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.     

Completeness of Integer Translates in Function Spaces on

We show that each of the Banach spacesC₀( ) andL^p( ), 2<p<∞, contains a function whose integer translates are complete. This function can also be chosen so that one of the following additional conditions hold: (1) Its non-negative integer translates are already complete. (2) Its integer translates form an orthonormal system inL²( ). (3) Its integer translates form a minimal system. A similar result holds for the corresponding Sobolev space, for certain weightedL²spaces, and in the multivariate setting. We also prove some results in the opposite direction.     

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.     

Lp Markov–Bernstein Inequalities on All Arcs of the Circle

Let 0<p<∞ and 0α<β2π. We prove that for n1 and trigonometric polynomials s_n of degree n, we have

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cn^p ∫^β_α |s_n(θ)|^p dθ, where c is independent of α, β, n, s_n. The essential feature is the uniformity in [α,β] of the estimate and the fact that as [α,β] approaches [0,2π], we recover the L_p Markov inequality. The result may be viewed as the complete L_p form of Videnskii's inequalities, improving earlier work of the second author.     

Global existence and blow-up of solutionsfor a higher-order kirchhoff-type equation with nonlinear dissipation

This paper deals with the higher-order Kirchhoff-type equation with nonlinear dissipationu_tt+(∫_Ω׀D^mu׀²dx)^q(−Δ)^mu+u_t׀u_t׀^r=׀u׀^pu,xΩ,t>0,in a bounded domain, where m < 1 is a positive integer, q, p, r < 0 arepositive constants. We obtain that the solution exists globally if p ≤ r, while ifp > max r, 2q , then for any initial data with negative initial energy, the solution blowsup at finite time in L^p+2 norm.     

A Singular Riesz Product in the Nevai Class and Inner Functions with the Schur Parameters in ∩p>2 l

There exist singular Riesz products dσ=∏^∞_κ=1 (1+Re(α_κζ^n_κ)) on the unit circle with the parameters (a_n)_n0 of orthogonal polynomials in L²(dσ) satisfying ∑^∞_n=0 |a_n|^p<+∞ for every p, p>2. The Schur parameters of the inner factor of the Cauchy integral ∫ (ζ−z)⁻¹ dσ(ζ), σ being such a Riesz product, belong to ∩_p>2 l^p.     

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.     

On the Strong Law of Large Numbers and the Law of the Logarithm for Weighted Sums of Independent Random Variables with Multidimensional Indices

Let (X, X ; ^d} be a field of independent identically distributed real random variables, 0 < p < 2, and {a _, ; ( , ) ^d × ^d, ≤ } a triangular array of real numbers, where ^d is the d-dimensional lattice. Under the minimal condition that sup _, |a _, | < ∞, we show that | |^{− 1/p} ∑ _≤ a _, X → 0 a.s. as | | → ∞ if and only if E(|X|^p(L|X|)^{d − 1}) < ∞ provided d ≥ 2. In the above, if 1 ≤ p < 2, the random variables are needed to be centered at the mean. By establishing a certain law of the logarithm, we show that the Law of the Iterated Logarithm fails for the weighted sums ∑ _≤ a _, X under the conditions that EX = 0, EX² < ∞, and E(X²(L|X|)^{d − 1}/L₂|X|) < ∞ for almost all bounded families {a _, ; ( , ) ^d × ^d, ≤ of numbers.     

Approximation ofk-Monotone Functions

It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inL_p, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inL_p, 1<p<∞, iso(n^−k/p).     

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.     

A one-parameter family of unimodal, concave, polynomial maps of the interval exhibiting multiplicities

This paper shows there exists a polynomial map, p, of the interval [0, 1] onto itself that is concave, symmetric about the point and such that, when parameterized {μp}, 0 ≤ μ ≤ 1, there exist three distinct values of the parameter μ₀ < μ₁ < μ₂ such that RLR³C = K(μ₀p) ≠ K(μ₁p) ≠ K(μ₂p) = RLR³C. There is also given an explicit construction of a C¹ family with the same properties.     

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.     

On the Cesaro Summability with Respect to the Walsh–Kaczmarz System

The Walsh system will be considered in the Kaczmarz rearrangement. We show that the maximal operator σ* of the (C,1)-means of the Walsh–Kaczmarz–Fourier series is bounded from the dyadic Hardy space H^p into L^p for every 1/2<p1. From this it follows by standard arguments that σ* is of weak type (1, 1) and bounded from L^q into L^q if 1<q∞.     

Singular integrals associated to the Laplacian on the affine groupax+b

We consider singular integral operators of the form (a)Z ₁L⁻¹Z₂, (b)Z ₁Z₂L⁻¹, and (c)L ⁻¹Z₁Z₂, whereZ ₁ andZ ₂ are nonzero right-invariant vector fields, andL is theL ²-closure of a canonical Laplacian. The operators (a) are shown to be bounded onL ^p for allp∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type (p, p) for anyp∈[1, ∞). Research supported by the Australian Research Council. Research carried out as a National Research Fellow.     

Kolmogorov and Linear Widths of Weighted Sobolev-Type Classes on a Finite Interval, II

Let I be a finite interval, r and ρ(t)=dist{t, ∂I}, tI. Denote by Δ^s₊W^r_p, α, 0α<∞, the class of functions x on I with the seminorm x^(r)ρ^α_Lp1 for which Δ^s_τx, τ>0, is nonnegative on I. We obtain two-sided estimates of the Kolmogorov widths d_n(Δ^s₊W^r_p, α)_Lq and of the linear widths d_n(Δ^s₊W^r_p, α)^lin_Lq, s=0, 1, …, r+1.     

Average CaseL∞-Approximation in the Presence of Gaussian Noise

We consider the average caseL_∞-approximation of functions fromC^r([0, 1]) with respect to ther-fold Wiener measure. An approximation is based onnfunction evaluations in the presence of Gaussian noise with varianceσ²>0. We show that the n th minimal average error is of ordern^{−(2r+1)/(4r+4)} ln^1/2 n, and that it can be attained either by the piecewise polynomial approximation using repetitive observations, or by the smoothing spline approximation using non-repetitive observations. This completes the already known results forL_q-approximation withq<∞ andσ0, and forL_∞-approximation withσ=0.     

Cesàro means of double Walsh-Fourier series

We investigate approximation properties of Cesàro (C; −α, −β)-means of double Walsh-Fourier series with α, β ∈ (0, 1). As an application, we obtain a sufficient condition for the convergence of the means σ _n,m ^/−α,−β (f; x, y) to f(x,y) in the L ^p -metric, p ∈ [1, ∞]. We also show that this sufficient condition cannot be improved.     

Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation

We consider a Cauchy problem for a semilinear heat equation

with p>p_S where p_S is the Sobolev exponent. If u(x,t)=(T−t)^−1/(p−1)φ((T−t)^−1/2x) for xR^N and t[0,T), where φ is a regular positive solution of

(P)

then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡(p−1)^−1/(p−1) for all p>1. Let p_L be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for p_S<p<p_L. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p>p_L.     

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.     

Best polynomial and durrmeyer approximation in Lp(S)

On a simplex SR^d, the best polynomial approximation is E_n()_Lp(S)=Inf{P_n−_Lp(S): P_n of total degree n}. The Durrmeyer modification, M_n, of the Bernstein operator is a bounded operator on L_p(S) and has many “nice” properties, most notably commutativity and self-adjointness. In this paper, relations between M_n−z.dfnc;_Lp(S) and E_[√n]()_Lp(S) will be given by weak inequalities will imply, for 0<α<1 and 1≤p≤∞, E_n()_Lp(S)=O(n^-2α)M_n−z.dfnc;_Lp(S)=O(n^-α). We also see how the fact that P(D)εL_p(S) for the appropriate P(D) affects directional smoothness.