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.     

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.     

The limiting angle of certain riemannian brownian motions

Let M be a Cartan-Hadamard manifold of dimension d ≧ 3, let p ? M and x = exp {r(x)θ(x)} be geodesic polar coordinates with pole p and let X be the Brownian motion on M. Let Sect_M(x) denote the sectional curvature of any plane section in M_x. We prove that for each c > 2, there is a 0 < β < 1 such that if - L²r(x)^2β ≦ Sect_M(x) ≦ -cr(x)^?2 for all x in the complement of a compact set, then lim_{t → ∞} θ(X_t) exists a.s. and defines a nontrivial invariant random variable. The Dirichlet problem at infinity and a conjecture of Greene and Wu are also discussed.     

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.     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

L 1→ L q Poincaré Inequalities for 0 < q < 1 Imply Representation Formulas

Given two doubling measures μ and ν in a metric space (S, ρ) of homogeneous type, let B ₀⊂S be a given ball. It has been a well-known result by now (see [1–4]) that the validity of an L ¹→L ¹ Poincaré inequality of the following form: for all metric balls B⊂B ₀⊂S, implies a variant of representation formula of fractional integral type: for ρ-a.e. x∈B ₀, One of the main results of this paper shows that an L ¹ to L ^q Poincaré inequality for some 0 < q < 1, i.e., for all metric balls B⊂B ₀, will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition, also implies the same formula. Analogous theorems related to high-order Poincaré inequalities and Sobolev spaces in metric spaces are also proved. Received December 27, 2000, Accepted May 28, 2001     

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.     

Oscillation criteria for fourth‐order differential equations with middle term

Oscillation criteria for self‐adjoint fourth‐order differential equations were established for various conditions on the coefficients r(x) > 0, q(x) and p(x). However, most of these results deal with the case when lim_{x → ∞}∫^x₁q(s) ds < +∞. In this note we give a new oscillation criterion in the case when this condition is not fulfilled, in particular when q(x)↗ + ∞ (even with exponential growth).     

Boundedness Properties of Pseudo-Differential and Calderón-Zygmund Operators on Modulation Spaces

In this article, we study the boundedness of pseudo-differential operators with symbols in S _ρ,δ ^m on the modulation spaces M ^p,q . We discuss the order m for the boundedness Op(S _ρ,δ ^m )⊂ℒ(M ^p,q ) to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space M ^p,q with q≠2. This unboundedness is still true even if we assume a generalized T(1) condition. These results are induced by the unboundedness of pseudo-differential operators on M ^p,q whose symbols are of the class S _1,δ ⁰ with 0<δ<1.      

Operator-valued Fourier Haar multipliers

Criteria are given to ensure the boundedness of Fourier Haar multiplier operators from Lp([0,1],X) to Lq([0,1],Y) where the Fourier Haar multiplier sequences come not from R, as in the classical setting, but rather from the space of bounded linear operators from a Banach space X into a Banach space Y.     

On holomorphic domination, I

Let X be a separable Banach space and u:X→R locally upper bounded. We show that there are a Banach space Z and a holomorphic function h:X→Z with u(x)<‖h(x)‖ for x∈X. As a consequence we find that the sheaf cohomology group Hq(X,O) vanishes if X has the bounded approximation property (i.e., X is a direct summand of a Banach space with a Schauder basis), O is the sheaf of germs of holomorphic functions on X, and q?1. As another consequence we prove that if f is a C¹-smooth -closed (0,1)-form on the space X=L₁[0,1] of summable functions, then there is a C¹-smooth function u on X with on X.     

The Dichotomy Between Traces on d-sets F in R^n and the Density of D（R^n／Г） in Function Spaces

A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are introduced and calculated.     

Boundedness of an oscillating multiplier on Triebel-Lizorkin spaces

Abstract In this paper, we study Triebel-Lizorkin space estimates for an oscillating multiplier mΩ,α,β. This operator was initially studied by Wainger and by Fefferman-Stein in the Lebesgue spaces. We obtain the boundedness results on the Triebel-Lizorkin space Fpα,q（R^n） for different p, q.     

The Fourier transform of order statistics with applications to Lorentz spaces

We present a formula for the Fourier transforms of order statistics in ℝ ⁿ showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in ℝ ⁿ . Fora ₁≥...≥a _n≥0 andq>0, denote by ℓ _w,q ⁿ then-dimensional Lorentz space with the norm ‖(x ₁,...,x _n)‖=(a ₁(x ₁ ^* ) ^q +...+a _n(x _n ^* ) ^q )^1/q , where (x ₁ ^* ,...,x _n ^* ) is the non-increasing permutation of the numbers |x ₁|,...,|x _n|. We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spaces intoL _q [10] to prove that, forn≥3 andq≤1, the space ℓ _w,q ⁿ is isometric to a subspace ofL _q if and only if the numbersa ₁,...,a _n form an arithmetic progression. Forq>1, all the numbersa _i must be equal so that ℓ _w,q ⁿ = ℓ _q ⁿ . Consequently, the Lorentz function spaceL _w,q(0, 1) is isometric to a subspace ofL _q if and only ifeither 0<q<∞ and the weightw is a constant function (so thatL _w,q=L_q),or q≤1 andw(t) is a decreasing linear function. Finally, we relate our results to the theory of positive definite functions. Both authors were supported in part by the NSF Workshop in Linear Analysis and Probability held at Texas A&M University in August 1993. The work was done during the first author’s visit to Texas A&M University.     

On the duals of Lp spaces with 0<p<1

Given a measure space < Ω,m,μ >, a locally bounded, Hausdorff topological linear space < X, τ > and a real number 0<p<1, one can define the space L_p(Ω,m,μ,X), which is, under certain assumptions, a Fréchet space if endowed with a suitable topology. M.M. Day [1] has given a necessary and sufficient condition, in terms of the properties of the measure space < Ω,m,μ >, for the dual of L_p(Ω,m,μ,C) to be trivial. In this paper a different proof along with a slight generalization is given for this result, using standard and elementary measure theoretic arguments. This revised version was published online in June 2006 with corrections to the Cover Date.     

SOME ASPECTS OF THE LIFTING PROPERTY OF THE SPACE £∞ (μ, X)

Abstract

Using a lifting of £_∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £_∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £_∞ ((O, 1), X) → £_∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £_∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X.

As an application we introduce the space £_∞ ^strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £_∞ ^strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £_∞(μ Y**)). Finally we investigate the measurability property of functions in £_∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y).     

Best m-ter m trigono metric approxi mation for the classes $ B_{p,{{\uptheta }}}^r $ of functions of low s moothness

We obtain an exact-order estimate for the best m-term trigonometric approximation of the Besov classes B_p,\uptheta ^r B_{p,{{\uptheta }}}^r of periodic functions of many variables of low smoothness in the space L _q , 1 < p ≤ 2 < q < ∞.     

Geometrical implications of the existence of very smooth bump functions in Banach spaces

We show that ifX is a Banach space and if there is a non-zero real-valuedC ^∞-smooth function onX with bounded support, then eitherX contains an isomorphic copy ofc ₀(N), or there is an integerk greater than or equal to 1 such thatX is of exact cotype 2k and, in this case,X contains an isomorphic copy ofl ^2k(N). We also show that ifX is a Banach space such that there is onX a non-zero real-valuedC ⁴-smooth function with bounded support and ifX is of cotypeq forq<4, thenX is isomorphic to a Hilbert space.     

Characterizations of new Cohen summing bilinear operators

We prove two characterizations of new Cohen summing bilinear operators. The first one is: Let X, Y and Z be Banach spaces, 1 < p < ∞, V : X × Y → Z a bounded linear operator and n ≥ 2 a natural number. Then V is new Cohen p-summing if and only if for all Banach spaces X₁,?…?, X_n and all p-summing operators U : X₁ × · · · × X_n → X, the operator V ? (U, I_Y) : X₁ × · · · × X_n × Y → Z is -summing. The second result is: Let H be a Hilbert space,, Y, Z Banach spaces and V : H × Y → Z a bounded bilinear operator and 1 < p < ∞. Then V is new Cohen p-summing if and only if for all Banach spaces E and all p-summing operators U : E → H, the operator V ? (U, I_Y) is (p, p*)-dominated.     

Entropy numbers of Sobolev embeddings of radial Besov spaces

Let be the radial subspace of the Besov space . We prove the independence of the asymptotic behavior of the entropy numbers

from the difference s₀−s₁ as long as the embedding itself is compact. In fact, we shall show that

This is in a certain contrast to earlier results on entropy numbers in the context of Besov spaces B_p,q^s(Ω) on bounded domains Ω.