L 2 existence theorems for the\bar \partial _b - Neumann problem on strongly pseudoconvex CR manifoldsproblem on strongly pseudoconvex CR manifolds

LetM be the boundary of a strongly pseudoconvex domain in \(\mathbb{C}^n \) ,n≥4 and ω be an open subset inM such that ?ω is the intersection ofM with a flat hypersurface. We establish theL ² existence theorems of the \(\bar \partial _b - Neumann\) problem on ω. In particular, we prove that the \(\bar \partial _b - Laplacian\) \(\square _b = \bar \partial _b \bar \partial _b^* + \bar \partial _b^* \bar \partial _b \) equipped with a pair of natural boundary conditions, the so-called \(\bar \partial _b - Neumann\) boundary conditions, has closed range when it acts on (0,q) forms, 1≤q≤n?3. Thus there exists a bounded inverse operator for \(\square _b \) , the \(\bar \partial _b - Neumann\) operatorN _b, and we have the following Hodge decomposition theorem on ω for \(\bar \partial _b \bar \partial _b^* N_b \alpha + \bar \partial _b^* \bar \partial _b N_b \alpha \) , for any (0,q) form α withL ²(ω) coefficients. The proof depends on theL ^p regularity of the tangential Cauchy-Riemann operators \(\bar \partial _b u = \alpha \) on ω?M under the compatibility condition \(\bar \partial _b \alpha = 0\) , where α is a (p, q) form on ω, where 1≤q≤n?2. The interior regularity ofN _b follows from the fact that \(\square _b \) is subelliptic in the interior of ω. The operatorN _b induces natural questions on the regularity up to the boundary ?ω. Near the characteristic point of the boundary, certain compatibility conditions will be present. In fact, one can show thatN _b is not a compact operator onL ²(ω).     

Hölderabschätzungen für Ableitungen von Lösungen der Gleichung\bar \partial u = f bei streng pseudokonvexem Rand

Using the local Kerzman kernel we prove regularity of solutions of \(\bar \partial \) u=f, where f is a \(\bar \partial \) -closed (0,1)-form in a strongly pseudoconvex domain G in ?^N. If f is in H^m,∞, then the solution is in \(\tilde C^{m,\mu } \) forμ<1, that is, the m-th derivatives are in C^o,μ/2 and in addition areμ-Hölder continuous on curves “parallel” to the holomorphic tangent bundle \(\tilde T\) ?G. If f is in C^m,α with o<α<1, then the solution is in \(\tilde C^{m,1 + \mu } \) forμ<α, that is, the m-th derivatives are in C^o,(1+μ/2, and they have first derivatives “parallel” to \(\tilde T\) ?G lying in \(\tilde C^{o,\mu } \) . We derive the same results for the global solution constructed by Grauert and Lieb, and similar estimates on complex manifolds.     

Anisotropic function spaces. I: Hardy's inequality,decompositions

В статье рассматрива ются анизотропные пр остранства Бесова \(B_p^{\bar s} \) и Соболева \(W_p^{\bar s} \) н а плоскости и на единич ном круге, где 1<р<∞ и \(1< p< \infty \) И \(\bar s = (s_1 ,s_2 )\) . Основная цель состои т в доказательстве анизотропных нераве нств Харди и в изучени и соответствующих про странств \(\dot B_p^{\bar s} \) и \(\dot W_p^{\bar s} \) типа Бесова—Соболе ва. Эти результаты буд ут использованы во втор ой работе для точного описания следов упом янутых пространств н а плоских кривых.     

On reconstructing separable reducedp-groups with a given socle

Let \(\bar B^* \) be a separable reduced (abelian)p-group which is torsion complete. We ask whether for \(G \subseteq \bar B^* \) there is \(H \subseteq _{pr} \bar B^* ,H[p] = G[p]\) ,H[p]=G[p],H not isomorphic toG. IfG is the sum of cyclic groups or is torsion complete, the answer is easily no. For otherG, we prove that the answer is yes assuming G.C.H. Even without G.C.H. the answer is yes if the density character ofG is equal to Min _n<ω|p ⁿG|, i.e., $$\mathop {Min}\limits_{n< \omega } |p^n G| = \mathop {Min}\limits_m \mathop \Sigma \limits_{n > m} |(p^n G)[p]/(p^{n + 1} G)[p]|$$ Of course, instead of two non-isomorphic we can get many, but we do not deal much with this.     

Recovery of a function from the coefficients of its Dirichlet series

Let L(λ) be an entire function of exponential type, letγ(t) be the function associated with L(λ) in the sense of Borel, let \(\bar D\) be the smallest closed convex set containing all the singular points ofγ(t), let λ₀, λ₁, ..., λ_n, ... be the simple zeros of L(λ), and let A \(\bar D\) be the space of functions analytic on \(\bar D\) with the topology of the inductive limit. With an arbitraryf (z) ∈ A( \(\bar D\) ) we can associate the series whereC is a closed contour containing \(\bar D\) , on and inside of whichf (z) is analytic. We give a method of recoveringf (z) from the Dirichlet coefficientsa _n.     

Positive\partial \bar \partial - closed currents and non-Kähler geometrycurrents and non-Kähler geometry

In this paper some new results on positive \(\partial \bar \partial - closed\) currents are applied to modifications \(f:\bar M \to M\) . The main result in this topic is that every smooth proper modification of a compact Kähler manifoldM is balanced. Moreover, under suitable hypotheses on the map, the Kähler degrees of \(\bar M\) corresponds to homological properties of the exceptional set of the modification. More examples ofp-Kähler manifolds are discussed in the last section of the paper.     

Die metrisierbaren linearen Teilräume des Raumes\mathfrak{O}_M von L. Schwartz

In Schwartz' terminology, a real or complex valued functionf, defined and infinitely differentiable on ? _n , belongs to \(\mathfrak{O}_M \) iff, as well as any of its derivatives, is at most of polynomial growth. The topology of \(\mathfrak{O}_M \) is defined by the seminorms sup{∣?(x)D ^p f(x)∣;x∈? ⁿ }, where ? belongs to \(\mathfrak{S}\) andD ^p is any derivative. It is well-known that \(\mathfrak{O}_M \) is non-metrisable. For any μ: ? ⁿ →?, let \(\mathfrak{B}_\mu \) be the space of all infinitely differentiable functionsf satisfying, for eachp, sup{∣(1+∣x∣²)^?μ(p) D ^p f(x)∣;x∈? ⁿ }<∞, with the obvious topology. These spaces, which are of very little use elsewhere in the theory of distributions, can be conveniently applied to characterise the metrisable linear subspaces of \(\mathfrak{O}_M \) : A linear subspace of \(\mathfrak{O}_M \) is metrisable if and only if it is, algebraically and topologically, a subspace of some \(\mathfrak{B}_\mu \) .     

Resplendency and recursive definability inω-stable theories

We prove Theorem A.Every resplendent model of an ω-stable theory is homogeneous. As an application we obtain Theorem B.Suppose T is ω-stable, M ? T is recursively saturated and P ∈ S (M) is such that for all finite \(\bar m\) ∈ M, p ↑ \(\bar m\) is realized in M. Then there is a \(\bar c\) ∈ M and a definition d of p over \(\bar c\) such that d is recursive in t ( \(\bar c\) /Ø).     

\mathfrak{E} of topological spaces II

In this paper, we obtain analogues, in the situation of \(\mathfrak{E}\) -extensions, of Magill's theorem on lattices of compactifications. We define an epireflective subcategory of the categoryT 2 of all Hausdorff spaces to be admissive (respectively finitely admissive) if for any \(\mathfrak{E}\) -regular spaceX, every Hausdorff quotient of \(\beta _\mathfrak{E} X\) which is Urysohn on \(\beta _\mathfrak{E} X - X\) (respectively which is finitary on \(\beta _\mathfrak{E} X - X\) ) and which is identity onX, has \(\mathfrak{E}\) . We notice that there are many proper epireflective subcategories ofT 2 containing all compact spaces and which are admissive; there are many such which are not admissive but finitely admissive. We prove that when \(\mathfrak{E}\) is a finitely admissive epireflective subcategory ofT 2, then the lattices of finitary \(\mathfrak{E}\) -extensions of two spacesX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic. Further if \(\mathfrak{E}\) is admissive, then the lattices of Urysohn \(\mathfrak{E}\) -extensions ofX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic.     

ОБ ИНтЕРпОлИРОВАНИИ В кОМплЕксНОИ ОБлАст И

LetD be a simply connected domain, the boundary of which is a closed Jordan curveγ; \(\mathfrak{M} = \left\{ {z_{k, n} } \right\}\) , 0≦k≦n; n=1, 2, 3, ..., a matrix of interpolation knots, \(\mathfrak{M} \subset \Gamma ; A_c \left( {\bar D} \right)\) the space of the functions that are analytic inD and continuous on \(\bar D; \left\{ {L_n \left( {\mathfrak{M}; f, z} \right)} \right\}\) the sequence of the Lagrange interpolation polynomials. We say that a matrix \(\mathfrak{M}\) satisfies condition (B _m ), \(\mathfrak{M}\) ∈(B _m ), if for some positive integerm there exist a setB _m containingm points and a sequencen _{p p=1} ^∞ of integers such that the series \(\mathop \Sigma \limits_{p = 1}^\infty \frac{1}{{n_p }}\) diverges and for all pairsn _i ,n _j ∈{n _p } _p=1 ^∞ the set \(\left( {\bigcap\limits_{k = 0}^{n_i } {z_{k, n_i } } } \right)\bigcap {\left( {\bigcup\limits_{k = 0}^{n_j } {z_{k, n_j } } } \right)} \) is contained inB _m . The main result reads as follows. {Let D=z: ¦z¦ \(\Gamma = \partial \bar D\) and let the matrix \(\mathfrak{M} \subset \Gamma \) satisfy condition (B_m). Then there exists a function \(f \in A_c \left( {\bar D} \right)\) such that the relation $$\mathop {\lim \sup }\limits_{n \to \infty } \left| {L_n \left( {\mathfrak{M}, f, z} \right)} \right| = \infty $$ holds almost everywhere on γ.     

On the cosets of the 2q-power group in the unit group modulop

Letp be a prime number ≡ 3 mod 4,G ^p the unit group of ?/p?, andg a generator ofG _p. Letq be an odd divisor ofp - 1 andG _p ^2q = {t ^2q;t ∈G _pthe subgroup of index2q inG _p. The groupG _p ² / _p ^2q consists of the classes \(\bar g^{2j} \) ,j = 0,...,q – 1. In this paper we study the ’excesses’ of the classes \(\bar g^{2j} \) in {l,...,(_p–l)/2}, i.e., the numbers \(\Phi _j = \left| {\left\{ {k;1 \leqslant k \leqslant \left( {p - 1} \right)/2,\bar k \in \bar g^{2j} } \right\}} \right| - \left| {\left\{ {k;\left( {p - 1} \right)/2 \leqslant k \leqslant p - 1,\bar k \in \bar g^{2j} } \right\}} \right|\) ,j = 0.....q — 1. First we express therelative class number h _2q of the subfieldK _2q? ?(e^2#x03C0;i/p ) of degree [K _2q: ?] =2q in terms of these excesses. We use this formula to establish certaincongruences for the Ф^j. E.g., ifq ∈ {3,5,11}, each number Фj is congruent modulo 4 to each other iff 2 dividesh _2q ^- . Finally we study thevariance of the excesses, i.e., the number \(\sigma ^2 = ((\Phi _0 - \hat \Phi )^2 + \ldots + (\Phi _{q - 1} - \hat \Phi )^2 )/(q - 1)\) , where \(\hat \Phi \) is the mean value of the numbers Ф_j. We obtain an explicit lower bound for σ² in terms ofh _2q ^- /h ₂ ^- . Moreover, we show that log σ² is asymptotically equal to 21og(h _2q ^- h ₂ ^- )/(q - 1) forp→∞. Three tables illustrate the results.     

Universelle Approximation durch Riesz-Transformierte der geometrischen Reihe

Let p={p_v} be a fixed sequence of complex numbers. Define \(p_n : = \mathop \Sigma \limits_{\nu = o}^n p_\nu \) and suppose that \(p_{m_k } \ne o\) for a subsequence M={m_k} of nonnegative integers. The matrix A=(α_kv) with the elements $$\alpha _{k\nu } = p_\nu /p_{m_k } if o \leqslant \nu \leqslant m_k ,\alpha _{k\nu } = oif \nu > m_k $$ generates a summability method (R,p,M) which is a refinement of the well known Riesz methods. The (R,p,M) methods have been introduced in [4]. In the present paper we are concerned with the summability of the geometric series \(\mathop \Sigma \limits_{\nu = o}^n z^\nu \) by (R,p,M) methods. We prove the following theorem. Suppose G is a simply connected domain with \(\{ z:|z|< 1\} \subset G,1 \varepsilon | G \) . Then there exists a universal, regular (R,p,M) method having the following properties: (1) \(\mathop \Sigma \limits_{\nu = o}^\infty z^\nu \) is compactly summable (R,p,M) to \(\tfrac{1}{{1 - z}}\) on G. (2) For every compact set B?¯G^c which has a connected complement and for every function f which is continuous on B and analytic in its interior there exists a subsequence M(B,f) of M such that \(\mathop \Sigma \limits_{\nu = o}^\infty z^\nu \) is uniformly summable (R,p,M(B,f)) to f(z) on B. (3) For every open set U?G^c which has simply connected components in ? and for every function f which is analytic on U there exists a subsequence M(U,f) of M such that \(\mathop \Sigma \limits_{\nu = o}^\infty z^\nu \) is compactly summable (R,p,M(U,f)) to f(z) on U.     

Erratum et Addendum zu der Arbeit: Über exzeptionelle Mengen

Let X be a complex space and A?X a compact subspace. Let \(\tilde X\) be the blowing up of A in X and \(\tilde A\) ? \(\tilde X\) the resulting hyper-surface. Then the normal bundle of \(\tilde A\) in { \(\tilde X\) is weakly negative iff the normal bundle of the k-th infinitesimal neighborhood of A in X is weakly negative for all k?0. This corrects a theorem in [5].     

A characterisation of multiplicity

LetR be a ring with non-zero identity and unitary leftR-modules, while \(\mathcal{N}_R \) is the subcategory of NoetherianR-modules. Given a length functionL on \(\mathcal{N}_R \) and central elements α₁,...,α _n ofR we can define the multiplicity length functione _R (L|α₁,...α _n |) on \(\mathcal{N}_R \) with the same properties as the classical multiplicity. Here, we characterise multiplicity as the greatest length function which can be defined inductively in terms of a certain type of function on \(\mathcal{N}_R \) .     

The extreme bodies in the set of plane convex bodies with a given width function

Let \(\bar K\) (w) denote the class of plane convex bodies having a width functionw. Examining the length measure of the boundary of a convex body in \(\bar K\) (w), a characterization is given for the extreme (indecomposable) bodies in \(\bar K\) (w). This is a generalization of the solution previously given by the author in Israel J. Math. (1974) for the case wherew′ is absolutely continuous.     

Estimate of a polynomial in generalized exponential functions in the convex hull of two domains

Пусть $$f_n (z) = \exp \{ \lambda _n z\} [1 + \psi _n (z)], n \geqq 1$$ гдеψ _n (z) — регулярны в н екоторой односвязно й областиS, λ _n — нули целой функц ии экспоненциальног о ростаL(λ) с индикатрис ой ростаh(?), причем $$|L\prime (\lambda _n )| > C(\delta )\exp \{ [h(\varphi _n ) - \varepsilon ]|\lambda _n |\} \varphi _n = \arg \lambda _n , \forall \varepsilon > 0$$ . Предположим, что на лю бом компактеK?S $$|\psi _n (z)|< Aq^{|\lambda |_n } , a< q< 1, n \geqq 1$$ гдеA иq зависит только отK. Обозначим через \(\bar D\) со пряженную диаграмму функцииL(λ), через \(\bar D_\alpha \) — смещение. \(\bar D\) на векторα. Рассмотр им множестваD ₁ иD ₂ так ие, чтоD ₁ иD ₂ и их вьшуклая обо лочкаE принадлежатS. Пусть \(\bar D_{\alpha _1 } \subset D_1 , \bar D_{\alpha _2 } \subset D_2 \) Доказывается, что сущ ествует некоторая об ластьG?E такая, что \(\mathop \cup \limits_{\alpha \in [\alpha _1 ,\alpha _2 ]} \bar D_\alpha \subset G\) и дляz∈G верна оценка $$\sum\limits_{v = 1}^n {|a_v f_v (z)|} \leqq B\max (M_1 ,M_2 ), M_j = \mathop {\max }\limits_{t \in \bar D_j } |\sum\limits_{v = 1}^n {a_v f_v (t)} |$$ , где константаB не зав исит от {a _v }.     

Elliptische Differentialgleichungen mit variablen Koeffizienten in Gebieten mit unbeschränktem Rand

We discuss the spectrum of a symmetric elliptic differential operator A with domain \(\mathop {H^m }\limits^o (\Omega ) \cap H^{2m} (\Omega )\) in regions Ω with unbounded boundary \(\dot \Omega \) , where are \(\bar \Omega \) uniformely of class C^2m and on \(\dot \Omega \) the normal condition x·ν(x)≦μ for sufficient small positiveμ. We prove the A-priori-estimate \(\parallel u\parallel _{m,\Omega } \leqq c\parallel (l + r) (A - k)u\parallel _{o,\Omega } \) and show for all k>k, k≧0 suitable, there are no eigenvalues of A and by characterizing weighted Sobolev spaces with negative norm the existence of solutions \((l + r)_2 ^{ - 1} u \in \mathop H\limits^0{^m} (\Omega ) \cap H^{2m} (\Omega )\) of the equation (A?k)u=f, (1+r)f∈L²(Ω).     

Power Majorization and Majorization of Sequences

Let \(\bar x\) , \(\bar y\ \in\ R_n\) be vectors which satisfy x₁ ≥ x₂ ≥ … ≥ x_n and y₁ ≥ y₂ >- … ≥ y_n and Σx_i = Σy_i. We say that \(\bar x\) is power majorized by \(\bar y\) if Σx_i ^p ≤ Σy_i ^p for all real p ? [0, 1] and Σx_i ^p ≥ Σy_i ^p for p ∈ [0, 1]. In this paper we give a classification of functions ? (which includes all possible positive polynomials) for which \(\bar\phi(\bar x) \leq \bar\phi(\bar y)\) (see definition below) when \(\bar x\) is power majorized \(\bar y\) . We also answer a question posed by Clausing by showing that there are vectors \(\bar x\) , \(\bar y\ \in\ R^n\) of any dimension n ≥ 4 for which there is a convex function ? such that \(\bar x\) is power majorized by \(\bar y\) and \(\bar\phi(\bar x)\ >\ \bar\phi(\bar y)\) .     

On positive definite measures

In this paper we study the Fourier transform of unbounded measures on a locally compact groupG. After a short introductory section containing background material, especially results established byL. Argabright andJ. Gil De Lamadrid we turn to the main subjects of the paper: first we characterize \(\Re \left( G \right), \mathfrak{J}\left( G \right)\) andB(G) cones in \(\mathfrak{W}\left( G \right)\) . After that we establish the subspace \(\mathfrak{W}_\Delta \left( G \right)\) of \(\mathfrak{W}\left( G \right)\) which contains \(\mathfrak{W}_p \left( G \right)\) , the linear span of all positive definite measures.