Nonlinear structure of some classical quasi-Banach spaces and -spaces

We investigate the Lipschitz structure of ℓ_p and L_p for 0<p<1 as quasi-Banach spaces and as metric spaces (with the metric induced by the p-norm) and show that they are not Lipschitz isomorphic. We prove that the -space L₀ is not uniformly homeomorphic to any other L_p space, that the L_p spaces for 0<p<1 embed isometrically into one another, and reduce the problem of the uniform equivalence amongst L_p spaces to their Lipschitz equivalence as metric spaces.     

An example of an almost greedy uniformly bounded orthonormal basis for

We construct a uniformly bounded orthonormal almost greedy basis for L_p(0,1), 1<p<∞. The example shows that it is not possible to extend Orlicz's theorem, stating that there are no uniformly bounded orthonormal unconditional bases for L_p(0,1), p≠2, to the class of almost greedy bases.     

Reznichenko families of trees and their applications

Examples of Talagrand, Gul'ko and Corson compacta resulting from Reznichenko families of trees are presented. The K_σδ property for weakly -analytic Banach spaces with an unconditional basis is proved.     

On some Nikolski - and Oswald-type inequalities

Inequalities of Nikolski (Trudy Mat. Inst. Steklova 38 (1951), 2.3, p. 255) in L_2π^p, p 1, and of Oswald (Izv. Vyssh. Uchebn. Zaved. Mat. 7 (1976), (3.4), p. 71; Theorem 1, p. 69), 0 < p < 1, are extended to the case of Orlicz spaces L_2π.     

Characterizations of spaces in the unit ball of

Let B denote the unit ball of . For 0<p<∞, the holomorphic function spaces Q_p and Q_p,0 on the unit ball of are defined as

and

In this paper, we give some derivative-free, mixture and oscillation characterizations for Q_p and Q_p,0 spaces in the unit ball of .     

Uniqueness of unconditional bases in quasi-banach spaces with applications to Hardy spaces

We prove some general results on the uniqueness of unconditional bases in quasi-Banach spaces. We show in particular that certain Lorentz spaces have unique unconditional bases answering a question of Nawrocki and Ortynski. We then give applications of these results to Hardy spaces by showing the spacesH _p (T ⁿ ) are mutually non-isomorphic for differing values ofn when 0<p<1. The research of the first two authors was partially supported by NSF-grant DMS 8901636.     

Weighted Hardy and Opial-type inequalities

An integral condition on weights u and v is given which is equivalent to the boundedness of the Hardy operator between the weighted Lebesgue spaces L_u^p and L_v^q with 0 < q < 1 < p < ∞. The Hardy inequalities are applied to give easily verified weight conditions which imply inequalities of Opial type.     

Uniqueness of unconditional bases in nonlocally convex c<Subscript>0</Subscript>-products

We show that the c ₀-product (X ⊕ X ⊕ ... ⊕ X ⊕ ...)₀ of a natural quasi-Banach space X with strongly absolute unconditional basis has a unique unconditional basis up to permutation. Our results apply to a wide range of cases, including most of the c ₀-products of the nonlocally convex classical quasi-Banach spaces.     

On equivalence of moduli of smoothness of polynomials in ,

It is well known that for functions , 1p∞. For general functions fL_p, it does not hold for 0<p<1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p=∞. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the L_p space for 0<p∞. We then show its analogues for the Ditzian–Totik modulus of smoothness and the weighted Ditzian–Totik modulus of smoothness for polynomials with .     

Commutators for Fourier multipliers on Besov Spaces

If T is any bounded linear operator on Besov spaces B_p^σ_j,q_j(Rⁿ)(j=0,1, and 0<σ₁<σ<σ₀), it is proved that the commutator [T,T_μ]=TT_μ−T_μT is bounded on B_p^σ,q(Rⁿ), if T_μ is a Fourier multiplier such that μ is any (possibly unbounded) symbol with uniformly bounded variation on dyadic coronas.     

Remarks Concerning Completeness of Translates in Function Spaces

This note explains how to translate the author's old result on cyclic vectors of the multiple shift operator into the language of completeness theorems for integer translates. This translation, together with those results, turns out to be a source for many completeness theorems. In particular, there follows the existence of functions f whose positive integer translates f(x−k), where k ₊ are complete in the spaces C^l₀( ), L^p( ), W^l_p( ), 2<p<∞, l=0, 1, …, as well as in their weighted and/or vector-valued analogues.     

Entropy numbers in sequence spaces with an application to weighted function spaces

We determine the exact asymptotic behaviour of entropy numbers of diagonal operators from ℓ_p to ℓ_q, 0<q<p∞, under mild regularity conditions on the generating diagonal sequence. On one hand, this is a quantitative version of Pitt's theorem for diagonal operators, and on the other hand it is a limiting case of results by Carl. An application to embeddings of weighted Besov and Triebel–Lizorkin spaces is also given.     

Weights in Serre's conjecture for via the Bernstein–Gelfand–Gelfand complex

Let be an n-dimensional mod p Galois representation. If ρ is modular for a weight in a certain class, called p-minute, then we restrict the Fontaine–Laffaille numbers of ρ; in other words, we specify the possibilities for the restriction of ρ to inertia at p. Our result agrees with the Serre-type conjectures for GL_n formulated by Ash, Doud, Pollack, Sinnott, and Herzig; to our knowledge, this is the first unconditional evidence for these conjectures for arbitrary n.     

On the asymptotic behaviour of a semigroup of linear operators

Let T = {T(t)}_{t ≥ 0} be a C₀-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ω_Lp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ω^p_R(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ω_β(T), β ≥ 0, of T. Our main results are as follows.

1. (i) Let T be a C₀-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω_{1 − ε}(T) < 0 for some ε > 0.

2. (ii) Let T be a C₀-semigroup on L^p such that the resolvent of the generator is uniformly bounded in the right half plane. Then ω_β(T) < 0 for all β>¦1/p − 1/p′¦, 1/p + 1/p′ = 1.

3. (iii) Let 1 ≤ p ≤ 2 and let T be a weakly L^p-stable C₀-semigroup on a Banach space X. Then for all β>1/p we have ω_β(T) ≤ 0.

Further, we give sufficient conditions in terms of ω^q_R(A) for the existence of L^p-solutions and W^1,p-solutions (1 ≤ p ≤ ∞) of the abstract Cauchy problem for a general class of operators A on X.     

On Positive and Copositive Polynomial and Spline Approximation inLp[−1, 1], 0<p<∞

For a functionfL_p[−1, 1], 0<p<∞, with finitely many sign changes, we construct a sequence of polynomialsP_nΠ_nwhich are copositive withfand such that f−P_npCω(f, (n+1)⁻¹)_p, whereω(f, t)_pdenotes the Ditzian–Totik modulus of continuity inL_pmetric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, thenωcannot be replaced byω²if 1<p<∞. In fact, we show that even for positive approximation and all 0<p<∞ the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained.     

Smoothness of stopping timesof diffusion processes

For an elliptic diffusion process, we prove that the exit time from an open set is in the fractional Sobolev spaces E^p_α (or D^p_α) provided that pα<1. The result is almost optimal.     

Wavelet characterization of weighted spaces

We give a characterization of weighted Hardy spaces H ^p (w), valid for a rather large collection of wavelets, 0 <p ≤ 1,and weights w in the Muckenhoupt class A _∞ We improve the previously known results and adopt a systematic point of view based upon the theory of vector-valued Calderón-Zygmund operators. Some consequences of this characterization are also given, like the criterion for a wavelet to give an unconditional basis and a criterion for membership into the space from the size of the wavelet coefficients.     

Embeddings in Spaces of Lipschitz Type, Entropy and Approximation Numbers, and Applications

We consider the embeddings of certain Besov and Triebel–Lizorkin spaces in spaces of Lipschitz type. The prototype of such embeddings arises from the result of H. Brézis and S. Wainger (1980, Comm. Partial Differential Equations5, 773–789) about the “almost” Lipschitz continuity of elements of the Sobolev spaces H^1+n/p_p( ⁿ) when 1<p<∞. Two-sided estimates are obtained for the entropy and approximation numbers of a variety of related embeddings. The results are applied to give bounds for the eigenvalues of certain pseudo-differential operators and to provide information about the mapping properties of these operators.     

On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle

Let u(r,θ) be biharmonic and bounded in the circular sector ¦θ¦ < π/4, 0 < r < ρ (ρ > 1) and vanish together with δu/δθ when ¦θ¦ = π/4. We consider the transform û(p,θ) = ∝₀¹r^{p − 1}u(r,θ)dr. We show that for any fixed θ₀ u(p,θ₀) is meromorphic with no real poles and cannot be entire unless u(r, θ₀) ≡ 0. It follows then from a theorem of Doetsch that u(r, θ₀) either vanishes identically or oscillates as r → 0.     

The asymptotic behavior of the solutions of the Cauchy problem generated by -accretive operators

The purpose of this paper is to study the asymptotic behavior of the solutions of certain type of differential inclusions posed in Banach spaces. In particular, we obtain the abstract result on the asymptotic behavior of the solution of the boundary value problem

where Ω is a bounded open domain in with smooth boundary ∂Ω, f(t,x) is a given L¹-function on ]0,∞[×Ω, γ1 and 1p<∞. Δ_p represents the p-Laplacian operator, is the associated Neumann boundary operator and β a maximal monotone graph in with 0β(0).