arXiv subjects

Renjin Jiang

Publications and source records attributed to Renjin Jiang.

33 records · Page 2Linked to original sources

arXiv2015

Heat Kernel Bounds on Metric Measure Spaces and Some Applications

Let $(X,d,μ)$ be a $RCD^\ast(K, N)$ space with $K\in \mathbb{R}$ and $N\in [1,\infty]$. For $N\in [1,\infty)$, we derive the upper and lower bounds of the heat kernel on $(X,d,μ)$ by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. When $N=\infty$, we also establish a sharp upper bound of the heat kernel by using the dimension free Harnack inequality. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the $L^p$ boundedness of (local) Riesz transforms.

math.MG↗

arXiv2015

Linear transport equations for vector fields with subexponentially integrable divergence

We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline integrability assumptions on the divergence of the velocity field $b$. For $W^{1,1}_{loc}$ vector fields $b$ satisfying $\frac{|b(x,t)|}{1+|x|}\in L^1(0,T; L^1)+L^1(0,T; L^\infty)$ and $$\operatorname{div} b\in L^1(0,T;L^\infty) + L^1\left(0,T; \operatorname{Exp}\left(\frac{L}{\log L}\right)\right),$$ we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every $γ>1$, we construct an example of a bounded autonomous velocity field $b$ with $$\operatorname{div} b\in \operatorname{Exp}\left(\frac{L}{\log^γL}\right) ,$$ for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the $BV$ setting are also addressed.

math.AP↗

arXiv2014

Hajlasz Gradients Are Upper Gradients

Let $(X, d, μ)$ be a metric measure space, with $μ$ a Borel regular measure. In this paper, we prove that, if $u\in L^1_{\mathop\mathrm{\,loc\,}}(X)$ and $g$ is a Hajłasz gradient of $u$, then there exists $\widetilde u$ such that $\widetilde u=u$ almost everywhere and $4g$ is a $p$-weak upper gradient of $\widetilde u$. This result avoids a priori assumption on the quasi-continuity of $u$ used in [Rev. Mat. Iberoamericana 16 (2000), 243-279]. As an application, an embedding of the Morrey-type function spaces based on Hajłasz-gradients into the corresponding function spaces based on upper gradients is obtained. We also introduce the notion of local Hajłasz gradient, and investigate the relations between local Hajłasz gradient and upper gradient.

math.FA↗

arXiv2014

The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces

Let $(X,d,μ)$ be a $RCD^\ast(K, N)$ space with $K\in mathbb{R}$ and $N\in [1,\infty)$. Suppose that $(X,d)$ is connected, complete and separable, and $\supp μ=X$. We prove that the Li-Yau inequality for the heat flow holds true on $(X,d,μ)$ when $K\ge 0$. A Baudoin-Garofalo inequality and Harnack inequalities for the heat flows are established on $(X,d,μ)$ for general $K\in \mathbb{R}$. Large time behaviors of heat kernels are also studied.

math.MG↗

arXiv2014

Towards spaces of harmonic functions with traces in square Campanato space and its scaling invariant

For $n\ge 1$ and $α\in (-1,1)$, let $H^{α,2}$ be the space of harmonic functions $u$ on the upper half space $\mathbb{R}^{n+1}_+$ satisfying $$\displaystyle\sup_{(x_0,r)\in \mathbb R^{n+1}_+}r^{-(2α+n)}\int_{B(x_0,r)}\int_0^r|\nabla_{x,t} u(x,t)|^2t\,dt\,dx<\infty,$$ and $\mathcal{L}_{2,n+2α}$ be the Campanato space on $\mathbb R^n$. We show that $H^{α,2}$ coincide with $e^{-t\sqrt{-Δ}}\mathcal{L}_{2,n+2α}$ for all $α\in (-1,1)$, where the case $α\in [0,1)$ was originally discovered by Fabes, Johnson and Neri [Indiana Univ. Math. J. 25 (1976), 159-170] and yet the case $α\in (-1,0)$ was left open. Moreover, for the scaling invariant version of $H^{α,2}$, $\mathcal{H}^{α,2}$, which comprises all harmonic functions $u$ on $\mathbb R^{n+1}_+$ satisfying $$\sup_{(x_0,r)\in\mathbb R^{n+1}_+}r^{-(2α+n)}\int_{B(x_0,r)} \int_0^r|\nabla_{x,t} u(x,t)|^2\,t^{1+2α} \,dt\,dx<\infty,$$ we show that $\mathcal{H}^{α,2}=e^{-t\sqrt{-Δ}}(-Δ)^\fracα{2}\mathcal{L}_{2,n+2α}$, where $(-Δ)^{\fracα{2}}\mathcal{L}_{2,n+2α}$ is the collection of all functions $f$ such that $(-Δ)^{-\fracα{2}}f$ are in $\mathcal{L}_{2,n+2α}$. Analogues for solutions to the heat equation are also established. As an application, we show that the spaces $\big((-Δ)^{\fracα{2}}\mathcal{L}_{2,n+2α}\big)^{-1}$ unify $Q_α^{-1}$, ${\mathrm{BMO}}^{-1}$ and $\dot{B}^{-1,\infty}_\infty$ naturally.

math.AP↗

arXiv2013

Cheeger-harmonic functions in metric measure spaces revisited

Let $(X,d,μ)$ be a complete metric measure space, with $μ$ a locally doubling measure, that supports a local weak $L^2$-Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on $(X,d,μ)$. Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented.

math.MG↗

arXiv2013

Korn inequality on irregular domains

In this paper, we study the weighted Korn inequality on some irregular domains, e.g., $s$-John domains and domains satisfying quasi-hyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincaré inequality.

math.CA↗

arXiv2013

Solvability of the divergence equation implies John via Poincaré inequality

Let $Ω\subset \rr^2$ be a bounded simply connected domain. We show that, for a fixed (every) $p\in (1,\fz),$ the divergence equation $\mathrm{div}\,\mathbf{v}=f$ is solvable in $W^{1,p}_0(Ω)^2$ for every $f\in L^p_0(Ω)$, if and only if $Ω$ is a John domain, if and only if the weighted Poincaré inequality $$\int_Ω|u(x)-u_Ω|^q\,dx\le C\int_Ω|\nabla u(x)|^q\dist(x,\partial Ω)^q\,dx$$ holds for some (every) $q\in [1,\fz)$. In higher dimensions similar results are proved under some additional assumptions on the domain in question.

math.CA↗

arXiv2013

Regularity and Capacity for the Fractional Dissipative Operator

This note is devoted to exploring some analytic-geometric properties of the regularity and capacity associated to the so-called fractional dissipative operator $\partial_t+(-Δ)^α$, naturally establishing a diagonally sharp Hausdorff dimension estimate for the blow-up set of a weak solution to the fractional dissipative equation $(\partial_t+(-Δ)^α)u(t,x)=F(t,x)$ subject to $u(0,x)=0$.

math.AP↗

arXiv2011

Lipschitz continuity of solutions of Poisson equations in metric measure spaces

Let $(X,d)$ be a pathwise connected metric space equipped with an Ahlfors $Q$-regular measure $μ$, $Q\in[1,\infty)$. Suppose that $(X,d,μ)$ supports a 2-Poincaré inequality and a Sobolev-Poincaré type inequality for the corresponding "Gaussian measure". The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation $Δu=f$, where $f\in L^p_\loc$. When $p>Q$, the local Lipschitz continuity of $u$ is established.

math.AP↗

arXiv2011

Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces

Let $(X,d)$ be a complete, pathwise connected metric measure space with locally Ahlfors $Q$-regular measure $μ$, where $Q>1$. Suppose that $(X,d,μ)$ supports a (local) $(1,2)$-Poincaré inequality and a suitable curvature lower bound. For the Poisson equation $Δu=f$ on $(X,d,μ)$, Moser-Trudinger and Sobolev inequalities are established for the gradient of $u$. The local Hölder continuity with optimal exponent of solutions is obtained.

math.AP↗

arXiv2010

Orlicz-Hardy Spaces Associated with Operators Satisfying Davies-Gaffney Estimates

Let ${\mathcal X}$ be a metric space with doubling measure, $L$ a nonnegative self-adjoint operator in $L^2({\mathcal X})$ satisfying the Davies-Gaffney estimate, $ω$ a concave function on $(0,\infty)$ of strictly lower type $p_ω\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for all $t\in (0,\infty).$ The authors introduce the Orlicz-Hardy space $H_{ω,L}({\mathcal X})$ via the Lusin area function associated to the heat semigroup, and the BMO-type space ${\mathop\mathrm{BMO}_{ρ,L}(\mathcal X)}$. The authors then establish the duality between $H_{ω,L}({\mathcal X})$ and $\mathrm{BMO}_{ρ,L}({\mathcal X})$; as a corollary, the authors obtain the $ρ$-Carleson measure characterization of the space ${\mathop\mathrm{BMO}_{ρ,L}(\mathcal X)}$. Characterizations of $H_{ω,L}({\mathcal X})$, including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let ${\mathcal X}={\mathbb R}^n$ and $ L=-Δ+V$ be a Schrödinger operator, where $V\in L^1_{\mathrm{\,loc\,}}({\mathbb R}^n)$ is a nonnegative potential. As applications, the authors show that the Riesz transform $\nabla L^{-1/2}$ is bounded from $H_{ω,L}({{\mathbb R}^n})$ to $L(ω)$; moreover, if there exist $q_1,\,q_2\in (0,\infty)$ such that $q_1<1<q_2$ and {\normalsize$[ω(t^{q_2})]^{q_1}$} is a convex function on $(0,\infty)$, then several characterizations of the Orlicz-Hardy space $H_{ω,L}({{\mathbb R}^n})$, in terms of the Lusin-area functions, the non-tangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when $ω(t)=t^p$ for all $t\in (0,\infty)$ and $p\in (0,1)$.

math.CA↗

arXiv2010

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let $L$ be a linear operator in $L^2({{\mathbb R}^n})$ and generate an analytic semigroup $\{e^{-tL}\}_{t\ge 0}$ with kernels satisfying an upper bound of Poisson type, whose decay is measured by $θ(L)\in (0,\infty].$ Let $ω$ on $(0,\infty)$ be of upper type 1 and of critical lower type $\widetilde p_0(ω)\in (n/(n+θ(L)), 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty)$. In this paper, the authors first introduce the VMO-type space $\mathrm{VMO}_{ρ,L}({\mathbb R}^n)$ and the tent space $T^{\infty}_{ω,\mathrm v}({\mathbb R}^{n+1}_+)$ and characterize the space $\mathrm{VMO}_{ρ,L}({\mathbb R}^n)$ via the space $T^{\infty}_{ω,\mathrm v}({\mathbb R}^{n+1}_+)$. Let $\widetilde{T}_ω ({\mathbb R}^{n+1}_+)$ be the Banach completion of the tent space $T_ω({\mathbb R}^{n+1}_+)$. The authors then prove that $\widetilde{T}_ω({\mathbb R}^{n+1}_+)$ is the dual space of $T^{\infty}_{ω,\mathrm v}({\mathbb R}^{n+1}_+)$. As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{ρ,L^\ast}({\mathbb R}^n)$ is the space $B_{ω,L}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$ and $B_{ω,L}({\mathbb R}^n)$ the Banach completion of the Orlicz-Hardy space $H_{ω,L}({\mathbb R}^n)$. These results generalize the known recent results by particularly taking $ω(t)=t$ for $t\in (0,\infty)$.

math.CA↗

arXiv2010

Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

Let $L$ be a divergence form elliptic operator with complex bounded measurable coefficients, $ω$ a positive concave function on $(0,\infty)$ of strictly critical lower type $p_ω\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty).$ In this paper, the authors introduce the generalized VMO spaces ${\mathop\mathrm{VMO}_ {ρ, L}({\mathbb R}^n)}$ associated with $L$, and characterize them via tent spaces. As applications, the authors show that $(\mathrm{VMO}_{ρ,L} ({\mathbb R}^n))^\ast=B_{ω,L^\ast}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$ and $B_{ω,L^\ast}({\mathbb R}^n)$ the Banach completion of the Orlicz-Hardy space $H_{ω,L^\ast}({\mathbb R}^n)$. Notice that $ω(t)=t^p$ for all $t\in (0,\infty)$ and $p\in (0,1]$ is a typical example of positive concave functions satisfying the assumptions. In particular, when $p=1$, then $ρ(t)\equiv 1$ and $({\mathop\mathrm{VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)$, where $H_{L^\ast}^1({\mathbb R}^n)$ was the Hardy space introduced by Hofmann and Mayboroda.

math.FA↗

arXiv2009

New Orlicz-Hardy Spaces Associated with Divergence Form Elliptic Operators

Let $L$ be the divergence form elliptic operator with complex bounded measurable coefficients, $ω$ the positive concave function on $(0,\infty)$ of strictly critical lower type $p_\oz\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty).$ In this paper, the authors study the Orlicz-Hardy space $H_{ω,L}({\mathbb R}^n)$ and its dual space $\mathrm{BMO}_{ρ,L^\ast}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$. Several characterizations of $H_{ω,L}({\mathbb R}^n)$, including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The $ρ$-Carleson measure characterization and the John-Nirenberg inequality for the space $\mathrm{BMO}_{ρ,L}({\mathbb R}^n)$ are also given. As applications, the authors show that the Riesz transform $\nabla L^{-1/2}$ and the Littlewood-Paley $g$-function $g_L$ map $H_{ω,L}({\mathbb R}^n)$ continuously into $L(ω)$. The authors further show that the Riesz transform $\nabla L^{-1/2}$ maps $H_{ω,L}({\mathbb R}^n)$ into the classical Orlicz-Hardy space $H_ω({\mathbb R}^n)$ for $p_ω\in (\frac{n}{n+1},1]$ and the corresponding fractional integral $L^{-γ}$ for certain $γ>0$ maps $H_{ω,L}({\mathbb R}^n)$ continuously into $H_{\widetildeω,L}({\mathbb R}^n)$, where $\widetildeω$ is determined by $ω$ and $γ$, and satisfies the same property as $ω$. All these results are new even when $ω(t)=t^p$ for all $t\in (0,\infty)$ and $p\in (0,1)$.

math.CA↗