An algebra containing the two-sided convolution operators

We present an intrinsically defined algebra of operators containing the right and left invariant Calderón–Zygmund operators on a stratified group. The operators in our algebra are pseudolocal and bounded on L^p (1<p<∞). This algebra provides an example of an algebra of singular integrals that falls outside of the classical Calderón–Zygmund theory.     

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data:

We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the L^p–L^q estimates of fundamental solutions for linear damped wave equations [K. Nishihara, L^p–L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The L^p–L^q estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and L^p–L^q estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, L^p–L^q estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method [Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701].     

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,μ) be an abstract Wiener space and let D_V:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V^*BV in H and in , as well as the realisations of the operators and in L^p(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:

(1) with for ;

(2) admits a bounded H^∞-functional calculus on ;

(3) with for ;

(4) admits a bounded H^∞-functional calculus on .

Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving L^p-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C₀-contraction semigroup on a Hilbert space H and let −L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L^p-domain characterisation for the operator L.

The n-width of the unit ball of H

Let E be a compact subset of the open unit disc Δ and let H^q be the Hardy space of analytic functions f on Δ for which stf¦^q has a harmonic majorant. We determine the value of the Kolmogorov, Gel'fand, and linear n-widths in L^p(E, μ) of the restriction to E of the unit ball of H^q when p q or when 1 q < p < ∞ and E is “small”.     

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.     

Product properties of Hilbert transforms

Let fεL^p(R), gεL^q(R) with 1<p<∞, 1<q<∞ and let Hf, Hg be their respective Hilbert transforms. We give a simple proof of the identity Hf · Hg − f · G = H(f · Hg + g · Hf) a.e. and of its inverse in the case (1/p) + (1/q) 1 which includes the cases already considered by Cossar and Tricomi. We next derive applications, especially to boundary values of analytic functions.     

Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

We show that the L^p spatial–temporal decay rates of solutions of incompressible flow in an 2D exterior domain. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in He, Xin [C. He, Z. Xin, Weighted estimates for nonstationary Navier–Stokes equations in exterior domain, Methods Appl. Anal. 7 (3) (2000) 443–458], and our previous results [H.-O. Bae, B.J. Jin, Asymptotic behavior of Stokes solutions in 2D exterior domains, J. Math. Fluid Mech., in press; H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains, submitted for publication]. For the spatial decay rate estimate, we first extend temporal decay rate result of the Navier–Stokes solutions for general L^p space when the initial velocity is in , 1<rq<∞ (1<r<q=∞).     

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.     

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.     

The Fine Structure of the n-Widths of H-Spaces in Lq(−1, 1)

The n-widths of the unit ball A_p of the Hardy space H^p in L_q( −1, 1) are determined asymptotically. It is shown that for 1 ≤ q < p ≤∞ there exist constants k₁ and k₂ such that [formula]≤ d_n(A_p, L_q(−1, 1)),dⁿ(A_p, L_q(−1, 1)), δ_n(A_p, L_q(−1, 1))[formula]where d_n, dⁿ, and δ_n denote the Kolmogorov, Gel′fand and linear n-widths, respectively. This result is an improvement of estimates previously obtained by Burchard and Höllig and by the author.     

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).     

Complexity of Weighted Approximation over

We study approximation of univariate functions defined over the reals. We assume that the rth derivative of a function is bounded in a weighted L_p norm with a weight ψ. Approximation algorithms use the values of a function and its derivatives up to order r−1. The worst case error of an algorithm is defined in a weighted L_q norm with a weight ρ. We study the worst case (information) complexity of the weighted approximation problem, which is equal to the minimal number of function and derivative evaluations needed to obtain error . We provide necessary and sufficient conditions in terms of the weights ψ and ρ, as well as the parameters r, p, and q for the weighted approximation problem to have finite complexity. We also provide conditions which guarantee that the complexity of weighted approximation is of the same order as the complexity of the classical approximation problem over a finite interval. Such necessary and sufficient conditions are also provided for a weighted integration problem since its complexity is equivalent to the complexity of the weighted approximation problem for q=1.     

On the Rate of Convergence of Two Bernstein–Bézier Type Operators for Bounded Variation Functions, II

The rates of convergence of two Bernstein–Bézier type operators B^(α)_n and L^(α)_n for functions of bounded variation have been studied for the case α1 by the author and A. Piriou (1998, J. Approx. Theory95, 369–387). In this paper the other case 0<α<1 is treated and asymptotically optimal estimations of B^(α)_n and L^(α)_n for functions of bounded variation are obtained. Besides, some interesting behaviors of the operators B^(α)_n and L^(α)_n (α>0) for monotone functions and functions of bounded variation are also given.     

Weighted Inequalities for the Generalized Maximal Operator in Martingale Spaces

The generalized maximal operator M in martingale spaces is considered. For 1 < p ≤ q < ∞, the authors give a necessary and sufficient condition on the pair ([^(m)]\hat \mu , v) for M to be a bounded operator from martingale space L ^p (v) into L ^q ([^(m)]\hat \mu ) or weak-L ^q ([^(m)]\hat \mu ), where [^(m)]\hat \mu is a measure on Ω × ℕ and v a weight on Ω. Moreover, the similar inequalities for usual maximal operator are discussed.     

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.     

Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk

Let f be an integrable function on the unit disk. The Hankel operator H_f is densely defined on the Bergman space A^p by H_fg = fg − P(fg), where g is a bounded analytic function and P is the Bergman projection (orthogonal projection from L² to A²) extended to L¹ via its integral formula. In this paper, the functions f for which H_f extends to a bounded operator from A^p to L^p are characterized, 1 < p < ∞. Also characterized are the functions f for which H_f extends to a compact or Schatten class operator on A². The proofs can be extended to handle any smoothly bounded domain in C in place of the unit disk.     

Best Approximations and Kolmogorov and Trigonometric Widths of the Classes <Emphasis Type="Italic">B</Emphasis><Stack><Subscript><Emphasis Type="Italic">p</Emphasis>,θ</Subscript><Superscript>Ω</Superscript></Stack> of Periodic Functions of Many Variables

We obtain estimates exact in order for the best approximations and Kolmogorov and trigonometric widths of the classes B _p,θ ^Ω of periodic functions of many variables in the space L ^q for certain values of the parameters p and q.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1557–1568, November, 2004.     

A note on the sunouchi operator with respect to the walsh—kaczmarz system

An analogue of the so—called Sunouchi operator with respect to the Walsh—Kaczmarz system will be investigated. We show the boundedness of this operator if we take it as a map from the dyadic Hardy space H ^p to L ^p for all 0<p≤1.. For the proof we consider a multiplier operator and prove its (H ^p H ^p)—boundedness for 0<p≤1. Since the multiplier is obviously bounded from L ² to L ², a theorem on interpolation of operators can be applied to show that our multiplier is of weak type (1,1) and of type (q q) for all 1<q<∞. The same statements follow also for the Sunouchi operator.     

The weak type inequality for the maximal operator of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series

The main aim of this paper is to prove that the maximal operator σ^* of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series is bounded from the Hardy space H_2/3 to the space weak-L_2/3.