Translation-Invariant Bilinear Operators with Positive Kernels

We study L ^r (or L ^{r, ∞}) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on \mathbb Rⁿ{\mathbb {R}^n}. We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on \mathbb R²ⁿ{\mathbb R^{2n}}, while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some sufficient conditions on nonnegative kernels yielding boundedness for the corresponding operators on products of Lebesgue spaces. We also prove that, unlike the linear case where boundedness from L ¹ to L ¹ and from L ¹ to L ^{1, ∞} are equivalent properties, boundedness from L ¹ × L ¹ to L ^1/2 and from L ¹ × L ¹ to L ^{1/2, ∞} may not be equivalent properties for bilinear translation-invariant operators with nonnegative kernels.     

Local geometry of L 1 ν L ∞ and L 1+L ∞

Extreme points of the unit sphere S (L ¹+L ^∞ ) of LL ¹+L ^∞ under the classical norm used in the interpolation theory were characterized in [8] and [11], while extreme points of S(L ¹ ∩ L ^∞) under the classical norm were characterized in [7]. In this paper extreme points of the unit sphere of L ¹+L ^∞ and L ¹ ∩ L ^∞ under the “dual” norms are characterized. Moreover, all the extreme points in L ¹ ∩ L ^∞ and L ¹+L ^∞ (under both kinds of norms) are examined if they are exposed, H-points, or strongly exposed. Smooth points in both these spaces for both the norms are also characterized. Finally, it is proved that in general the spaces L ^p +L ^q and L ^p ∩ L ^q are not isometric to Orlicz spaces, provided that 1<p<q<+∞. The paper was written while the first named author was visiting The University of Memphis The third named author is supported by KBN-Grant 2 PO3A 050 09.     

Weyl quantization of Lebesgue spaces

We study boundedness and compactness properties for the Weyl quantization with symbols in L^q (?^2d ) acting on L^p (?^d ). This is shown to be equivalent, in suitable Banach space setting, to that of the Wigner transform. We give a short proof by interpolation of Lieb's sufficient conditions for the boundedness of the Wigner transform, proving furthermore that these conditions are also necessary. This yields a complete characterization of boundedness for Weyl operators in L^p setting; compactness follows by approximation. We extend these results defining two scales of spaces, namely L_*^q (?^2d ) and L_?^q (R^2d ), respectively smaller and larger than the L^q (?^2d ),and showing that the Weyl correspondence is bounded on L_*^q (R^2d ) (and yields compact operators), whereas it is not on L_?^q (R^2d ). We conclude with a remark on weak‐type L^p boundedness (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A remark on partly dissipative reaction diffusion systems on IRn

The long time behavior of the solutions of some partly dissipative reaction diffusion systems is studied. We prove the existence of a compact （L^2 × L^2 - H^1 × L^2） attractor for a partly dissipative reaction diffusion system in Rn. This improves a previous result obtained by A. Rodrigues-Bernal and B. Wang concerning the existence of a compact （L^2 × L^2 - L^2 × L^2） attractor for the same system.     

Vector measure Maurey–Rosenthal-type factorizations and -sums of -spaces

Consider a vector measure of bounded variation m with values in a Banach space and an operator T:XL¹(m), where L¹(m) is the space of integrable functions with respect to m. We characterize when T can be factorized through the space L²(m) by means of a multiplication operator given by a function of L²(|m|), where |m| is the variation of m, extending in this way the Maurey–Rosenthal Theorem. We use this result to obtain information about the structure of the space L¹(m) when m is a sequential vector measure. In this case the space L¹(m) is an ℓ-sum of L¹-spaces.     

Characterization of Operators on the Dual of Hypergroups which Commute with Translations and Convolutions

For a locally compact group G, L^1 (G) is its group algebra and L^∞(G) is the dual of L^1 (G).Lau has studied the bounded linear operators T:L^∞(G)→L^∞(G) which commute with convolutions and translations. For a subspace H of L^∞(G), we know that M(L^∞(G),H), the Banach algebra of all bounded linear operators on L^∞(G) into H which commute with convolutions, has been studied by Pyre and Lau. In this paper, we generalize these problems to L(K)^*, the dual of a hypergroup algebra L(K) in a very general setting, i.e. we do not assume that K admits a Haar measure. It should be noted that these algebras include not only the group algebra L^1(G) but also most of the semigroup algebras.Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however,we succeed in getting some interesting results.     

Remarks on Normal Generation of Special Line Bundles on Algebraic Curves

Let L be a very ample line bundle with h ¹(L) ≥2 on a curve of genus g. We prove that L is normally generated if deg(L) ≥2g ? 1 ? 4h ¹(L) for large enough genus g.     

Dunford-pettis operators onL 1 and the Radon-Nikodym property

Using the duality between Dunford-Pettis operators onL ¹ and Pettis-Cauchy martingales, we prove that the Dunford-Pettis operators fromL ¹ intoL ¹ form a lattice. We show also that a Banach spaceX has the Radon-Nikodym property provided the Dunford-Pettis members of ℒ(L ¹,X) are representable. The lifting of dual valued Dunford-Pettis operators is investigated. Some remarks are included.     

Commutators of Littlewood-Paley operators

Let b ∈ Lloc(Rn) and L denote the Littlewood-Paley operators including the LittlewoodPaley g function,Lusin area integral and gλ* function. In this paper,the authors prove that the Lp boundedness of commutators [b,L] implies that b ∈ BMO(Rn) . The authors therefore get a characterization of the Lp-boundedness of the commutators [b,L]. Notice that the condition of kernel function of L is weaker than the Lipshitz condition and the Littlewood-Paley operators L is only sublinear,so the results obtained in the p...     

On the convergence of Fourier series of computable Lebesgue integrable functions

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of L^p‐computable functions (computable Lebesgue integrable functions) with a size notion, by introducing L^p‐computable Baire categories. We show that L^p‐computable Baire categories satisfy the following three basic properties. Singleton sets {f } (where f is L^p‐computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of L^p‐computable functions is not meager. We give an alternative characterization of meager sets via Banach‐Mazur games. We study the convergence of Fourier series for L^p‐computable functions and show that whereas for every p > 1, the Fourier series of every L^p‐computable function f converges to f in the L^p norm, the set of L¹‐computable functions whose Fourier series does not diverge almost everywhere is meager (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

L^p- L^q decay estimates of solutions to Cauchy problems of thermoviscoelastic systems

L^p- L^q decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. Then with the help of the information of characteristic roots for the coefficient matrix of the system, L^p- L^q decay estimate of parabolic type of solution to the Cauchy problem is obtained.     

Finitely purely atomic measures: Coincidence and rigidity properties

This paper is a natural continuation of the paper [2] by the same author. We shall prove that several coincidence and rigidity phenomena which usually do not appear are possible only in case the underlying measure space is trivial (i.e. is a finite union of atoms). Examples: coincidence of twoL ^p spaces, reflexivity ofL ¹, Radon—Nikodym property ofL ^∞, coincidence of Dunford, Pettis or Bochner integrability, coincidence of theL ^p space and of the weakL ^p space.     

Lp-Computability

In this paper we investigate conditions for L^p-computability which are in accordance with the classical Grzegorczyk notion of computability for a continuous function. For a given computable real number p ≥ 1 and a compact computable rectangle I ? ?^q, we show that an L^p function f ∈ L^p(I) is L^P-computable if and only if (i) f is sequentially computable as a linear functional and (ii) the L^p-modulus function of f is effectively continuous at the origin of ?^q.     

Uniqueness of Solutions on the Whole Time Axis to the Navier-Stokes Equations in Unbounded Domains

We consider the uniqueness of bounded continuous L^{3, ∞}-solutions on the whole time axis to the Navier-Stokes equations in 3-dimensional unbounded domains. Here, L^{p, q} denotes the scale of Lorentz spaces. Thus far, uniqueness of such solutions to the Navier-Stokes equations in unbounded domain, roughly speaking, is known only for a small solution in BC(?; L^{3, ∞}) within the class of solutions which have sufficiently small L^∞(L^{3, ∞})-norm. In this paper, we discuss another type of uniqueness theorem for solutions in BC(?; L^{3, ∞}) using a smallness condition for one solution and a precompact range condition for the other one. The proof is based on the method of dual equations.     

Cyclic Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Bold">R</Emphasis>), 1 \le <Emphasis Type="Italic">p</Emphasis> < \infty

Functions whose translates span L ^p (R) are called L ^p-cyclic functions. For a fixed p \memb [1, \infty], we construct Schwartz-class functions which are L ^r -cyclic for r > p and not L ^r - cyclic for r \le p. We then construct Schwartz-class functions which are L ^r -cyclic for r \ge p and not L ^r -cyclic for r < p. The constructions differ for p \memb (1, 2) and p > 2.     

A remark on a maximal function over a Cantor set of directions

LetM _e ⁰ be the maximal operator over segments of length 1 with directions belonging to a Cantor set. It has been conjectured that this operator is bounded onL ². We consider a sequence of operators over finite sets of directions converging toM _e ⁰ . We improve the previous estimate for the (L ²,L ²)-norm of these particular operators. We also prove thatM _e ⁰ is bounded from some subsets ofL ² toL ². These subsets are composed of positive functions whose Fourier transforms have a very weak decay or are supported in a vertical strip. Partially supported by Spanish DGICYT grant no. PB90-0187.     

The effect of spatial quadrature on finite element galerkin approximations to hyperbolic integro-differential equations

The purpose of this paper is to study the effect of numerical quadrature on the finite element approximations to the solutions of hyperbolic intego-differential equations. Both semidiscrete and fully discrete schemes are analyzed and optimal estimates are derived in L ^∞(H ¹)L ^∞(L ²) norms and quasi-optimal estimate in L ^∞(L ^∞) norm using energy arguments. Further, optimal L(L ²)-estimates are shown to hold with minimal smoothness assumptions on the initial functions. The analysis in the present paper not only improves upon the earlier results of Baker and Dougalis [SIAM J. Numer. Anal. 13 (1976), pp. 577-598] but also confirms the minimum smoothness assumptions of Rauch [SIAM J. Numer. Anal. 22 (1985), pp. 245-249] for purely second order hyperbolic equation with quadrature.     

On uniqueness of invariant measures for finite‐ and infinite‐dimensional diffusions

We prove uniqueness of “invariant measures,” i.e., solutions to the equation L*μ = 0 where L = Δ + B · ∇ on ℝⁿ with B satisfying some mild integrability conditions and μ being a probability measure on ℝⁿ. This solves an open problem posed by S. R. S. Varadhan in 1980. The same conditions are shown to imply that the closure of L on L¹(μ) generates a strongly continuous semigroup having μ as its unique invariant measure. The question whether an extension of L generates a strongly continuous semigroup on L¹(μ) and whether such an extension is unique is addressed separately and answered positively under even weaker local integrability conditions on B. The special case when B is a gradient of a function (i.e., the “symmetric case”) in particular is studied and conditions are identified ensuring that L*μ = 0 implies that L is symmetric on L²(μ) or L*μ = 0 has a unique solution. We also prove infinite‐dimensional analogues of the latter two results and a new elliptic regularity theorem for invariant measures in infinite dimensions. © 1999 John Wiley & Sons, Inc.     

Orlicz-Hardy spaces associated with operators

Let L be a linear operator in L2(Rn) and generate an analytic semigroup {e-tL}t 0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0, ∞) be of upper type 1 and of critical lower type p0(ω) ∈ (n/(n + θ(L)), 1] and ρ(t) = t-1/ω-1(t-1) for t ∈ (0, ∞). We introduce the Orlicz-Hardy space Hω, L(Rn) and the BMO-type space BMOρ, L(Rn) and establish the John-Nirenberg inequality for BMOρ, L(Rn) functions and the duality relation between Hω, L(Rn) and BMOρ, L...     

Dual frames in connected with generalized sampling in shift-invariant spaces

The aim of this article is to derive stable generalized sampling in a shift-invariant space by using some special dual frames in L²(0,1). These sampling formulas involve samples of filtered versions of the functions in the shift-invariant space. The involved samples are expressed as the frame coefficients of an appropriate function in L²(0,1) with respect to some particular frame in L²(0,1). Since any shift-invariant space with stable generator is the image of L²(0,1) by means of a bounded invertible operator, our generalized sampling is derived from some dual frame expansions in L²(0,1).