An improvement on Waring?CGoldbach problem for unlike powers

Let p _i be prime numbers. In this paper, it is proved that for any integer k?R5, with at most $O\big(N^{1-\frac{1}{3k\times2^{k-2}}+\varepsilon}\big)$ exceptions, all positive even integers up to N can be expressed in the form $p_{2}^{2}+p_{3}^{3}+p_{5}^{5}+p_{k}^{k}$ . This improves the result $O\big(\frac{N}{\log^{c}N}\big)$ for some c>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1?C4] for the problem in the form $p_{2}^{2}+p_{3}^{3}+p_{4}^{4}+p_{5}^{5}$ . This method can also be used for some other similar forms.     

Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators

Let ${\Phi}$ be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index ${p_\Phi^- \in(0,\,1]}$ . Let L be an injective operator of type ω having a bounded H _∞ functional calculus and satisfying the k-Davies–Gaffney estimates with ${k \in {\mathbb Z}_+}$ . In this paper, the authors first introduce an Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform ${D_{\gamma}L^{-\delta/(2k)}}$ is bounded from the Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz space ${L^{\widetilde{\Phi}}(\mathbb{R}^n)}$ when ${p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}$ , ${0 < \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz–Hardy space ${H^{\widetilde \Phi}(\mathbb{R}^n)}$ when ${p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}$ , ${1\le \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the weak Orlicz–Hardy space ${WH^\Phi(\mathbb{R}^n)}$ when ${\gamma = \delta}$ and ${p_\Phi=n/(n + \lfloor \gamma \rfloor)}$ or ${p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}$ with ${p_\Phi^-}$ attainable, where ${\widetilde{\Phi}}$ is an Orlicz function whose inverse function ${\widetilde{\Phi}^{-1}}$ is defined by ${\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}$ for all ${t \in (0,\,\infty)}$ , ${p_\Phi}$ denotes the strictly critical lower type index of ${\Phi}$ , ${\lfloor \gamma \rfloor}$ the maximal integer not more than ${\gamma}$ and ${(p_-(L),\,p_+(L))}$ the range of exponents ${p \in[1,\, \infty]}$ for which the semigroup ${\{e^{-tL}\}_{t >0 }}$ is bounded on ${L^p(\mathbb{R}^n)}$ .     

The Generalization of the Decomposition of Functions by Energy Operators

This work starts with the introduction of a family of differential energy operators. Energy operators $({\varPsi}_{R}^{+}, {\varPsi}_{R}^{-})$ were defined together with a method to decompose the wave equation in a previous work. Here the energy operators are defined following the order of their derivatives $(\varPsi^{-}_{k}, \varPsi^{+}_{k}, k=\{0,\pm 1,\pm 2,\ldots\})$ . The main part of the work demonstrates for any smooth real-valued function f in the Schwartz space $(\mathbf{S}^{-}(\mathbb{R}))$ , the successive derivatives of the n-th power of f ( $n \in \mathbb{Z}$ and n≠0) can be decomposed using only $\varPsi^{+}_{k}$ (Lemma); or if f in a subset of $\mathbf{S}^{-}(\mathbb{R})$ , called $\mathbf{s}^{-}(\mathbb{R})$ , $\varPsi^{+}_{k}$ and $\varPsi^{-}_{k}$ ( $k\in \mathbb{Z}$ ) decompose in a unique way the successive derivatives of the n-th power of f (Theorem). Some properties of the Kernel and the Image of the energy operators are given along with the development. Finally, the paper ends with the application to the energy function.     

The Left Hilbert BVP for h-Regular Functions in Clifford Analysis

Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e ₁, e ₂, . . . , e _n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ .     

Postulation of Disjoint Unions of Lines and Double Lines in {\mathbb{P}^3}

A double line ${C \subset \mathbb{P}^3}$ is a connected divisor of type (2, 0) on a smooth quadric surface. Fix ${(a, c) \in \mathbb{N}^2\ \backslash\ \{(0, 0)\}}$ . Let ${X \subset \mathbb{P}^3}$ be a general disjoint union of a lines and c double lines. Then X has maximal rank, i.e. for each ${t \in \mathbb{Z}}$ either ${h^1(\mathcal{I}_X(t)) = 0}$ or ${h^0(\mathcal{I}_X(t)) = 0}$ .     

Structure of \mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k}) for some fields \mathbb {k}=\mathbb {Q}(\sqrt{2p_1p_2},i) with \mathbf {C}l_2(\mathbb {k})\simeq (2, 2, 2)

Let \(p_1 \equiv p_2 \equiv 5\pmod 8\) be different primes. Put \(i=\sqrt{-1}\) and \(d=2p_1p_2\) , then the bicyclic biquadratic field \(\mathbb {k}=\mathbb {Q}(\sqrt{d},i)\) has an elementary abelian 2-class group of rank \(3\) . In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-abelian Galois group \(\mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k})\) of the second Hilbert 2-class field \(\mathbb {k}_2^{(2)}\) of \(\mathbb {k}\) . We study the capitulation problem of the 2-classes of \(\mathbb {k}\) in its seven unramified quadratic extensions \(\mathbb {K}_i\) and in its seven unramified bicyclic biquadratic extensions \(\mathbb {L}_i\) .     

Irrational Factor of Order k and ITS Connections With k-Free Integers

We introduce an irrational factor of order k defined by \({I_{k}(n) ={\prod_{i=1}^{l}} p_{i}^{\beta_{i}}}\) , where \({n = \prod_{i=1}^{l} p_{i}^{\alpha_{i}}}\) is the factorization of n and \({\beta_{i} = \left\{\begin{array}{ll}\alpha_i, \quad \quad {\rm if} \quad \alpha_i < k \\ \frac{1}{\alpha_i},\quad \quad {\rm if} \quad \alpha_i \geqq k \end{array}\right.}\) . It turns out that the function \({\frac{I_{k} (n)}{n}}\) well approximates the characteristic function of k-free integers. We also derive asymptotic formulas for \({\prod_{v=1}^{n} I_{k}(v)^{\frac{1}{n}}, \sum_{n \leqq x} I_{k}(n)}\) and \({\sum_{n \leqq x} (1 - \frac{n}{x}) I_{k}(n)}\) .     

Planar Beurling Transform and Bergman Type Spaces

In this paper we describe the actions of the operator $S_\mathbb{D }$ or its adjoint $S_\mathbb{D }^*$ on the poly-Bergman spaces of the unit disk $\mathbb{D }.$ Let $k$ and $j$ be positive integers. We prove that $(S_\mathbb{D })^{j}$ is an isometric isomorphism between the true poly-Bergman subspace $\mathcal{A }_{(k)}^2(\mathbb{D })\ominus N_{(k),j}$ onto the true poly-Bergman space $\mathcal{A }_{(j+k)}^2(\mathbb{D }),$ where the linear space $N_{(k),j}$ have finite dimension $j.$ The action of $(S_\mathbb{D })^{j-1}$ on the canonical Hilbert base for the Bergman subspace $\mathcal{A }^2(\mathbb{D })\ominus \mathcal{P }_{j-1},$ gives a Hilbert base $\{ \phi _{ j , k } \}_{ k }$ for $\mathcal{A }_{(j)}^2(\mathbb{D }).$ It is shown that $\{ \phi _{ j , k } \}_{ j, k }$ is a Hilbert base for $L^2(\mathbb{D },d A)$ such that whenever $j$ and $k$ remain constant we obtain a Hilbert base for the true poly-Bergman space $\mathcal{A }_{(j)}^2(\mathbb{D })$ and $\mathcal{A }_{(-k)}^2(\mathbb{D }),$ respectively. The functions $\phi _{ j , k }$ are polynomials in $z$ and $\overline{z}$ and are explicitly given in terms of the $(2,1)$ -hypergeometric polynomials. We prove explicit representations for the true poly-Bergman kernels and the Koshelev representation for the poly-Bergman kernels of $\mathbb{D }.$ The action of $S_\Pi $ on the true poly-Bergman spaces of the upper half-plane $\Pi $ allows one to introduce Hilbert bases for the true poly-Bergman spaces, and to give explicit representations of the true poly-Bergman and poly-Bergman kernels.     

Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains

We examine the fourth order problem $\Delta ^2 u = \lambda f(u) $ in $ \Omega $ with $ \Delta u = u =0 $ on $ {\partial \Omega }$ , where $ \lambda > 0$ is a parameter, $ \Omega $ is a bounded domain in $\mathbb{R }^N$ and where $f$ is one of the following nonlinearities: $ f(u)=e^u$ , $ f(u)=(1+u)^p $ or $ f(u)= \frac{1}{(1-u)^p}$ where $ p>1$ . We show the extremal solution is smooth, provided $$\begin{aligned} N < 2 + 4 \sqrt{2} + 4 \sqrt{ 2 - \sqrt{2}} \approx 10.718 \text{ when} f(u)=e^u, \end{aligned}$$ and $$\begin{aligned} N < \frac{4p}{p-1} + \frac{4(p+1)}{p-1} \left( \sqrt{ \frac{2p}{p+1}} + \sqrt{ \frac{2p}{p+1} - \sqrt{ \frac{2p}{p+1}}} - \frac{1}{2} \right) \end{aligned}$$ when $ f(u)=(u+1)^p$ . New results are also obtained in the case where $ f(u)=(1-u)^{-p}$ . These are substantial improvements to various results on critical dimensions obtained recently by various authors. To do that, we derive a new stability inequality satisfied by minimal solutions of the above equation, which is more amenable to estimates as it allows a method of proof reminiscent of the second order case.     

Harmonic divisors and rationality of zeros of Jacobi polynomials

Let $P_{n}^{ ( \alpha,\beta ) } ( x ) $ be the Jacobi polynomial of degree n with parameters α,β. The main result of the paper states the following: If b≠1,3 and c are non-zero relatively prime natural numbers then $P_{n}^{ ( k+ ( d-3 ) /2,k+ ( d-3 ) /2 ) } ( \sqrt{b/c} ) \neq0$ for all natural numbers d,n and $k\in\mathbb{N}_{0}$ . Moreover, under the above assumption, the polynomial $Q ( x ) = \frac{b}{c} ( x_{1}^{2}+\cdots+x_{d-1}^{2} ) + ( \frac{b}{c}-1 ) x_{d}^{2}$ is not a harmonic divisor, and the Dirichlet problem for the cone {Q(x)<0} has polynomial harmonic solutions for polynomial data functions.     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

Uniqueness of \mathbb{C}^{ * } - and \mathbb{C}_{{\text{ + }}} -actions on Gizatullin Surfaces

A Gizatullin surface is a normal affine surface V over $ \mathbb{C} $ , which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with $ \mathbb{C}_{{\text{ + }}} $ -actions on V considered up to a “speed change”. Non-Gizatullin surfaces are known to admit at most one $ \mathbb{A}^{1} $ -fibration V → S up to an isomorphism of the base S. Moreover, an effective $ \mathbb{C}^{ * } $ -action on them, if it does exist, is unique up to conjugation and inversion t $ \mapsto $ t ^?1 of $ \mathbb{C}^{ * } $ . Obviously, uniqueness of $ \mathbb{C}^{ * } $ -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when $ \mathbb{A}^{{\text{1}}} $ -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base $ S \cong \mathbb{A}^{{\text{1}}} $ . We exhibit as well large subclasses of Gizatullin $ \mathbb{C}^{ * } $ -surfaces for which a $ \mathbb{C}^{ * } $ -action is essentially unique and for which there are at most two conjugacy classes of $ \mathbb{A}^{{\text{1}}} $ -fibrations over $ \mathbb{A}^{{\text{1}}} $ .     

Uniqueness of integrable solutions to {\nabla \zeta=G \zeta, \zeta|_\Gamma = 0} for integrable tensor coefficients G and applications to elasticity

Let ${\Omega \subset \mathbb{R}^{N}}$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ${\partial\Omega}$ . We show that the solution to the linear first-order system $$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if ${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$ and ${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$ . As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for ${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$ with Curl ${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$ , Curl ${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$ for some p, q > 1 with 1/p + 1/q = 1 as well as det ${P \geq c^+ > 0}$ . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$ satisfy sym ${(\nabla\Phi^\top\nabla\Psi) = 0}$ for some ${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$ with det ${\nabla\Psi \geq c^+ > 0}$ . Then, there exist a constant translation vector ${a \in \mathbb{R}^{3}}$ and a constant skew-symmetric matrix ${A \in \mathfrak{so}(3)}$ , such that ${\Phi = A\Psi + a}$ .     

On uniformly bounded spherical functions in Hilbert space

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms ${G \ni a \mapsto ka \in G}$ , ${k \in K}$ . Let ${{\mathfrak{H}}}$ be a complex Hilbert space and let ${{\mathcal L}({\mathfrak{H}})}$ be the algebra of all bounded linear operators on ${{\mathfrak{H}}}$ . A mapping ${u \colon G \to {\mathcal L}({\mathfrak{H}})}$ is termed a K-spherical function if it satisfies (1) ${|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}$ for any ${a,b\in G}$ , where |K| denotes the cardinality of K, and (2) ${u (0) = {\rm id}_{\mathfrak {H}},}$ where ${{\rm id}_{\mathfrak {H}}}$ designates the identity operator on ${{\mathfrak{H}}}$ . The main result of the paper is that for each K-spherical function ${u \colon G \to {\mathcal {L}}({\mathfrak {H}})}$ such that ${\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}$ there is an invertible operator S in ${{\mathcal L}({\mathfrak{H}})}$ with ${\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}$ such that the K-spherical function ${{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}$ defined by ${{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}$ satisfies ${{\tilde{u}}(-a) = {\tilde{u}}(a)^*}$ for each ${a \in G}$ . It is shown that this last condition is equivalent to insisting that ${{\tilde{u}}(a)}$ be normal for each ${a \in G}$ .     

Extinction profile of the logarithmic diffusion equation

Let ${N \geq 3}$ and u be the solution of u _t = Δ log u in ${\mathbb{R}^N \times (0, T)}$ with initial value u ₀ satisfying ${B_{k_1}(x, 0) \leq u_{0} \leq B_{k_2}(x, 0)}$ for some constants k ₁ > k ₂ > 0 where ${B_k(x, t) = 2(N - 2)(T - t)_{+}^{N/(N - 2)}/(k + (T - t)_{+}^{2/(N - 2)}|x|^{2})}$ is the Barenblatt solution for the equation and ${u_0 - B_{k_0} \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 if ${N \geq 4}$ . We give a new different proof on the uniform convergence and ${L^1(\mathbb{R}^N)}$ convergence of the rescaled function ${\tilde{u}(x, s) = (T - t)^{-N/(N - 2)}u(x/(T - t)^{-1/(N - 2)}, t), s = -{\rm log}(T - t)}$ , on ${\mathbb{R}^N}$ to the rescaled Barenblatt solution ${\tilde{B}_{k_0}(x) = 2(N - 2)/(k_0 + |x|^{2})}$ for some k ₀ > 0 as ${s \rightarrow \infty}$ . When ${N \geq 4, 0 \leq u_0(x) \leq B_{k_0}(x, 0)}$ in ${\mathbb{R}^N}$ , and ${|u_0(x) - B_{k_0}(x, 0)| \leq f \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 and some radially symmetric function f, we also prove uniform convergence and convergence in some weighted L ¹ space in ${\mathbb{R}^N}$ of the rescaled solution ${\tilde{u}(x, s)}$ to ${\tilde{B}_{k_0}(x)}$ as ${s \rightarrow \infty}$ .     

Lower bounds on Ricci curvature and quantitative behavior of singular sets

Let Y ⁿ denote the Gromov-Hausdorff limit $M^{n}_{i}\stackrel{d_{\mathrm{GH}}}{\longrightarrow} Y^{n}$ of v-noncollapsed Riemannian manifolds with ${\mathrm{Ric}}_{M^{n}_{i}}\geq-(n-1)$ . The singular set $\mathcal {S}\subset Y$ has a stratification $\mathcal {S}^{0}\subset \mathcal {S}^{1}\subset\cdots\subset \mathcal {S}$ , where $y\in \mathcal {S}^{k}$ if no tangent cone at y splits off a factor ? ^k+1 isometrically. Here, we define for all η>0, 0<r≤1, the k-th effective singular stratum $\mathcal {S}^{k}_{\eta,r}$ satisfying $\bigcup_{\eta}\bigcap_{r} \,\mathcal {S}^{k}_{\eta,r}= \mathcal {S}^{k}$ . Sharpening the known Hausdorff dimension bound $\dim\, \mathcal {S}^{k}\leq k$ , we prove that for all y, the volume of the r-tubular neighborhood of $\mathcal {S}^{k}_{\eta,r}$ satisfies ${\mathrm {Vol}}(T_{r}(\mathcal {S}^{k}_{\eta,r})\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},\eta)r^{n-k-\eta}$ . The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let $\mathcal {B}_{r}$ denote the set of points at which the C ²-harmonic radius is ≤r. If also the $M^{n}_{i}$ are Kähler-Einstein with L ₂ curvature bound, $\| Rm\|_{L_{2}}\leq C$ , then ${\mathrm {Vol}}( \mathcal {B}_{r}\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},C)r^{4}$ for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the $M^{n}_{i}$ , we obtain a slightly weaker volume bound on $\mathcal {B}_{r}$ which yields an a priori L _p curvature bound for all p<2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.     

Interior W 1,p Regularity and Hölder Continuity of Weak Solutions to a Class of Divergence Kolmogorov Equations with Discontinuous Coefficients

We consider a class of Kolmogorov equation $$Lu={\sum^{p_0}_{i,j=1}{\partial_{x_i}}(a_{ij}(z){\partial_{x_j}}u)}+{\sum^{N}_{i,j=1}b_{ij}x_{i}{\partial_{x_j}}u-{\partial_t}u}={\sum^{p_0}_{j=1}{\partial_{x_j}}F_{j}(z)}$$ in a bounded open domain ${\Omega \subset \mathbb{R}^{N+1}}$ , where the coefficients matrix (a _ij (z)) is symmetric uniformly positive definite on ${\mathbb{R}^{p_0} (1 \leq p_0 < N)}$ . We obtain interior W ^1,p (1 < p < ∞) regularity and Hölder continuity of weak solutions to the equation under the assumption that coefficients a _ij (z) belong to the ${VMO_L\cap L^\infty}$ and ${({b_{ij}})_{N \times N}}$ is a constant matrix such that the frozen operator ${L_{z_0}}$ is hypoelliptic.     

The second pinching theorem for hypersurfaces with constant mean curvature in a sphere

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng–Terng, Wei–Xu, Zhang, and Ding–Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $S^{n+1}$ . Denote by $S$ the squared norm of the second fundamental form of $M$ . We prove that there exist two positive constants $\gamma (n)$ and $\delta (n)$ depending only on $n$ such that if $|H|\le \gamma (n)$ and $\beta (n,H)\le S\le \beta (n,H)+\delta (n)$ , then $S\equiv \beta (n,H)$ and $M$ is one of the following cases: (i) $S^{k}\Big (\sqrt{\frac{k}{n}}\Big )\times S^{n-k}\Big (\sqrt{\frac{n-k}{n}}\Big )$ , $\,1\le k\le n-1$ ; (ii) $S^{1}\Big (\frac{1}{\sqrt{1+\mu ^2}}\Big )\times S^{n-1}\Big (\frac{\mu }{\sqrt{1+\mu ^2}}\Big )$ . Here $\beta (n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)} \sqrt{n^2H^4+4(n-1)H^2}$ and $\mu =\frac{n|H|+\sqrt{n^2H^2+ 4(n-1)}}{2}$ .     

Howe correspondence and Springer correspondence for real reductive dual pairs

We consider a real reductive dual pair (G′, G) of type I, with rank ${({\rm G}^{\prime}) \leq {\rm rank(G)}}$ . Given a nilpotent coadjoint orbit ${\mathcal{O}^{\prime} \subseteq \mathfrak{g}^{{\prime}{*}}}$ , let ${\mathcal{O}^{\prime}_\mathbb{C} \subseteq \mathfrak{g}^{{\prime}{*}}_\mathbb{C}}$ denote the complex orbit containing ${\mathcal{O}^{\prime}}$ . Under some condition on the partition λ′ parametrizing ${\mathcal{O}^{\prime}}$ , we prove that, if λ is the partition obtained from λ by adding a column on the very left, and ${\mathcal{O}}$ is the nilpotent coadjoint orbit parametrized by λ, then ${\mathcal{O}_\mathbb{C}= \tau (\tau^{\prime -1}(\mathcal{O}_\mathbb{C}^{\prime}))}$ , where ${\tau, \tau^{\prime}}$ are the moment maps. Moreover, if ${chc(\hat\mu_{\mathcal{O}^{\prime}}) \neq 0}$ , where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to ${\mu_{\mathcal{O}^{\prime}}}$ with corresponds to ${\mathcal{O}_\mathbb{C}}$ via the Springer correspondence.     

A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ , $ \mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ \mathfrak{k} \subset \mathfrak{g} $ being of the form $ \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ , where $ \mathfrak{k}_{0} \subset \mathfrak{g} $ is a finite-dimensional reductive in $ \mathfrak{g} $ subalgebra and $ C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ is the centralizer of $ \mathfrak{k}_{0} $ in $ \mathfrak{g} $ . We prove a general nonvanishing and $ \mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when $ \mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.