H^*_{p,q},,,mathcal{Q}_{p,q}$ and $mathcal{P}_{p,q}$ of the real line. In the same settings, we characterize the dual of $H^s_{p,q}$
Let $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally $log$-Holder continuous condition and $L$ a non-negative self-adjoint operator on $L^2(mathbb R^n)$ whose heat kernels satisfying the Gaussian upper bound estima