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1.
Let $X$ be a compact connected Riemann surface and $G$ a connected reductive complex affine algebraic group. Given a holomorphic principal $G$ -bundle $E_G$ over $X$ , we construct a $C^\infty $ Hermitian structure on $E_G$ together with a $1$ -parameter family of $C^\infty $ automorphisms $\{F_t\}_{t\in \mathbb R }$ of the principal $G$ -bundle $E_G$ with the following property: Let $\nabla ^t$ be the connection on $E_G$ corresponding to the Hermitian structure and the new holomorphic structure on $E_G$ constructed using $F_t$ from the original holomorphic structure. As $t\rightarrow -\infty $ , the connection $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . In particular, as $t\rightarrow -\infty $ , the curvature of $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the curvature of the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . The family $\{F_t\}_{t\in \mathbb R }$ is constructed by generalizing the method of [6]. Given a holomorphic vector bundle $E$ on $X$ , in [6] a $1$ -parameter family of $C^\infty $ automorphisms of $E$ is constructed such that as $t\rightarrow -\infty $ , the curvature converges, in $C^0$ topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.  相似文献   

2.
Consider the stationary Navier–Stokes equations in an exterior domain $\varOmega \subset \mathbb{R }^3 $ with smooth boundary. For every prescribed constant vector $u_{\infty } \ne 0$ and every external force $f \in \dot{H}_2^{-1} (\varOmega )$ , Leray (J. Math. Pures. Appl., 9:1–82, 1933) constructed a weak solution $u $ with $\nabla u \in L_2 (\varOmega )$ and $u - u_{\infty } \in L_6(\varOmega )$ . Here $\dot{H}^{-1}_2 (\varOmega )$ denotes the dual space of the homogeneous Sobolev space $\dot{H}^1_{2}(\varOmega ) $ . We prove that the weak solution $u$ fulfills the additional regularity property $u- u_{\infty } \in L_4(\varOmega )$ and $u_\infty \cdot \nabla u \in \dot{H}_2^{-1} (\varOmega )$ without any restriction on $f$ except for $f \in \dot{H}_2^{-1} (\varOmega )$ . As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that $\Vert f\Vert _{\dot{H}^{-1}_2(\varOmega )}$ and $|u_{\infty }|$ are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1–82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as $u_{\infty } \rightarrow 0$ in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case $u_{\infty } \ne 0$ .  相似文献   

3.
Let $\Phi $ be a continuous $n\times n$ matrix-valued function on the unit circle $\mathbb T $ such that the $(k-1)$ st singular value of the Hankel operator with symbol $\Phi $ is greater than the $k$ th singular value. In this case, it is well-known that $\Phi $ has a unique superoptimal meromorphic approximant $Q$ in $H^{\infty }_{(k)}$ ; that is, $Q$ has at most $k$ poles in the unit disc $\mathbb D $ (in the sense that the McMillan degree of $Q$ in $\mathbb D $ is at most $k$ ) and $Q$ minimizes the essential suprema of singular values $s_{j}\left((\Phi -Q)(\zeta )\right)\!, j\ge 0$ , with respect to the lexicographic ordering. For each $j\ge 0$ , the essential supremum of $s_{j}\left((\Phi -Q)(\zeta )\right)$ is called the $j$ th superoptimal singular value of degree $k$ of $\Phi $ . We prove that if $\Phi $ has $n$ non-zero superoptimal singular values of degree $k$ , then the Toeplitz operator $T_{\Phi -Q}$ with symbol $\Phi -Q$ is Fredholm and has index $$ \mathrm{ind}T_{\Phi -Q}=\dim \ker T_{\Phi -Q}=2k+\dim \mathcal E , $$ where $\mathcal E =\{ \xi \in \ker H_{Q}: \Vert H_{\Phi }\xi \Vert _{2}=\Vert (\Phi -Q)\xi \Vert _{2}\}$ and $H_{\Phi }$ denotes the Hankel operator with symbol $\Phi $ . This result can in fact be extended from continuous matrix-valued functions to the wider class of $k$ -admissible matrix-valued functions, i.e. essentially bounded $n\times n$ matrix-valued functions $\Phi $ on $\mathbb T $ for which the essential norm of the Hankel operator $H_{\Phi }$ is strictly less than the smallest non-zero superoptimal singular value of degree $k$ of $\Phi $ .  相似文献   

4.
We obtain sharp two-sided inequalities between $L^p$ -norms $(1<p<\infty )$ of functions $\textit{Hf}$ and $H^*f$ , where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty ).$ In an equivalent form, it gives sharp constants in the two-sided relationships between $L^p$ -norms of functions $H\varphi -\varphi $ and $\varphi $ , where $\varphi $ is a nonnegative nonincreasing function on $(0,+\infty )$ with $\varphi (+\infty )=0.$ In particular, it provides an alternative proof of a result obtained by Kruglyak and Setterqvist (Proc Am Math Soc 136:2005–2013, 2008) for $p=2k \,\,(k\in \mathbb N )$ and by Boza and Soria (J Funct Anal 260:1020–1028, 2011) for all $p\ge 2$ , and gives a sharp version of this result for $1<p<2$ .  相似文献   

5.
Let $F$ be a global function field over a finite constant field and $\infty $ a place of $F$ . The ring $A$ of functions regular away from $\infty $ in $F$ is a Dedekind domain. For such $A$ Goss defined a $\zeta $ -function which is a continuous function from $\mathbb{Z }_p$ to the ring of entire power series with coefficients in the completion $F_\infty $ of $F$ at $\infty $ . He asks what one can say about the distribution of the zeros of the entire function at any parameter of $\mathbb{Z }_p$ . In the simplest case $A$ is the polynomial ring in one variable over a finite field. Here the question was settled completely by J. Sheats, after previous work by J. Diaz-Vargas, B. Poonen and D. Wan: for any parameter in $\mathbb{Z }_p$ the zeros of the power series have pairwise different valuations and they lie in  $F_\infty $ . In the present article we completely determine the distribution of zeros for the simplest case different from polynomial rings, namely $A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)$ —this $A$ has class number $1$ , it is the affine coordinate ring of a supersingular elliptic curve and the place $\infty $ is $\mathbb{F }\,\!{}_2$ -rational. The answer is slightly different from the above case of polynomial rings. For arbitrary $A$ such that $\infty $ is a rational place of $F$ , we describe a pattern in the distribution of zeros which we observed in some computational experiments. Finally, we present some precise conjectures on the fields of rationality of these zeroes for one particular hyperelliptic $A$ of genus  $2$ .  相似文献   

6.
Tensor truncation techniques are based on singular value decompositions. Therefore, the direct error control is restricted to $\ell ^{2}$ or $L^{2}$ norms. On the other hand, one wants to approximate multivariate (grid) functions in appropriate tensor formats in order to perform cheap pointwise evaluations, which require $\ell ^{\infty }$ or $L^{\infty }$ error estimates. Due to the huge dimensions of the tensor spaces, a direct estimate of $\left\| \cdot \right\| _{\infty }$ by $\left\| \cdot \right\| _{2}$ is hopeless. In the paper we prove that, nevertheless, in cases where the function to be approximated is smooth, reasonable error estimates with respect to $\left\| \cdot \right\| _{\infty }$ can be derived from the Gagliardo–Nirenberg inequality because of the special nature of the singular value decomposition truncation.  相似文献   

7.
Triebel (J Approx Theory 35:275–297, 1982; 52:162–203, 1988) investigated the boundary values of the harmonic functions in spaces of the Triebel–Lizorkin type ${\mathcal F^{\alpha,q}_{p}}$ on ${\mathbb{R}^{n+1}_+}$ by finding an characterization of the homogeneous Triebel–Lizorkin space ${{\bf \dot{F}}^{\alpha,q}_p}$ via its harmonic extension, where ${0 < p < \infty, 0 < q \leq \infty}$ , and ${\alpha < {\rm min}\{-n/p, -n/q\}}$ . In this article, we extend Triebel’s result to α < 0 and ${0 < p, q \leq \infty}$ by using a discrete version of reproducing formula and discretizing the norms in both ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf{\dot{F}}}^{\alpha,q}_p}$ . Furthermore, for α < 0 and ${1 < p,q \leq \infty}$ , the mapping from harmonic functions in ${\mathcal{F}^{\alpha,q}_{p}}$ to their boundary values forms a topological isomorphism between ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf \dot{F}}^{\alpha,q}_p}$ .  相似文献   

8.
9.
Let $\mathcal F ^a_\lambda $ be the PBW degeneration of the flag varieties of type $A_{n-1}$ . These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$ . We prove that $\mathcal F ^a_\lambda $ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\lambda $ of $\mathcal F ^a_\lambda $ . The varieties $R_\lambda $ can be viewed as towers of successive $\mathbb{P }^1$ -fibrations, thus providing an analogue of the classical Bott–Samelson–Demazure–Hansen desingularization. We prove that the varieties $R_\lambda $ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel–Weil type theorem for $\mathcal F ^a_\lambda $ . Using the Atiyah–Bott–Lefschetz formula for $R_\lambda $ , we compute the $q$ -characters of the highest weight $\mathfrak sl _n$ -modules.  相似文献   

10.
In this paper, we show the existence of positive $T$ -periodic solutions of second-order functional differential equations $u^{\prime \prime }(t)-\rho ^2u(t)+\lambda g(t)f(u(t-\tau (t)))=0,\ \ t\in \mathbb R , $ where $\rho >0$ is a constant, $g\in C(\mathbb R ,[0,\infty ))$ , $\tau \in C(\mathbb R ,\mathbb R )$ are $T$ -periodic functions, $f\in C([0,\infty ),[0,\infty ))$ and $\lambda $ is a positive parameter. Our approach based on global bifurcation theorem.  相似文献   

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