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Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. We show that Sobolev estimates for the complex Green operator hold simultaneously for forms of symmetric bidegrees, that is, they hold for $(p,q)$--forms if and only if they hold for $(m-p,m-1-q)$--forms. Here $m$ equals the CR dimension of $M$ plus one. Symmetries of this type are known to hold for compactness estimates. We further show that with the usual microlocalization, compactness estimates for the positive part percolate up the complex, i.e. if they hold for $(p,q)$--forms, they also hold for $(p,q+1)$--forms. Similarly, compactness estimates for the negative part percolate down the complex. As a result, if the complex Green operator is compact on $(p,q_{1})$--forms and on $(p,q_{2})$--forms ($q_{1}\leq q_{2}$), then it is compact on $(p,q)$--forms for $q_{1}\leq q\leq q_{2}$. It is interesting to contrast this behavior of the complex Green operator with that of the $\overline{\partial}$--Neumann operator on a pseudoconvex domain., Comment: Added a reference to related work, removed a reference that was not quoted. To appear in Transactions of the American Mathematical Society

For 0 < α < 1 let V (α) denote the supremum of the numbers v\ud such that every α-H¨older continuous function is of bounded variation on a set of Hausdorff dimension v. Kahane and Katznelson (2009) proved the estimate 1/2 ≤ V (α) ≤ 1/(2−α) and asked whether the upper bound is sharp. We show that in fact V (α) = max{1/2, α}. Let dimH and dimM denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on V (α) is a consequence of the following theorem. Let {B(t) : t ∈ [0, 1]} be a fractional Brownian motion of Hurst index α. Then, almost surely, there exists no set\ud A ⊂ [0, 1] such that dimMA > max{1 − α, α} and B : A → R is of bounded variation. Furthermore, almost surely, there exists no set A ⊂ [0, 1] such that dimMA > 1 − α and B : A → R is β-H¨older continuous for some β > α. The zero set and the set of record times of B witness that the above theorems give the optimal dimensions. We also prove similar restriction theorems for deterministic self-affine functions and generic α-H¨older continuous functions. Finally, let {B(t) : t ∈ [0, 1]} be a two-dimensional Brownian motion. We prove that, almost surely, there is a compact set D ⊂ [0, 1] such that dimH D ≥ 1/3 and B: D → R2 is non-decreasing in each coordinate. It remains open whether 1/3 is best possible.\ud

Let ( M n + 1 , g , e − f d μ ) (M^{n+1},g,e^{-f}d\mu ) be a complete smooth metric measure space with 2 ≤ n ≤ 6 2\leq n\leq 6 and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded f f -minimal hypersurfaces in M M with uniform upper bounds on f f -index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in R n + 1 \mathbb {R}^{n+1} with 2 ≤ n ≤ 6 2\leq n\leq 6 . We also prove some estimates on the f f -index of f f -minimal hypersurfaces.

We prove that the group G = H o m ( Z N , Z ) G=\mathrm {Hom}(\mathbb {Z}^{\mathbb {N}}, \mathbb {Z}) of all homomorphisms from the Baer-Specker group Z N \mathbb {Z}^{\mathbb {N}} to the group Z \mathbb {Z} of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G G is discrete. As G G is non-discrete, it is not reflexive. Since G G can be viewed as a closed subgroup of the Tychonoff product Z c \mathbb {Z}^{\mathfrak {c}} of continuum many copies of the integers Z \mathbb {Z} , this provides an example of a group described in the title, thereby resolving a problem by Galindo, Recoder-Núñez and Tkachenko. It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.

For an action α of Z by homeomorphisms of a compact metric space, D. Lind introduced a dynamical zeta function and conjectured that this function has a natural boundary when d > 2. In this note, under the assumption that α is a mixing action by continuous automorphisms of a compact connected abelian group of finite topological dimension, it is shown that the upper growth rate of periodic points is zero and that the unit circle is a natural boundary for the dynamical zeta function.

Let X, Y be compact Hausdorff spaces and B£(X), B*(Y)f 0 < a, |3 < 12 (the first uncountable ordinal), the associated Banach spaces of bounded real-valued Baire functions of classes a and 0. If B*(X) =£ B*AX) (which is the case if a =£ 0 and X is not dispersed), then B*(X) is neither linearly isometric to BQ(Y) nor equivalent to B^(Y) in several other ways. B^(X) is linearly isometric to B%(Y) if and only if X is Baire isomorphic to Y. For 1 < a < ft the maximal ideal space of B£(X) for a nondispersed compact space X As not an F-space. 1. Let I bea compact (more generally, completely regular) Hausdorff space and C(X) the space of continuous real-valued functions on X. Let B0(X) = C(X)9 and inductively define Ba(X) for each ordinal a < Q, (Q denotes the first uncountable ordinal) to be the space of pointwise limits of sequences of functions in U | < a B^(X). Let B*(X) be the space of bounded functions contained in Ba(X). With the pointwise operations Ba(X) and B*(X) are lattice-ordered algebras. With the supremum norm B*(X) is a Banach algebra (see [4, §41]). The Baire sets of X of multiplicative class a, denoted by Za(X)9 are defined to be the zero sets of functions in B*(X). Those of additive class a, denoted by CZa(X), are defined as the complements of sets in Za(X). Finally, those of ambiguous class a, denoted by Aa(X), are the sets which are simultaneously in Za(X) and CZa(X). With the set-theoretic operations of union and intersection, Aa(X) is a Boolean algebra for each a < 12. The sets of exactly ambiguous class a, denoted by EAa(X\ are those in ^ a ( Z ) \ U ^ < a Ac(X). The sets of exactly additive and exactly multiplicative class a are defined analogously. The class of all Baire subsets of X is AMS (MOS) subject classifications (1970). Primary 06A65, 26A21, 28A05, 46E1S, 46E30, 46J10, 54C50, 54H05; Secondary 04A15.

Suppose e: M—• X is a map from a finite dimensional manifold to a locally compact space. Homotopy conditions on e are developed under which it can be extended to a proper map et M' —• X by adding boundary to M. This is applied in several settings to obtain geometric conclusions from homotopy data. For example: an embedding X C M of a manifold factor in a manifold of codimension > 3 is locally flat if and only if the complement M — X is locally 1connected at points of X. A completion of a map e: M —> X is a manifold M D M, with Mt M C bM', and an extension e: M —-> M which is proper. A map is proper if preimages of compact sets are compact. MAIN THEOREM (THE END THEOREM [11]). Suppose X is a locally compact locally 1-connected metric space. If M is a manifold of dimension > 6, e: M —• Xis a map which is proper on bM, and the end of e is onto, tame, and l'LC, then e has a completion. We define the terms used in the theorem. A neighborhood of the end is an open set UCM such that e\M U is proper. The end of e is onto if e(U) = X for every neighborhood U of the end. The end of e is 1-LC if for every x E X and neighborhoods U C M of the end, and V C X of x, there are neighborhoods if C U, V C V such that points (loops) in if d e~(V) can be joined by by arcs (resp. are nullhomotopic) in U O e~~(V). The end of e is tame if there are homotopies pulling M back from the end which are small in X. More precisely, for every e > 0 and neighborhood U of the end there is a neighborhood of the end V C U and a homotopy H: M x I —> M such that H0 = IM, HX(M) CM~V, Ht(M~ U)CM~V,md for each m G M the diameter of the arc eH: {m} x ƒ —> X is less than e. Tameness is a necessary condition for a completion, since such homotopies can be obtained by pushing M' in a collar of M' M C bM. If X is a point, this theorem is exactly the theorem of Browder, Livesay, and Levine [1], later extended to nonsimply connected ends by Siebenmann [12]. The proof in outline is similar to the classical X = pt. case. Homotopy and Received by the editors August 9, 1978. AMS (MOS) subject classifications (1970). Primary 57A99, 54C55, 57B05. 1 Partially supported by the National Science Foundation. ©American Mathematical Society 1979 0002-9904/7 9/0000-001S /$01.7 5