Harmonic approximation and Sarason's-type theorem

This chapter takes up several independent topics in Euclidean Fourier analysis, all having some bearing on the subject of partial differential equations. Section 1 elaborates on the relationship between the Fourier transform and the Schwartz space, the subspace of $L^{1}(\mathbb{R}^{N})$ consisting of smooth functions with the property that the product of any iterated partial derivative of the function with any polynomial is bounded. It is possible to make the Schwartz space into a metric space, and then one can consider the space of continuous linear functionals; these continuous linear functionals are called “tempered distributions.” The Fourier transform carries the space of tempered distributions in one-one fashion onto itself. Section 2 concerns weak derivatives, and the main result is Sobolev’s Theorem, which tells how to recover information about ordinary derivatives from information about weak derivatives. Weak derivatives are easy to manipulate, and Sobolev’s Theorem is therefore a helpful tool for handling derivatives without continually having to check the validity of interchanges of limits. Sections 3–4 concern harmonic functions, those functions on open sets in Euclidean space that are annihilated by the Laplacian. The main results of Section 3 are a characterization of harmonic functions in terms of a mean-value property, a reflection principle that allows the extension to all of Euclidean space of any harmonic function in a half space that vanishes at the boundary, and a result of Liouville that the only bounded harmonic functions in all of Euclidean space are the constants. The main result of Section 4 is a converse to properties of Poisson integrals for half spaces, showing that harmonic functions in a half space are given as Poisson integrals of functions or of finite complex measures if their $L^{p}$ norms over translates of the bounding Euclidean space are bounded. Sections 5–6 concern the Calderon–Zygmund Theorem, a far-reaching generalization of the theorem concerning the boundedness of the Hilbert transform. Section 5 gives the statement and proof, and two applications are the subject of Section 6. One of the applications is to Riesz transforms, and the other is to the Beltrami equation, whose solutions are “quasiconformal mappings.” Sections 7–8 concern multiple Fourier series for smooth periodic functions. The theory is established in Section 7, and an application to traces of integral operators is given in Section 8.

are left prime, that h-1 g is analytic and nonsingular for I z I < 1 and that G is nonsingular. Nevertheless (1) is not uniquely specified. However (Hannan, 1969), if, for prescribed q, r, [A(q) *. B(r)] is of rankp then (1) is uniquely specified. This will be called the rank condition. Some structures (1) cannot be brought to a form where this condition is satisfied (Hannan, 1971), so that the condition is overidentifying. The set, Cq,r, of all structures satisfying the rank condition for given q and r is mapped, by using the elements of A(j) and B(j) and the on and above diagonal elements of C, onto an open set in Euclidean space, if it is required that h-1 g is nonsingular for I z I < 1, and hence constitutes an analytic manifold. For q and r fixed the set of structures (1) not in Cq,r is evidently of lower dimension than Cq,r Tuan (1978, end of ? 1) states a number of objections to the use Of Cq,r and we wish to discuss these, and the general problem of parameterizing (1). For this last purpose Tuan (1978) used a family of canonical forms, called by him the quasiautoregressive-moving average representation. With any structure (1) is associated a set of integers mj (j = 1, ...,p) which determine the form of this representation (Tuan, 1978, p. 101). Thus given any structure (1) there is associated a set of mj and a matrix of polynomials u(z), with unit determinant, such that ug and uh are in the canonical form. Of course the matrix function u is an extremely complicated function of g and h. Using K as a symbol for the m1 (j = 1, .. . p), we may also map the set CK of all structures with these mj into an open set in Euclidean space, if h-' g is nonsingular for I z I < 1. Now there is no overidentification. However, there is a major problem if there are prior constraints imposed on (1) for it seems almost impossible, in general, to translate these constraints into constraints on the canonical forms because u is such a complicated function of g and h. On the other hand, the set of all structures (1) for given q and r is very complicated to parameterize. However, for the reason mentioned at the end of the previous paragraph, almost all of that set is constituted by Cq,r, which is easily parameterized. Tuan (1978) objects to Cq,r because of overidentification and because constraints may cause the rank condition to fail, and for other reasons to be discussed below. It is most unlikely that constraints would make [A (q) .B(r)] identically of rank less than p,

This paper establishes an explicit characterization of those real-valued functions on a finely open set in Euclidean space which are continuously differentiable with respect to the fine topology of classical potential theory. It improves and extends previous work of several authors. Differentiability of all orders is considered, and some consequences of the characterization are deduced.

We obtain (i) lower and upper bounds for the heat content of an open set in Rm with R-smooth boundary and finite Lebesgue measure, (ii) a necessary and sufficient geometric condition for finiteness of the heat content in Rm, and corresponding lower and upper bounds, (iii) lower and upper bounds for the heat loss of an open set in Rm with finite Lebesgue measure.

In addition to the notation given on pages xi and xii we shall use the following: T is a nonnegative number, and the interval [0,T] is interpreted as an interval of time; the points on this interval are, as a rule, denoted by t, s. D denotes an open set in Euclidean space, \(\bar{D}\) the closure of D, and ∂D the boundary of D. Q denotes an open set in Ed+1; the points of Q are expressed as (t,x) where t∈E1, x∈E d . ∂′Q denotes the parabolic boundary of Q (see Section 4.5).

We study the approximation of J-holomorphic maps continuous to the boundary from a domain in into an almost complex manifold by maps J- holomorphic to the boundary, giving partial results in the non-integrable case. For the integrable case, we study arcs in complex manifolds and establish the existence of neighborhoods biholomorphic to open sets in Euclidean space for several classes of arcs. As an application we obtain C k approximation of holomorphic maps continuous to the boundary into complex manifolds by maps holomorphic to the boundary, provided the boundary is nice enough.

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsShif BerhanuSHIF BERHANU graduated with a B.Sc. degree from Addis Ababa University, Ethiopia, in 1980. In 1987 he received his Ph.D. from Rutgers University under the direction of Francois Trevès. He is a professor of mathematics at Temple University. Berhanu's main area of research involves complex vector fields, that is, firstorder partial differential operators with complex-valued coefficients. Currently, he, P. Cordaro, and J. Hounie are collaborating to write a book on the subject of complex vector fields.Ahmed MohammedAHMED MOHAMMED received his B.Sc. in 1983 and M.Sc. in 1986 from Addis Ababa University, Ethiopia. He obtained his Ph.D. in 1995 from Temple University under the guidance of Professor Cristian E. Gutierrez. Between 1995 and 2000, he taught in the Department of Mathematics and Computer Science at Kuwait University, Kuwait. He held a visiting position at Temple University from 2000 to 2002. Currently he is an associate professor in the Department of Mathematical Sciences at Ball State University. If he is not doing mathematics in his office or teaching a class, chances are you'll find him at the Starbucks in the university book store.

We give a simple proof of the fact that compact, connected topological spaces have the “average distance property”. For a metric space (X, d), this asserts the existence of a unique number a = a(X) such that, given finitely many points x1, …, xn ∈ X, then there is some y ∈ X withWe examine the possible values of a(X) , for subsets of finite dimensional normed spaces. For example, if diam(X) denotes the diameter of some compact, convex set in a euclidean space, then a(X) ≤ diam(X)/√2 . On the other hand, a(X)/diam(X) can be arbitrarily close to 1 , for non-convex sets in euclidean spaces of sufficiently large dimension.

These series of notes serve as an introduction to some of both the classical and modern techniques in Reifenberg theory. At its heart, Reifenberg theory is about studying general sets or measures which can be, in one sense or another, approximated on all scales by well behaved spaces, typically just Euclidean space itself. Such sets and measures turn out not to be arbitrary, and often times come with special structure inherited from what they are being approximated by. We will begin by recalling and proving the standard Reifenberg theorem, which says that sets in Euclidean space which are well approximated by affine subspaces on all scales must be homoemorphic to balls. These types of results have applications to studying the regular parts of solutions of nonlinear equations. The proof given is designed to move cleanly over to more complicated scenarios introduced later. The rest of the lecture notes are designed to introduce and prove the Rectifiable Reifenberg Theorem, including an introduction to the relevant concepts. The Rectifiable Reifenberg Theorem roughly says that if a measure $\mu$ is summably close on all scales to affine subspaces $L^k$, then $\mu=\mu^++\mu^k$ may be broken into pieces such that $\mu^k$ is $k$-rectifiable with uniform Hausdorff measure estimates, and $\mu^+$ has uniform bounds on its mass. These types of results have applications to studying the singular parts of solutions of nonlinear equations. The proof given is designed to give a baby introduction to ways of thinking in more modern PDE analysis, including an introduction to Neck regions and their Structure Theory.

Many topological spaces are simply sets of points (atoms) endowed with a topology. Some spaces, however, have elements that are functions, matrices, or other non-atomic items. Another special type of space has elements that are themselves subsets of another space; these spaces are called hyperspaces. Hyperspaces are metric spaces, and the metric defined on them is called the Hausdorff metric. Hyperspaces wllose points are the closed subsets and hyperspaces whose points are the closed connected subsets (of metric spaces) have been extensively studied. Also, hyperspaces of closed and convex subsets of a bounded convex set in Euclidean space are of great interest in geometry. See Lay [3]. In recent years, geometers and topological dynamicists have explored spaces of closed and bounded subsets of the plane in connection with the study of fractals. One of the major results in the theory is that the hyperspace of closed subsets of a closed interval of real numbers is homeomorphic with the Hilbert cube, and with the space IX, the countable product of unit intervals. For more general information about the Hausdorff metric and spaces of subsets, see Devaney [2] and Sieradski [6]. The main result of this paper is a topological characterization of the space of closed subsets of a convergent sequence of points. The proof given here provides a homneornorphic embedding of the space in the plane, E2. The result was first proved by Pelczynski [5]. He studied arbitrary compact zero-dimensional metric spaces, so his proofs are much more widely applicable, but they are also somewhat technical. Our proof depends only on some well-known results in the theory of metric spaces, and is therefore accessible to advanced undergraduate mathematics majors. We also prove that for no convergent sequence of real numbers is there an isometric embedding of the hyperspace in Euclidean space, E,1, for any n. For other results and discussion, see Nadler [4]. Let (X, d) denote a compact metric space. The hyperspace (2X, D) of X is the metric space whose points are the closed nonempty subsets of X and whose metric is the Hausdorff metric D given by

In the first and second parts of this sequence we dealt with applications of graph theory to distance distribution in certain sets in euclidean spaces, to potential theory, to estimations of the transfinite diameter [1] and to value distribution of "triangle functional " (e.g. perimeter, area of triangles) [2]. The basic tool is provided in all these applications by the result formulated as Lemma 2. This, an essentially pure logical result, proves to be a very flexible and versatile instrument in applications.Here the same method is used in an abstract setting. First we deduce certain results for the density of a given family of subsets of an abstract set S in another family of subsets of the same S. Then we apply the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces, in particular in a normed linear function space. Applications of this method to functional on Hilbert spaces were given by Katona [3].

Densities and liftings for derived algebras

Coverings of planar and three-dimensional sets with subsets of smaller diameter

An elementary construction of the normal cycle of a compact definable set in Euclidean space (and more generally of a compactly supported constructible function) is given. Here “definable” means definable in some o-minimal structure. The construction is based on the notion of support function and uses only basic o-minimal geometry.

Chromatic numbers of graphs associated with point sets in Euclidean space