A sharp form of an embedding into exponential and double exponential spaces

Let Ω be a bounded domain in . In the well-known paper (Indiana Univ. Math. J. 20 (1971) 1077) Moser found the smallest value of K such that

     

A note on embedding into product spaces

Using factorization properties of an operator ideal over a Banach space, it is shown how to embed a locally convex space from the corresponding Grothendieck space ideal into a suitable power of E, thus achieving a unified treatment of several embedding theorems involving certain classes of locally convex spaces.     

On embedding expanders into ℓ p spaces

In this note we show that the minimum distortion required to embed alln-point metric spaces into the Banach space ℓ _p is between (c ₁/p) logn and (c ₂/p) logn, wherec ₂>c ₁>0 are absolute constants and 1≤p<logn. The lower bound is obtained by a generalization of a method of Linial et al. [LLR95], by showing that constant-degree expanders (considered as metric spaces) cannot be embedded any better. Research supported by Czech Republic Grant GAČR 201/94/2167 and Charles University grants No. 351 and 361.     

On embedding trees into uniformly convex Banach spaces

We investigate the minimum value ofD =D(n) such that anyn-point tree metric space (T, ρ) can beD-embedded into a given Banach space (X, ∥·∥); that is, there exists a mappingf :T →X with 1/D ρ(x,y) ≤ ∥f(x) −f(y)∥ ≤ρ(x,y) for anyx,y εT. Bourgain showed thatD(n) grows to infinity for any superreflexiveX (and this characterized super-reflexivity), and forX =ℓ _p, 1 <p < ∞, he proved a quantitative lower bound of const·(log logn)^min(1/2,1/p). We give another, completely elementary proof of this lower bound, and we prove that it is tight (up to the value of the constant). In particular, we show that anyn-point tree metric space can beD-embedded into a Euclidean space, with no restriction on the dimension, withD =O(√log logn). This paper contains results from my thesis [Mat89] from 1989. Since the subject of bi-Lipschitz embeddings is becoming increasingly popular, in 1997 I finally decided to publish this English version. Supported by Czech Republic Grant GAČR 0194 and by Charles University grants No. 193, 194.     

A constructive and functorial embedding of locally compact metric spaces into locales

The paper establishes, within constructive mathematics, a full and faithful functor M from the category of locally compact metric spaces and continuous functions into the category of formal topologies (or equivalently locales). The functor preserves finite products, and moreover satisfies f?g if, and only if, M(f)?M(g) for continuous . This makes it possible to transfer results between Bishop's constructive theory of metric spaces and constructive locale theory.     

On the distortion required for embedding finite metric spaces into normed spaces

We investigate the minimum dimensionk such that anyn-point metric spaceM can beD-embedded into somek-dimensional normed spaceX (possibly depending onM), that is, there exists a mappingf: M→X with $$\frac{1}{D}dist_M (x,y) \leqslant \left| {f(x) - f(y)} \right| \leqslant dist_M (x,y) for any$$ Extending a technique of Arias-de-Reyna and Rodríguez-Piazza, we prove that, for any fixedD≥1,k≥c(D)n ^1/2D for somec(D)>0. For aD-embedding of alln-point metric spaces into the samek-dimensional normed spaceX we find an upper boundk≤12Dn ^1/[(D+1)/2]lnn (using thel _∞ ^k space forX), and a lower bound showing that the exponent ofn cannot be decreased at least forD?[1,7)∪[9,11), thus the exponent is in fact a jumping function of the (continuously varied) parameterD.     

A sharp extrapolation theorem for lorentz spaces

Basing on geometric properties, we give a complete characterization of the Lorentz spaces that can be sharp extrapolation spaces for different types of the behavior of the growth of the norms of an operator in tending to a critical exponent. The results of this article are connected with the calculation of extrapolation constructions in about the same way as theorems of the Calderón-Mityagin type.     

A sharp form of the Sobolev trace theorems

Trace theorems for Besov spaces are proved using piecewise polynomial approximation theory and the K-method of interpolating Banach spaces. These theorems are limiting cases of standard embedding results.     

A sharp form of Nevanlinna's second fundamental theorem

Letf be meromorphic in the plane. We find a sharp upper bound for the error term

A sharp isoperimetric inequality for rank one symmetric spaces

 We prove an isoperimetric inequality for compact, regular domains in rank one symmetric spaces, which is sharp for geodesic balls. Besides volume and area of a given domain, some weak information about the second fundamental form of its boundary is involved. Received: 2 September 2002 / Revised version: 10 December 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 53C35, 52A40, 51M25     

A sharp form of Moser-Trudinger inequality in high dimension

Let Ω be a bounded smooth domain in Rn(n?3). This paper deals with a sharp form of Moser-Trudinger inequality. Let

     

On exponential bases for the Sobolev spaces over an interval

We suppose that K is a countable index set and that $\text{Λ = {λ}{}_{\mathrm{k}}\text{¦ k ? K}}$ is a sequence of distinct complex numbers such that forms a Riesz (strong) basis for L²[a, b], a < b. Let Σ = {σ₁, σ₂,…, σ_m} consist of m complex numbers not in Λ. Then, with p(λ) = Π_{k = 1}^m (λ ? σ_k), forms a Riesz (strong) bas Sobolev space H^m[a, b]. If we take σ₁, σ₂,…, σ_m to be complex numbers already in Λ, then, defining p(λ) as before, forms a Riesz (strong) basis for the space H^?m[a, b]. We also discuss the extension of these results to “generalized exponentials” tⁿe^λ_kt.     

Convolution equations in spaces of sequences with an exponential growth constraint

One describes the sets of the solutions of the convolution equations S*x=0 (on the set or on ₊={n:n0}) in the spaces of sequences of the type X=^X(, ), where. One proves that any 1-invariant subspace E,EX, coincides with KezS for some S and, after the Laplace transform can be represented in the form f·A(K_{(, )}), where K_{(, )}={z:kⁿ}_{n z} ^: }+{xX:x_k=0, k(, ), whose zeros do not accumulate to the circumference ¦¦=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSP, Vol. 149, pp. 107–115, 1986.The author expresses his sincere gratitude to N. K. Nikol'skii for the formulation of the problem and for his interest in the paper.