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Jie Xiao

Publications and source records attributed to Jie Xiao.

At least 127 records · Page 7Linked to original sources
arXiv2016

A quasiconformal composition problem for the Q-spaces

Given a quasiconformal mapping $f:\mathbb R^n\to\mathbb R^n$ with $n\ge2$, we show that (un-)boundedness of the composition operator ${\bf C}_f$ on the spaces $Q_α(\mathbb R^n)$ depends on the index $α$ and the degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of the index $α$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel-Lizorkin and Besov spaces. Consequently, Tukia-Väisälä's quasiconformal extension $f:\mathbb R^n\to\mathbb R^n$ of an arbitrary quasisymmetric mapping $g:\mathbb R^{n-p}\to \mathbb R^{n-p}$ is shown to preserve $Q_α (\mathbb R^n)$ for any $(α,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\}$. Moreover, $Q_α(\mathbb R^n)$ is shown to be invariant under inversions for all $0<α<1$.

math.FA↗
arXiv2016

A Maximum Problem of S.-T. Yau for Variational p-Capacity

Through using the semidiameter (in connection to: the mean radius and surface radius) of a convex closed hypersurface in $\mathbb R^{n\ge 2}$ as an sharp upper bound of the variational $(1,n)\ni p$-capacity radius, this paper settles a restriction/variant of S.-T. Yau's \cite[Problem 59]{Yau} from the surface area to the variational $p$-capacity whose limit as $p\to 1$ actually induces the surface area.

math.DG↗
arXiv2015

Hall algebras of cyclic quivers and $q$-deformed Fock spaces

Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel--Hall algebra ${\mathscr D}(Q)$ of a quiver $Q$ as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation $L(Λ_0)$ of ${\mathscr D}(Δ_n)$ of the cyclic quiver $Δ_n$ provides a realization of the $q$-deformed Fock space $\bigwedge^\infty$ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of $Δ_n$, we obtain a decomposition of the basic representation $L(Λ_0)$ which induces the Kashiwara--Miwa--Stern decomposition of $\bigwedge^\infty$ and a construction of the canonical basis of $\bigwedge^\infty$ defined by Leclerc and Thibon in terms of certain monomial basis elements in ${\mathscr D}(Δ_n)$.

math.QA↗
arXiv2015

p-capacity vs surface-area

This paper is devoted to exploring the relationship between the $[1,n)\ni p$-capacity and the surface-area in $\mathbb R^{n\ge 2}$ which especially shows: if $Ω\subset\mathbb R^n$ is a convex, compact, smooth set with its interior $Ω^\circ\not=\emptyset$ and the mean curvature $H(\partialΩ,\cdot)>0$ of its boundary $\partialΩ$ then $$ \left(\frac{n(p-1)}{p(n-1)}\right)^{p-1}\le\frac{\left(\frac{\hbox{cap}_p(Ω)}{\big(\frac{p-1}{n-p}\big)^{1-p}σ_{n-1}}\right)}{\left(\frac{\hbox{area}(\partialΩ)}{σ_{n-1}}\right)^\frac{n-p}{n-1}}\le\left(\sqrt[n-1]{\int_{\partialΩ}\big(H(\partialΩ,\cdot)\big)^{n-1}\frac{dσ(\cdot)}{σ_{n-1}}}\right)^{p-1}\quad\forall\quad p\in (1,n) $$ whose limits $1\leftarrow p\ \&\ p\rightarrow n$ imply $$ 1=\frac{cap_1(Ω)}{\hbox{area}(\partialΩ)}\ \ \& \ \int_{\partialΩ}\big(H(\partialΩ,\cdot)\big)^{n-1}\frac{dσ(\cdot)}{σ_{n-1}}\ge 1, $$ thereby not only discovering that the new best known constant is roughly half as far from the one conjectured by Pólya-Szegö in \cite[(2)]{P} but also extending the Pólya-Szegö inequality in \cite[(5)]{P}, with both the conjecture and the inequality being stated for the electrostatic capacity of a convex solid in $\mathbb R^3$.

math.DG↗
arXiv2015

Geometric realizations of Lusztig's symmetries

In this paper, we give geometric realizations of Lusztig's symmetries. We also give projective resolutions of a kind of standard modules. By using the geometric realizations and the projective resolutions, we obtain the categorification of the formulas of Lusztig's symmetries.

math.RT↗
arXiv2015

Uniqueness of nonnegative weak solution to $u^p\le(-Δ)^\fracα{2}u$ on $\mathbb R^N$

This note shows that under $(p,α, N)\in (1,\infty)\times(0,2)\times\mathbb Z_+$ the fractional order differential inequality $$ (\dagger)\quad u^p \le (-Δ)^{\fracα{2}} u\quad\hbox{in}\quad\mathbb R^{N} $$ has the property that if $N\leα$ then a nonnegative solution to $(\dagger)$ is unique, and if $N>α$ then the uniqueness of a nonnegative weak solution to $(\dagger)$ occurs when and only when $p\le N/(N-α)$, thereby innovatively generalizing Gidas-Spruck's result for $u^p+Δu\le 0$ in $\R^N$ discovered in \cite{GS}.

math.AP↗
arXiv2014

Anisotropic Sobolev Capacity with Fractional Order

In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure $μ$ that naturally induces an embedding of the anisotropic fractional Sobolev class $\dotΛ_{α,K}^{1,1}$ into the $μ$-based-Lebesgue-space $L^{n/β}_μ$ with $0<β\le n$. Also, we investigate the anisotropic fractional $α$-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as $α\rightarrow 0^+$, will be provided.

math.MG↗
arXiv2014

Holomorphic Campanato Spaces on the Unit Ball

As outlined below, this paper is devoted to a Carleson-type-measure-based study of the holomorphic Campanato $2$-space on the open unit ball $\mathbb B_n$ of $\mathbb C^n$, comprising all Hardy $2$-functions whose oscillations in non-isotropic metric balls on the compact unit sphere $\mathbb S_n$ are proportional to some power of the radius other than the dimension $n\ge 1$.

math.CV↗
arXiv2014

Isocapacity Estimates for Hessian Operators

Through a new powerful potential-theoretic analysis, this paper is devoted to discovering the geometrically equivalent isocapacity forms of Chou-Wang's Sobolev type inequality and Tian-Wang's Moser-Trudinger type inequality for the fully nonlinear $1\leq k\leq n/2$ Hessian operators.

math.FA↗
arXiv2014

Preduals of quadratic Campanato spaces associated to operators with heat kernel bounds

Let $L$ be a nonnegative, self-adjoint operator on $L^2(\mathbb{R}^n)$ with the Gaussian upper bound on its heat kernel. As a generalization of the square Campanato space $\mathcal{L}^{2,λ}_{-Δ}(\mathbb R^n)$, in \cite{DXY} the quadratic Campanato space $\mathcal{L}_L^{2,λ}(\mathbb{R}^n)$ is defined by a variant of the maximal function associated with the semigroup $\{e^{-tL}\}_{t\geq 0}$. On the basis of \cite{DX} and \cite{YY} this paper addresses the preduality of $\mathcal{L}_L^{2,λ}(\mathbb{R}^n)$ through an induced atom (or molecular) decomposition. Even in the case $L=-Δ$ the discovered predual result is new and natural.

math.AP↗
arXiv2014

Two Predualities and Three Operators over Analytic Campanato Spaces

This article is devoted to not only characterizing the first and second preduals of the analytic Campanato spaces ($\mathcal{CA}_p)$ on the unit disk, but also investigating boundedness of three operators: superposition ($\mathsf{S}^ϕ$); backward shift ($\mathsf{S_b}$); Schwarzian derivative ($\mathsf{S}$), acting on $\mathcal{CA}_p$.

math.CV↗
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↗
arXiv2014

Analytic Campanato Spaces and Their Compositions

This paper is devoted to characterizing the analytic Campanato spaces $\mathcal{AL}_{p,η}$ (including the analytic Morrey spaces, the analytic John-Nirenberg space, and the analytic Lipschitz/Hölder spaces) on the complex unit disk $\mathbb D$ in terms of the Möbius mapping and the Littlewood-Paley form, and consequently their compositions with the analytic self-maps of $\mathbb D$.

math.CV↗
arXiv2013

Towards Conformal Capacities in Euclidean Spaces

This paper addresses the so-called conformal capacities in $\mathbb R^n$, $n\ge 3$, through comparing three existing definitions (due to Betsakos, Colesanti-Cuoghi, Anderson-Vamananmurthy-Fuglede respectively) and studying their associated iso-capacitary inequalities with connection to half-diameter, mean-width, mean-curvature and ADM-mass, Hadamard type variational formula, Minkowski type problem, and Yau type problem.

math.DG↗