Boundedness of fractional operators in weighted variable exponent spaces with non doubling measures
Osvaldo Gorosito, Gladis Pradolini, Oscar Salinas
在具有下Ahlfors测度的变指数Lebesgue空间中，得到了有界区域上中心分数次极大函数和分数次积分算子的加权范数不等式.
In the context of variable exponent Lebesgue spaces equipped with a lower Ahlfors measure we obtain weighted norm inequalities over bounded domains for the centered fractional maximal function and the fractional integral operator.
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Boundedness of maximal functions on non-doubling manifolds with ends
设$M$是一个端点在\cite{GS}中构造的流形，$Δ$是$M$上的Laplace-Beltrami算子.本文证明了Hardy-Littlewood极大函数和与热半群$\M_Δf(x)=\sup_{t> 0} |\exp (-tΔ)f(x)| $相关的极大函数在$L^p(M)$上的弱型$(1,1)$和$L^p$有界性，对于$1 < p \le \infty$ .这些结果的意义来自于$M$不满足加倍条件的事实。
Local boundedness for weak solutions to fractional porous medium equation
本文研究了分数多孔介质方程在快速扩散情形下弱解的局部有界性，通过最优尾部假设建立了相关结论。
Boundedness of the fractional maximal operator on variable exponent Lebesgue spaces: a short proof
本文给出了分式极大算子有界性的一个简单证明，从而为C. Capone，D. Cruz Uribe和A. Fiorenza在\cite{CCUF}。
On Some Quantitative Unique Continuation Properties of Fractional Schrödinger Equations: Doubling, Vanishing Order and Nodal Domain Estimates
在这篇文章中，我们确定了一个广义Dirichlet-to-Neumann映射（这是与分数薛定谔方程）上的一个紧凑的，光滑的黎曼流形，$(M,g)$，没有边界的特征函数消失的极大序模的界限。此外，这些结果仅稍加修改就可推广到具有$C^1$势的方程。在这里，Carleman估计是一个关键工具。这些产生一个定量的三球不等式，这意味着定量的体积和边界加倍的估计，从而导致控制的极大序模的消失。利用边界加倍性质，证明了解析流形上广义Dirichlet-to-Neumann映射的本征函数节点域的$\mathcal{H}^{n-1}$ -测度的上界.
Boundedness of fractional operators associated with Schrödinger operators on weighted variable Lebesgue spaces via extrapolation
本文研究了加权变Lebesgue空间上与Schrödinger算子相关的分数阶算子有界性，通过发展外推理论，从经典不等式出发推导出分数阶积分及其交换子的加权不等式与Lipschitz变积分空间估计。
Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations
本文考虑一类$d$维（$d=1,2$）抛物-椭圆型Keller-Segel方程.我们证明了其解决方案的时间有界性在超临界范围$α>d\left(1-c\right)$，其中$c$是一个明确的常数取决于我们的问题的参数一致。此外，我们建立了$\|u(t)-u_\infty\|_{L^\infty}\rightarrow0$的充分条件，其中$u_\infty\equiv 1$是唯一的非平凡齐次解。最后，我们给出了一个唯一性结果。
We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator on the Heisenberg-Weyl group $\mathbb{H}^n$. Amongst other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates.
The global boundedness and asymptotic behavior are investigate for the solution of time-space fractional non-local reaction-diffusion equation (TSFNRDE) $$ \frac{\partial^{α}u}{\partial t^{α}}=-(-Δ)^{s} u+μu^{2}(1-kJ*u)-γu, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty),$$ where $s\in(0,1),α\in(0,1), N \leq 2$. The operator $\partial_{t}^{α}$ is the Caputo fractional derivative, which $-(-Δ)^{s}$ is the fractional Laplacian operator. For appropriate assumptions on $J$, it is proved that for homogeneous Dirichlet boundary condition, this problem admits a global bounded weak solution for $N=1$, w…
Local boundedness of solutions to nonlocal equations modeled on the fractional p-Laplacian
我们的状态和证明估计的局部有界性的非局部，可能退化，抛物型积分微分方程的形式\begin{equation*} \partial_tu(x,t)+\mbox{P.V.}\int\limits_{\mathbb R^n}K(x,y,t) |u(x,t)-u(y,t) |^{p-2}(u(x,t)-u(y,t))\, dy,\end{equation*} $(x,t)\in\mathbb R^n\times\mathbb R$，其中$\mbox{P.V.} $的意思是在主值意义下，$p\in (1,\infty)$和内核服从$K(x,y,t)\approx |x-y |^{n+ps}$的一些$s\in (0,1)$，一致在$(x,y,t)\in\mathbb R^n\times \mathbb R^n\times\mathbb R$。
On Gaussian Lipschitz spaces and the boundedness of Fractional Integrals and Fractional Derivatives on them
本文引入了关于Gauss测度的Lipschitz空间，并研究了其上的分数次积分和分数次导数算子的有界性.所用的方法是一般的，足以提供经典情形的另一种证明，而且还可以推广到Laguerre和Jacobi展开的情形
Boundedness of fractional integrals and fractional derivatives on Laguerre Lipschitz spaces
该研究受高斯Lipschitz空间相关结果启发，通过发展拉盖尔框架下的泊松积分理论，证明了与拉盖尔多项式展开相关的一类分数次积分与导数在拉盖尔Lipschitz空间上的有界性。
In this paper, the sharp maximal theorem is generalized to mixed-norm ball Banach function spaces, which is defined as Definition 2.7. As an application, we give a characterization of BMO via the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Moreover, the characterization of homogeneous Lipschitz space is also given by the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Finally, two applications of Corollary 6.4 are given.
The global boundedness and asymptotic behavior are investigated for the solutions of a nonlocal time fractional reaction-diffusion equation (NTFRDE) $$ \frac{\partial^{α}u}{\partial t^{α}}=Δu+μu^{2}(1-kJ*u)-γu, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty)$$ with $0<α<1,β, μ,k>0,N\leq 2$ and $u(x,0)=u_{0}(x)$. Under appropriate assumptions on $J$ and the property of time fractional derivative, it is proved that for any nonnegative and bounded initial conditions, the problem has a global bounded classical solution if $k^{*}=0$ for $N=1$ or $k^{*}=(μC^{2}_{GN}+1)η^{-1}$ for $N=2$, where $C_{GN}$…
For the time-space fractional degenerate Keller-Segel equation \begin{equation*} \begin{cases} \partial _{t}^{β}u=-(-Δ)^{\fracα{2}}(ρ(v)u),& t>0\\ (-Δ)^{\fracα{2}} v+v=u,& t>0 \end{cases} \end{equation*} $x\inΩ, Ω\subset \mathbb{R}^{n}, β\in (0,1),α\in (1,2)$, we consider for $n\geq 3$ the problem of finding a time-independent upper bound of the classical solution such that as $θ>0,C>0$ \begin{equation*} \left \| u(\cdot ,t)-\overline{u_{0}} \right \|_{L^{\infty }(Ω)}+\left \| v(\cdot ,t)-\overline{u_{0}} \right \|_{W^{1,\infty }(Ω)}\leq Ce^{(-θ)^{1/β}t}, \end{equation*} where $\overline{u_{0}…
Uniform Hölder regularity with small exponent in competition-fractional diffusion systems
对于一类具有分数阶Laplacian幂的竞争扩散非线性系统，包括分数阶Gross-Pitaevskii系统，我们证明了当种间竞争参数发散时，对于充分小的正指数，一致有界性蕴涵着Hölder有界性.这意味着强收敛的家庭的解决方案的隔离，他们的支持发生。
Here, we consider periodic homogenization for time-fractional Hamilton--Jacobi equations. By using the perturbed test function method, we establish the convergence, and give estimates on a rate of convergence. A main difficulty is the incompatibility between the function used in the doubling variable method, and the non-locality of the Caputo derivative. Our approach is to provide a lemma to prove the rate of convergence without the doubling variable method with respect to the time variable, which is a key ingredient.
The global boundedness and asymptotic behavior are investigated for the solutions of a nonlocal time fractional p-Laplacian reaction-diffusion equation (NTFPLRDE) $$ \frac{\partial^{α}u}{\partial t^{α}}=Δ_{p} u+μu^{2}(1-kJ*u) -γu, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty)$$ with $0<α<1,β, μ,k>0,N\leq 2$ and $Δ_{p}u =div(\left| \bigtriangledown u \right|^{p-2}\bigtriangledown u)$. Under appropriate assumptions on $J$ and the conditions of $1<p<2$, it is proved that for any nonnegative and bounded initial conditions, the problem has a global bounded classical solution if $k^{*}=0$ for $N=1$ …
Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators
本文证明了多线性分数次积分算子与多线性分数次极大算子之间的加权范数不等式和点态估计。作为这些估计的结果，我们得到了多线性分数次积分算子的加权弱不等式和加权强不等式。特别地，我们将\cite{CPSS}中的一些结果推广到多线性情形。另一方面，我们证明了与Young函数$B$相关联的多线性分式极大算子${\cal M}_{α,B}$和多线性极大算子${\cal M}_ψ={\cal M}_{0,ψ}$，$ψ(t)=B(t^{1-α/(nm)})^{{nm}/{(nm-α)}}$之间的加权逐点估计。作为这些估计的一个应用，我们得到了${\cal M}_{α,B}$的$L^p-L^q$有界性结果的一个直接证明，对于$B(t)=t$和$B_k(t)=t(1+\log^+t)^k$，当$1/q=1/p-α/n$ .我们还给出了充分条件的权重参与的有界性结果的${\cal M}_{α,B}$，推广了那些在\cite{M}为$B(t)=t$。最后，我们证明了多线性分式极大算子在Banach函数空间中的有界性结果。
In this paper, we show that the global solution of the surface anisotropic two-dimensional quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion established by the author in [2] is bounded in Sobolev spaces uniformly with respect to time.
Hölder Continuity for Fully Fractional Parabolic Equations with Space-time Nonlocal Operators
本文研究含空间分数阶扩散与Marchaud型分数阶时间导数的全分数阶抛物型方程，在不假设解和非齐次项有界的条件下，引入时间依赖非局部尾项，利用非局部De Giorgi迭代技术证明弱解的局部有界性与局部Hölder连续性。
We study generalized fractional $p$-Laplacian equations to prove local boundedness and Hölder continuity of weak solutions to such nonlocal problems by finding a suitable fractional Sobolev-Poincaré inquality.
Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators
本文研究分数阶$p$-拉普拉斯型算子相关抛物方程局部弱解的局部有界性，通过新型水平集截断等方法推广了线性情形结果，对所有$p>1$建立了含时间$L^1$-范数非局部抛物尾项的局部有界性。
We establish some new properties of the Dunkl-Wiener amalgam spaces defined on the real line. These results allow us to obtain the boundedness of Dunkl-type fractional integral and fractional maximal operators in the Dunkl-Fofana spaces.
We consider the Nemytskii operators $u\to |u|$ and $u\to u^{\pm}$ in a bounded domain $Ω$ with $C^2$ boundary. We give elementary proofs of the boundedness in $H^s(Ω)$ with $0\le s<3/2$.
New fractional type weights and the boundedness of some operators
本文建立了两类分数型变指标权，分别刻画多线性分数型算子及其交换子在加权变指标 Lebesgue 空间上的有界性，并推广了已有的矩阵权结果。
We consider parabolic PDEs associated with fractional type operators drifted by non-linear singular first order terms. When the drift enjoys some boundedness properties in appropriate Lebesgue and Besov spaces, we establish by exploiting a priori Besov-type estimates, the H{ö}lder continuity of the solutions. In particular, we handle the almost critical case in whole generality.
Characterization of the boundedness of the discrete Fractional maximal operator between weighted in Lorentz sequence spaces
本文完整刻画了离散分数阶极大算子在加权Lorentz序列空间间的有界性，通过尖锐重排估计将其归结为上确界算子的有界性，并利用近期结果得到最终判定。
Well-posedness of Dirichlet boundary value problems for reflected fractional $p$-Laplace-type inhomogeneous equations in compact doubling metric measure spaces
研究紧致加倍度量测度空间中反射分数阶 p-Laplace 型非齐次方程的 Dirichlet 问题，证明了存在性、唯一性、比较与稳定性，并建立了内部正则性和 Kellogg 型性质。
On the Boundedness of Multilinear Fractional Strong Maximal Operator with multiple weights
本文研究了多线性分式强极大算子$\mathcal{M}_{\mathcal{R},α}$与矩形相关或与更一般的重权基$A_{(\vec{p},q),\mathcal{R}}$相关的有界性.在矩形情形下，我们首先给出了$\mathcal{M}_{\mathcal{R},α}$的一个端点估计，它不仅推广了Schenen，Marcinkiewicz和Zygmund的著名线性结果，而且推广了Grafakos，Liu，Pérez和Torres的多重线性结果（$α=0$）到情形$0<α<mn.$然后，在一个权情形下，我们给出了$\mathcal{M}_{\mathcal{R},α}$和$A_{(\vec{p},q),\mathcal{R}}$之间的几个等价刻画，通过应用与我们以前使用的方法不同的方法。此外，给出了$\mathcal{M}_{\mathcal{R},α}$的二权范数不等式成立的一个充分条件，并在$m=1$的情形下，建立了强极大算子的一个向量值二加权不等式.在一般基的基础上，我们进一步研究了多重权$A_{(\vec{p},q),\mathcal{R}}$条件的性质，包括等价刻画和单调性，在本质上扩展了前人的认识.最后，对多重强Muckenhoupt权进行了综述，系统地论证了多重权与矩形相关的性质。
A remark on fractional integrals on modulation spaces
本文给出了分数次积分算子在调制空间上有界的充要条件。
In this article, the authors establish a general (two-weight) boundedness criterion for a pair of functions, $(F,f)$, on $\mathbb{R}^n$ in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz--)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón--Zygmund operators, Littlewood--Paley $g$-functions, Lusin area functions, Littlewood--Paley $g^\ast_λ$-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) bou…
We prove fractional Sobolev-Poincaré inequalities, capacitary versions of fractional Poincaré inequalities, and pointwise and localized fractional Hardy inequalities in a metric space equipped with a doubling measure. Our results generalize and extend earlier work where such inequalities have been considered in the Euclidean spaces or in the non-fractional setting in metric spaces. The results concerning pointwise and localized variants of fractional Hardy inequalities are new even in the Euclidean case.
We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds, in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajlasz-Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.
Let $M$ be a complete non-compact manifold satisfying the volume doubling condition, with doubling index $N$ and reverse doubling index $n$, $n\le N$, both for large balls. Assume a Gaussian upper bound for the heat kernel, and an $L^2$-Poincaré inequality outside a compact set. If $2<n$, then we show that for $p\in (2,n)$, $(R_p)$: $L^p$-boundedness of the Riesz transform, $(G_p)$: $L^p$-boundedness of the gradient of the heat semigroup, and $(RH_p)$: reverse $L^p$-Hölder inequality for the gradient of harmonic functions, are equivalent to each other. Our characterization implies that for $p\…
Let $p\in[1,\infty]$, $q\in[1,\infty)$, $s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $α\in\mathbb{R}$. In this article, the authors first find a reasonable version $\widetilde{I}_β$ of the (generalized) fractional integral $I_β$ on the special John--Nirenberg--Campanato space via congruent cubes, $JN_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$, which coincides with the Campanato space $\mathcal{C}_{α,q,s}(\mathbb{R}^n)$ when $p=\infty$. To this end, the authors introduce the vanishing moments up to order $s$ of $I_β$. Then the authors prove that $\widetilde{I}_β$ is bounded from $JN_{(p,q,s)_α}^…
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ and $H_X({\mathbb R}^n)$ the Hardy space associated with $X$, and let $α\in(0,n)$ and $β\in(1,\infty)$. In this article, assuming that the (powered) Hardy--Littlewood maximal operator satisfies the Fefferman--Stein vector-valued maximal inequality on $X$ and is bounded on the associate space of $X$, the authors prove that the fractional integral $I_α$ can be extended to a bounded linear operator from $H_X({\mathbb R}^n)$ to $H_{X^β}({\mathbb R}^n)$ if and only if there exists a positive constant $C$ such that, for any ball $B\sub…
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ satisfying some mild assumptions and let $α\in(0,n)$ and $β\in(1,\infty)$. In this article, when $α\in(0,1)$, the authors first find a reasonable version $\widetilde{I}_α$ of the fractional integral $I_α$ on the ball Campanato-type function space $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ with $q\in[1,\infty)$, $s\in\mathbb{Z}_+^n$, and $d\in(0,\infty)$. Then the authors prove that $\widetilde{I}_α$ is bounded from $\mathcal{L}_{X^β,q,s,d}(\mathbb{R}^n)$ to $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ if and only if there exists a positive c…
Parabolic Muckenhoupt Weights Characterized by Parabolic Fractional Maximal and Integral Operators with Time Lag
本文引入带时滞的非对角双权抛物 Muckenhoupt 类，刻画带时滞抛物分数次极大算子与新型分数次积分的加权有界性，并给出 Sobolev 嵌入和热方程先验估计。
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