Compact Hankel Operators on Generalized Bergman Spaces of the Polydisc

Let ${\vartheta}$ be a measure on the polydisc ${\mathbb{D}^n}$ which is the product of n regular Borel probability measures so that ${\vartheta([r,1)^n\times\mathbb{T}^n) >0 }$ for all 0 < r < 1. The Bergman space ${A^2_{\vartheta}}$ consists of all holomorphic functions that are square integrable with respect to ${\vartheta}$ . In one dimension, it is well known that if f is continuous on the closed disc ${\overline{\mathbb{D}}}$ , then the Hankel operator H _f is compact on ${A^2_\vartheta}$ . In this paper we show that for n ≥ 2 and f a continuous function on ${{\overline{\mathbb{D}}}^n}$ , H _f is compact on ${A^2_\vartheta}$ if and only if there is a decomposition f = h + g, where h belongs to ${A^2_\vartheta}$ and ${\lim_{z\to\partial\mathbb{D}^n}g(z)=0}$ .     

Ultrapowers ofL 1(μ) and the subsequence splitting principle

We study the ultrapowers $L_1 (\mu )_\mathfrak{U} $ of aL ₁(μ) space, by describing the components of the well-known representation $L_1 (\mu )_\mathfrak{U} = L_1 (\mu _\mathfrak{U} ) \oplus _1 L_1 (\nu _\mathfrak{U} )$ , and we give a representation of the projection from $L_1 (\mu )_\mathfrak{U} $ onto $L_1 (\mu _\mathfrak{U} )$ . Moreover, the subsequence splitting principle forL ₁(μ) motivates the following question: if $\mathfrak{V}$ is an ultrafilter on ? and $[f_i ] \in L_1 (\mu )_\mathfrak{V} $ , is it possible to find a weakly convergent sequence (g _i ) ?L ₁(μ) following $\mathfrak{V}$ and a disjoint sequence (h _i ) ?L ₁(μ) such that [f _i ]=[g _i ]+[h _i ]? If $\mathfrak{V}$ is a selective ultrafilter, we find a positive answer by showing that $f = [f_i ] \in L_1 (\mu )_\mathfrak{V} $ belongs to $L_1 (\mu _{_\mathfrak{V} } )$ if and only if its representatives {f _i } are weakly convergent following $\mathfrak{V}$ and $f \in L_1 (\nu _\mathfrak{V} )$ if and only if it admits a representative consisting of pairwise disjoint functions. As a consequence, we obtain a new proof of the subsequence splitting principle. If $\mathfrak{V}$ is not a p-point then the above characterizations of $L_1 (\nu _{_\mathfrak{V} } )$ and $L_1 (\nu _{_\mathfrak{V} } )$ fail and the answer to the question is negative.     

Creature forcing and large continuum: the joy of halving

For ${f,g\in\omega^\omega}$ let ${c^\forall_{f,g}}$ be the minimal number of uniform g-splitting trees needed to cover the uniform f-splitting tree, i.e., for every branch ν of the f-tree, one of the g-trees contains ν. Let ${c^\exists_{f,g}}$ be the dual notion: For every branch ν, one of the g-trees guesses ν(m) infinitely often. We show that it is consistent that ${c^\exists_{f_\epsilon,g_\epsilon}{=}c^\forall_{f_\epsilon,g_\epsilon}{=}\kappa_\epsilon}$ for continuum many pairwise different cardinals ${\kappa_\epsilon}$ and suitable pairs ${(f_\epsilon,g_\epsilon)}$ . For the proof we introduce a new mixed-limit creature forcing construction.     

Nonvanishing of Rankin–Selberg L-functions for Hilbert modular forms

Let F be a totally real number field of degree n over $\mathbb{Q}$ with ring of integers $\mathcal{O}$ and narrow class number one. Let S _2k (Γ) be the vector space of cuspidal Hilbert modular forms of parallel weight 2k for $\varGamma=\mathrm{SL}_{2}(\mathcal{O})$ , and let B _2k be an orthogonal Hecke eigenbasis for this space. For any fixed Hecke eigenform f∈S _2k (Γ) and any ε>0, we prove that $$\# \biggl\{ g \in B_{2k}: L \biggl(f \times g, \frac{1}{2} \biggr) \ne0 \biggr\} \gg k^{n- \varepsilon}, $$ where L(f×g,s) is the Rankin–Selberg L–function of f and g.     

A note on the Petri loci

Let ${\mathcal {M}_g}$ be the coarse moduli space of complex projective nonsingular curves of genus g. We prove that when the Brill?CNoether number ??(g, r, n) is non-negative every component of the Petri locus ${P^r_{g,n} \subset \mathcal {M}_g}$ whose general member is a curve C such that ${W^{r+1}_n(C) = \emptyset}$ , has codimension one in ${\mathcal {M}_g}$ .     

Prescribed diagonal Schouten tensor in locally conformally flat manifolds

We consider the pseudo-euclidean space ${(\mathbb{R}^n, g)}$ , with n ≥  3 and ${g_{ij} = \delta_{ij} \varepsilon_i, \varepsilon_i = \pm 1}$ and tensors of the form ${T = \sum \nolimits_i \varepsilon_i f_i (x) dx_i^2}$ . In this paper, we obtain necessary and sufficient conditions for a diagonal tensor to admit a metric ${\bar{g}}$ , conformal to g, so that ${A_{\bar g}=T}$ , where ${A_{\bar g}}$ is the Schouten Tensor of the metric ${\bar g}$ . The solution to this problem is given explicitly for special cases for the tensor T, including a case where the metric ${\bar g}$ is complete on ${\mathbb{R}^n}$ . Similar problems are considered for locally conformally flat manifolds. As an application of these results we consider the problem of finding metrics ${\bar g}$ , conformal to g, such that ${\sigma_2 ({\bar g })}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })}}$ is equal to a given function. We prove that for some functions, f ₁ and f ₂, there exist complete metrics ${\bar{g} = g/{\varphi^2}}$ , such that ${\sigma_2 ({\bar g }) = f_1}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })} = f_2}$ .     

Bounds on Surface Stress-Driven Shear Flow

The background method is adapted to derive rigorous limits on surface speeds and bulk energy dissipation for shear stress-driven flow in two- and three-dimensional channels. By-products of the analysis are nonlinear energy stability results for plane Couette flow with a shear stress boundary condition: when the applied stress is gauged by a dimensionless Grashoff number $\operatorname{Gr}$ , the critical $\operatorname{Gr}$ for energy stability is 139.5 in two dimensions, and 51.73 in three dimensions. We derive upper bounds on the friction (a.k.a. dissipation) coefficient $C_{f} = \tau/\overline{u}^{2}$ , where τ is the applied shear stress and $\overline{u}$ is the mean velocity of the fluid at the surface, for flows at higher $\operatorname{Gr}$ including developed turbulence: C _f ≤1/32 in two dimensions and C _f ≤1/8 in three dimensions. This analysis rigorously justifies previously computed numerical estimates.     

On Uniform Approximation in R2 of Continuous Functions of Two Variables with Bounded Variation in the Sense of Hardy

Introduce the notation: $\mathbb{Z}$ is the set of integers, $\bar {\mathbb{Z}}={\mathbb{Z}} \cup \{-\infty, +\infty\},{\mathbb{R}}_+^2 =\{x=(x_1,x_2) \in {\mathbb{R}}^2; x_1>0,x_2>0\}$ , $g_{k,m} (x,\alpha,h)= \int\limits_0^1 {g_1 (\frac{(k+u)h_1 - x_1}{\alpha_1})g_2(\frac{(m+u)h_2 - x_2}{\alpha_2})}du$ , where $g_i :\mathbb{R} \to \mathbb{R},x \in \mathbb{R}^2 ,\alpha ,h \in \mathbb{R}_ + ^2 $ . Under certain conditions on the functions g ₁, g ₂, we prove that the system of functions $g_{k,m} (x,\alpha^(n), h^(n)) (k,m \in \bar {\mathbb{Z}})$ , where $\alpha ^{\left( n \right)} ,h^{\left( n \right)} \in \mathbb{R}_ + ^2 $ are arbitrary infinitesimal sequences, is complete in the space C $\mathbb{R}^2 $ of uniformly continuous bounded functions f equipped with the norm $||f|| = \mathop {\sup }\limits_{x \in \mathbb{R}^2 } |f(x)|$ . Starting with the functions g _k,m , it is possible to construct a method for uniform approximating in $\mathbb{R}^2 $ any continuous function of bounded variation in the sense of Hardy. An error estimate is derived in terms of the second order moduli of continuity. Based on the obtained results, we discuss in detail the accuracy of uniform approximation of functions of several variables by linear functions. The error estimates are derived by using second order moduli of continuity. We pay a particular attention to sharpness of constants. Bibliography: 8 titles.     

Codimension 1 Subvarieties of {\mathcal{M}_g } and Real Gonality of Real Curves

Let ${\mathcal{M}_g }$ be the moduli space of smooth complex projective curves of genus g. Here we prove that the subset of ${\mathcal{M}_g }$ formed by all curves for which some Brill-Noether locus has dimension larger than the expected one has codimension at least two in ${\mathcal{M}_g }$ . As an application we show that if ${X \in \mathcal{M}_g }$ is defined over $\mathbb{R}$ then there exists a low degree pencil ${u:X \to \mathbb{P}^1 }$ defined over $\mathbb{R}.$      

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.     

Towards a classification of modular compactifications of \mathcal {M}_{g,n}

The moduli space of smooth curves admits a beautiful compactification $\mathcal{M}_{g,n} \subset \overline{\mathcal{M}}_{g,n}$ by the moduli space of stable curves. In this paper, we undertake a systematic classification of alternate modular compactifications of $\mathcal{M}_{g,n}$ . Let $\mathcal{U}_{g,n}$ be the (non-separated) moduli stack of all n-pointed reduced, connected, complete, one-dimensional schemes of arithmetic genus g. When g=0, $\mathcal{U}_{0,n}$ is irreducible and we classify all open proper substacks of $\mathcal{U}_{0,n}$ . When g≥1, $\mathcal{U}_{g,n}$ may not be irreducible, but there is a unique irreducible component $\mathcal{V}_{g,n} \subset\mathcal{U}_{g,n}$ containing $\mathcal{M}_{g,n}$ . We classify open proper substacks of $\mathcal {V}_{g,n}$ satisfying a certain stability condition.     

Completeness of the Grid Translates of Functions

If $f\in L^{p}(\mathbb{R}^{d})$ is a bounded real valued continuous function which has a unique maximum or a unique minimum at a point $x_{0}\in \mathbb{R}^{d}$ and if the inverse image of the neighborhoods of f(x ₀) shrinks regularly to x ₀, then $\mathrm{ span }\{f^{m}(x-2^{-m}\varSigma_{i=1}^{d} j_{i} e_{i})\mid m\in\mathbb{N}, j_{i}\in\mathbb{Z}\}$ is a dense subset of $L^{p}(\mathbb{R}^{d}), 1\le p<\infty$ where f ^m (x)=f(x) ^m and {e _i } is the natural basis of $\mathbb{R}^{d}$ . The result extends to all homogeneous groups, Riemannian symmetric spaces of noncompact type, Damek-Ricci spaces etc.     

Chebyshev's inequalities for functions monotone in the mean

LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ?,n} →R. Choosew ∈ F withw _i = w(i) > 0. PutW _i = w₁ + ? + w_i. Givenf ∈ F, define \(\bar f\) ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if \(\bar f\) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$      

On Rationality of W-algebras

We study the problem of classification of triples ( $ \mathfrak{g} $ ; f; k), where g is a simple Lie algebra, f its nilpotent element and k ∈ $ \mathbb{C} $ , for which the simple W-algebra W _k ( $ \mathfrak{g} $ ; f) is rational.     

Convex underestimators of polynomials

Convex underestimators of a polynomial on a box. Given a non convex polynomial ${f\in \mathbb{R}[{\rm x}]}$ and a box ${{\rm B}\subset \mathbb{R}^n}$ , we construct a sequence of convex polynomials ${(f_{dk})\subset \mathbb{R}[{\rm x}]}$ , which converges in a strong sense to the “best” (convex and degree-d) polynomial underestimator ${f^{*}_{d}}$ of f. Indeed, ${f^{*}_{d}}$ minimizes the L ₁-norm ${\Vert f-g\Vert_1}$ on B, over all convex degree-d polynomial underestimators g of f. On a sample of problems with non convex f, we then compare the lower bounds obtained by minimizing the convex underestimator of f computed as above and computed via the popular α BB method and some of its other refinements. In most of all examples we obtain significantly better results even with the smallest value of k.     

Greedy expansions in Hilbert spaces

We study the rate of convergence of expansions of elements in a Hilbert space H into series with regard to a given dictionary D. The primary goal of this paper is to study representations of an element f ∈ H by a series f ～ ∑ _j=1 ^∞ c _j (f)g _j (f), $g_j \left( f \right) \in \mathcal{D}$ . Such a representation involves two sequences: {g _j (f)} _j=1 ^∞ and {c _j (f) _j=1 ^∞ . In this paper the construction of {g _j (f)} _j=1 ^∞ is based on ideas used in greedy-type nonlinear approximation, hence the use of the term greedy expansion. An interesting open problem questions, “What is the best possible rate of convergence of greedy expansions for f ∈ A ₁(D)?” Previously it was believed that the rate of convergence was slower than $m^{ - \tfrac{1} {4}}$ . The qualitative result of this paper is that the best possible rate of convergence of greedy expansions for $f \in A_1 \left( \mathcal{D} \right)$ is faster than $m^{ - \tfrac{1} {4}}$ . In fact, we prove it is faster than $m^{ - \tfrac{2} {7}}$ .     

Approximation by radial basis functions with finitely many centers

Interpolation by translates of “radial” basis functions Φ is optimal in the sense that it minimizes the pointwise error functional among all comparable quasiinterpolants on a certain “native” space of functions $\mathcal{F}_\Phi $ . Since these spaces are rather small for cases where Φ is smooth, we study the behavior of interpolants on larger spaces of the form $\mathcal{F}_{\Phi _0 } $ for less smooth functions Φ₀. It turns out that interpolation by translates of Φ to mollifications of functionsf from $\mathcal{F}_{\Phi _0 } $ yields approximations tof that attain the same asymptotic error bounds as (optimal) interpolation off by translates of Φ₀ on $\mathcal{F}_{\Phi _0 } $ .     

On restricting Cauchy–Pexider functional equations to submanifolds

Sufficient geometric conditions are given which determine when the Cauchy–Pexider functional equation f(x)g(y) = h(x + y) restricted to x, y lying on a hypersurface in ${\mathbb{R}^d}$ has only solutions which extend uniquely to exponential affine functions ${\mathbb{R}^d \to \mathbb{C}}$ (when f, g, h are assumed to be measurable and non-trivial). The Cauchy–Pexider-type functional equations ${\prod_{j=0}^df_j(x_j)=F(\sum_{j=0}^dx_j)}$ for ${x_0, \ldots,x_d}$ lying on a curve and ${f_1(x_1)f_2(x_2)f_3(x_3)=F(x_1+x_2+x_3)}$ for x ₁, x ₂, x ₃ lying on a hypersurface are also considered.