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Register a new userThe Hardy-Lorentz Spaces $H^{p,q}(R^n)$
$H^{p,q}(R^n)$ المساحات هاردي-لورنتز
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في هذا البحث نناقش مساحات Hardy-Lorentz $H^{p,q}(R^n)$، حيث $0<ple 1$، $0<qle infty$. نناقش تحليل الذري للعناصر في هذه المساحات، وخصائص التدرج الخاصة بها، وسلوك الأحداث المفردة والعمليات الأخرى التي تؤثر عليها.
In this paper we consider the Hardy-Lorentz spaces $H^{p,q}(R^n)$, with $0<ple 1$, $0<qle infty$. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them.
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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 obtain a decomposition for any distribution of the variable weak Hardy space into good and bad parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(cdot)}(mathbb R^n)$ and the space $L^{infty}(mathbb R^n)$: begin{equation*} (H^{p(cdot)}(mathbb R^n),L^{infty}(mathbb R^n))_{theta,infty} =W!H^{p(cdot)/(1-theta)}(mathbb R^n),quad thetain(0,1), end{equation*} where $W!H^{p(cdot)/(1-theta)}(mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$ with $p_-:=mathopmathrm{ess,inf}_{xinrn}p(x)in(1,infty)$ is proved to coincide with the variable Lebesgue space $W!L^{p(cdot)}(mathbb R^n)$.
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 $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally log-Holder continuous condition and $L$ 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. In this article, the authors introduce the variable weak Hardy space $W!H_L^{p(cdot)}(mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space $W!T^{p(cdot)}(mathbb R^n)$ which is also obtained in this article. In particular, when $L$ is non-negative and self-adjoint, the authors obtain the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$. As an application of the molecular characterization, when $L$ is the second-order divergence form elliptic operator with complex bounded measurable coefficient, the authors prove that the associated Riesz transform $ abla L^{-1/2}$ is bounded from $W!H_L^{p(cdot)}(mathbb R^n)$ to the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$. Moreover, when $L$ is non-negative and self-adjoint with the kernels of ${e^{-tL}}_{t>0}$ satisfying the Gauss upper bound estimates, the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$ is further used to characterize the space via non-tangential maximal functions.
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 $H^s_{p,q}$ for large $p$ and $q$. We also introduce a Garsia-type space $mathcal{G}_{p,q}$ and characterize its dual space.
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 variable exponent Lebesgue space $L^{p(cdot)/p_0}(mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(cdot)}(mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A. D. Wang, the authors characterize $H^{p(cdot)}(mathbb R^n)$ via the first order Riesz transforms when $p_-in (frac{n-1}n,infty)$, and via compositions of all the first order Riesz transforms when $p_-in(0,frac{n-1}n)$.
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