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.     

On 1-isometries of affine quadrics over finite fields

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume $4 \leqslant dim V \leqslant \infty \wedge |\mathbb{F}| \in \mathbb{N}$ . A 1-isometry of the central quadric $\mathcal{F}: = \{ x \in V|q(x) = 1\}$ is a permutation ? of $\mathcal{F}$ such that (*) $$q(x - y) = \nu \Leftrightarrow q(x^\varphi - y^\varphi ) = \nu \forall x,y \in \mathcal{F}$$ holds true for a fixed element ν of $\mathbb{F}$ . For arbitraryν ∈ $\mathbb{F}$ we prove that? is induced (in a certain sense) by a semi-linear bijection $(\sigma ,\varrho ):(V,\mathbb{F}) \to (V,\mathbb{F})$ such thatq oσ =? oq, provided $\mathcal{F}$ contains lines and the exceptional case $(\nu = 2 \Lambda |\mathbb{F}| = 3 \Lambda \dim V = 4 \Lambda |\mathcal{F}| = 24)$ is excluded. In the exceptional case and as well in case of dim V = 3 there are counterexamples. The casesν ≠ 2 and v=2 require different techniques.     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Hermitian-einstein metrics on parabolic stable bundles

Let $\overline M $ be a compact complex manifold of complex dimension two with a smooth Kähler metric and D a smooth divisor on $\overline M $ . If E is a rank 2 holomorphic vector bundle on $\overline M $ with a stable parabolic structure along D, we prove the existence of a metric on $E'{\text{ = }}E|_{\overline M \backslash D} $ (compatible with the parabolic structure) which is Hermitian-Einstein with respect to the restriction of the Kähler metric to $\overline M $ ěD. A converse is also proved.     

Rate of convergence of Fourier series on the classes of \bar \Psi -integrals-integrals

We introduce the notion of $\bar \Psi $ -integrals of 2π-periodic summable functions f, f ε L, on the basis of which the space L is decomposed into subsets (classes) $L^{\bar \Psi } $ . We obtain integral representations of deviations of the trigonometric polynomials U _n(f;x;Λ) generated by a given Λ-method for summing the Fourier series of functions $f{\text{ }}\varepsilon {\text{ }}L^{\bar \Psi } $ . On the basis of these representations, the rate of convergence of the Fourier series is studied for functions belonging to the sets $L^{\bar \Psi } $ in uniform and integral metrics. Within the framework of this approach, we find, in particular, asymptotic equalities for upper bounds of deviations of the Fourier sums on the sets $L^{\bar \Psi } $ , which give solutions of the Kolmogorov-Nikol'skii problem. We also obtain an analog of the well-known Lebesgue inequality.     

Strongly embedded subspaces of p-convex Banach function spaces

Let $X(\mu )$ be a p-convex ( $1\le p<\infty $ ) order continuous Banach function space over a positive finite measure  $\mu $ . We characterize the subspaces of  $X(\mu )$ which can be found simultaneously in  $X(\mu )$ and a suitable $L^1(\eta )$ space, where $\eta $ is a positive finite measure related to the representation of  $X(\mu )$ as an $L^p(m)$ space of a vector measure  $m$ . We provide in this way new tools to analyze the strict singularity of the inclusion of  $X(\mu )$ in such an $L^1$ space. No rearrangement invariant type restrictions on  $X(\mu )$ are required.     

Global regularity of the\overline \partial-neumann problem on an annulus between two pseudoconvex manifolds which satisfy property (P)problem on an annulus between two pseudoconvex manifolds which satisfy property (P)

LetX be a complex manifold of dimensionn≥3. Let Ω₁, Ω₂ be two open pseudoconvex submanifolds with smooth boundary such that Ω₁ ? Ω₂ ?X . Let Ω = Ω₂ \ $\overline \Omega_1 $ . Assume thatbΩ₁ andbΩ₁ satisfy Catlin's condition (P). Then the compactness estimate for (p, q)-forms with 0<q<n?1 holds for the $\overline \partial$ -Neumann problem on Ω. This result implies that given a $\overline \partial$ -closed (p, q)-form α with 0<q<n?1, which isC ^∞ on $\overline \Omega$ and which is cohomologous to zero on Ω, the canonical solutionu of the equation $\overline \partial$ u=α is smooth on $\overline \Omega$ .     

Calderón constants of finite-dimensional couples

The Calderón constant æ( $\bar X$ ) is a numerical invariant of finite-dimensional Banach couple $\bar X = (X_0 ,X_1 )$ measuring its interpolation property with respect to linear operators acting in $\bar X$ . In the paper we prove the duality relation æ( $\bar X$ )≈ æ( $\bar X$ ^*)and calculate the asymptotic behavior of æ( $\bar X$ ) as dim $\bar X \to \infty $ for a few “classical” Banach couples.     

A-Systems, Independent Functions, and Sets Bounded in Spaces of Measurable Functions

Let $U \subset L_o ([0,1],\mathcal{M},m)$ be a set of Lebesgue measurable functions. Suppose also that two seminormed spaces of real number sequences are given: $\mathcal{A}$ and $\mathcal{B}$ . We study $\left( {\mathcal{A},\mathcal{B}} \right)$ -sets U defined by the classes $\mathcal{A}$ and $\mathcal{B}$ as follows: $\forall a = (a_n ) \in \mathcal{A}, \forall (f_n (t)) \in u^\mathbb{N} $ (or for sequences similar to $(f_n (t))$ ) $\exists E = E(a) \subset [0,1], mE = 1$ such that $\{ a_n f_n (t)\} 1_E (t)\} \in \mathcal{B}, t \in [0,1]$ . We consider three versions of the definition of $\left( {\mathcal{A},\mathcal{B}} \right)$ -sets, one of which is based on functions independent in the probability sense. The case ${\mathcal{B}}=l_\infty$ is studied in detail. It is shown that $({\mathcal{A}},l_\infty)$ -independent sets are sets bounded or order bounded in some well-known function spaces (L _p , L _p,q , etc.) constructed with respect to the Lebesgue measure. A characterization of such sets in terms of seminormed spaces of number sequences is given. The (l ₁,c _°)- and $(\mathcal{A},l_1 )$ -sets were studied by E. M. Nikishin.     

Regions of Values for the Systems \{ f(z_0 ),f'(z_0 ),c_2 \} and \{ f(r),f'(r),f(z_0 )\} in the Class of Typically Real Functions

Let T be the class of functions $f(z)$ having the following properties: these functions are regular and typically real in the disk $\left| z \right| < 1$ and have the expansions $f(z) = z + c_2 z^2 + c_3 z^3 + ....$ . We give algebraic and geometric characterizations of regions of values for the functionals in the class T mentioned in the title. In the same class of functions, we find regions of values for $f'(z_0 )$ with fixed $c_2$ and $f(z_0 )$ and for $f(z_0 )$ with fixed $f(r)$ and $f'(r)$ . Bibliography: 8 titles.     

Embedded Spaces of Trigonometric Splines and Their Wavelet Expansion

With each infinite grid X: ? < x _?1 < x ₀ < x ₁ < ? we associate the system of trigonometric splines $\{ \mathfrak{T}_j^B \}$ of class C ¹(α, β), the linear space $$T^B (X)\mathop = \limits^{def} \{ \tilde u|\tilde u = \sum\limits_j {c_j \mathfrak{T}_j^B } \quad \forall c_j \in \mathbb{R}^1 \} ,$$ and the functionals g ⁽ⁱ⁾ ∈ (C ¹(α, β))* with the biorthogonality property: $\left\langle {g(i),\mathfrak{T}_j^B } \right\rangle = \delta _{i,j}$ (here $\alpha \mathop = \limits^{def} \lim _{j \to - \infty } x_j ,\quad \beta \mathop = \limits^{def} \lim _{j \to + \infty } x_j$ ). For nested grids $\bar X \subset X$ , we show that the corresponding spaces $T^B (\bar X)$ are embedded in $T^B (X)$ and obtain decomposition and reconstruction formulas for the spline-wavelet expansion $T^B (X) = T^B (\bar X)\dot + W$ derived with the help of the system of functionals indicated above.     

The elliptic functions and integrals of the “{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}” problem

We describe the functions needed in the determination of the rate of convergence of best $L^\infty $ rational approximation to $\exp ( - x)$ on [0,∞) when the degree n of the approximation tends to ∞ (“1/9” problem).     

A Criterion for Weak Generalized Localization in the Class L_1 for Multiple Trigonometric Series from the Viewpoint of Isometric Transformations

In this paper, we study the problem of the variation (if any) of the sets of convergence and divergence everywhere or almost everywhere of a multiple Fourier series (integral) of a function $f \in L_p $ , $p \geqslant 1$ , $f(x) = 0$ , on a set of positive measure $\mathfrak{A} \subset \mathbb{T}^N = [ - \pi ,\pi )^N $ , $N \geqslant 2$ , depending on the rotation of the coordinate system, i.e., depending on the element $\tau \in \mathcal{F}$ , where $\mathcal{F}$ is the rotation group about the origin in $\mathbb{R}^N $ . This problem has been reduced to the study of the change in the geometry of the sets $\tau ^{ - 1} (\mathfrak{A}) \cap \mathbb{T}^N $ (where $\tau ^{ - 1} \in \mathcal{F}$ satisfies $\tau ^{ - 1} \cdot \tau = 1$ ) and $\mathbb{T}^N \backslash {\text{supp}}(f \circ \tau )$ depending on the “rotation,” i.e., on $\tau \in \mathcal{F}$ . In the present paper, we consider two settings of this problem (depending on the sense in which the Fourier series of the function $f \circ \tau $ is understood) and give (for both cases) possible solutions of the problem in the class $L_1 (\mathbb{T}^N )$ , $N \geqslant 2$ .     

The number of non-isomorphic models in quasi-varieties of semigroups

Define \( n_K (\lambda )\) tobe eitherω, or the number of non-isomorphic algebras in \(K\) ] having cardinality λ, whichever cardinal is larger. It is proved here that if \(K\) ] is a quasi-variety (universal Horn class) of semigroups, then \(n_K\) is one of four functions. Each of these functions satisfies: \(n_K (\omega ) = \omega\) or \(n_K (\omega ) = 2^\omega\) . If \(n_K (\lambda )< 2^\lambda\) for some infinite λ then \(K\) ] is a residually finitevariety.     

On the Zeros of Laplace Transforms

Suppose that f is a positive, nondecreasing, and integrable function in the interval $(0,1)$ . Then, by Pólya's theorem, all the zeros of the Laplace transform $$F(z) = \int_0^1 {e^{zt} f(t)dt} $$ lie in the left-hand half-plane $\operatorname{Re} z \leqslant 0$ . In this paper, we assume that the additional condition of logarithmic convexity of f in a left-hand neighborhood of the point 1 is satisfied. We obtain the form of the left curvilinear half-plane and also, under the condition $f( + 0) >0$ , the form of the curvilinear strip containing all the zeros of $f(z)$ .     

Representation of Integers by Positive Quaternary Quadratic Forms

Let $f(x,y,x,w) = x^2 + y^2 + z^2 + Dw^2$ , where $D >1$ is an integer such that $D \ne d^2$ and ${{\sqrt n } \mathord{\left/ {\vphantom {{\sqrt n } {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} \right. \kern-0em} {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}$ . Let $rf(n)$ be the number of representations of n by f. It is proved that $r_f (n) = \pi ^2 \frac{n}{{\sqrt D }}\sigma _f (n) + O\left( {\frac{{n^{1 + \varepsilon - c(\theta )} }}{{\sqrt D }}} \right),$ where $\sigma _f (n)$ is the singular series, $c(\theta ) >0$ , and ε is an arbitrarily small positive constant. Bibliography: 14 titles.     

On permutations of central quadrics which preserve an inner distance

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume dimV ≥ 4 and ¦ $\mathbb{F}$ ¦ ≥ 4. We consider a permutation ? of the central affine quadric $\mathcal{F}$ := {x εV ¦q(x) = 1} such that $$(*)x \cdot y = \mu \Leftrightarrow x^\varphi \cdot y^\varphi = \mu \forall x,y\varepsilon \mathcal{F}$$ holds true, where μ is a fixed element of $\mathbb{F}$ and where “·” is the scalar product associated withq. We prove that ? is induced (in a certain sense) by a semi-linear bijection (σ,?): (V, $\mathbb{F}$ ) → (V, $\mathbb{F}$ ) such thatq o ?o q, provided $\mathcal{F}$ contains lines and the pair (μ, $\mathbb{F}$ ) has additional properties if there ar no planes in $\mathcal{F}$ . The cases μ,≠ 0 and μ = 0 require different techniques.     

A Characterization of Projective Spaces by a Set of Planes

This note deals with the following question: How many planes of a linear space (P, $\mathfrak{L}$ ) must be known as projective planes to ensure that (P, $\mathfrak{L}$ ) is a projective space? The following answer is given: If for any subset M of a linear space (P, $\mathfrak{L}$ ) the restriction (M, $\mathfrak{L}$ )(M)) is locally complete, and if for every plane E of (M, $\mathfrak{L}$ (M)) the plane $\bar E$ generated by E is a projective plane, then (P, $\mathfrak{L}$ ) is a projective space (cf. 5.6). Or more generally: If for any subset M of P the restriction (M, $\mathfrak{L}$ (M)) is locally complete, and if for any two distinct coplanar lines G1, G2 ∈ $\mathfrak{L}$ (M) the lines $\bar G_1 ,\bar G_2 \varepsilon \mathfrak{L}$ generated by G1, G2 have a nonempty intersection and $\overline {G_1 \cup {\text{ }}G_2 }$ satisfies the exchange condition, then (P, $\mathfrak{L}$ ) is a generalized projective space.