Sharp bounds for integral means

We introduce a bound M of f, ‖f‖_∞?M?2‖f‖_∞, which allows us to give for 0?p<∞ sharp upper bounds, and for −∞<p<0 sharp lower bounds for the average of |f|^p over E if the average of f over E is zero. As an application we give a new proof of Grüss's inequality estimating the covariance of two random variables. We also give a new estimate for the error term in the trapezoidal rule.     

Monotonicity and best approximation in Orlicz-Sobolev spaces with the Luxemburg norm

Criteria for strict monotonicity, upper (lower) locally uniform monotonicity and uniform monotonicity of Orlicz-Sobolev spaces with the Luxemburg norm are given. Some applications to best approximation are presented.     

Weyl product algebras and modulation spaces

We discuss algebraic properties of the Weyl product acting on modulation spaces. For a certain class of weight functions ω we prove that is an algebra under the Weyl product if p∈[1,∞] and 1?q?min(p,p^′). For the remaining cases p∈[1,∞] and min(p,p^′)<q?∞ we show that the unweighted spaces Mp,q are not algebras under the Weyl product.     

On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces

Generalized Orlicz–Lorentz sequence spaces λ_φ generated by Musielak‐Orlicz functions φ satisfying some growth and regularity conditions (see [28] and [33]) are investigated. A regularity condition δ^λ ₂ for φ is defined in such a way that it guarantees many positive topological and geometric properties of λ_φ. The problems of the Fatou property, the order continuity and the Kadec–Klee property with respect to the uniform convergence of the space λ_φ are considered. Moreover, some embeddings between λ_φ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of λ_φ, their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [19], [4] and [17]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

MONOTONE POINTS IN ORLICZ-BOCHNER SEQUENCE SPACES

In Orlicz-Bochner sequence spaces endowed with Orlicz norm and Luxemburg norm, points of lower monotonicity, upper monotonicity, lower local uniform monotonicity and upper local uniform monotonicity are characterized.     

Local monotonicity structure of Calderón-Lozanovskiǐ spaces

We first prove that if x is an element on the unit sphere of arbitrary Köthe space E, x is strickly positive μ-a.e. and x is an LM-point, then x is an UM-point. Criteria for lower and upper monotone points in Calderón-Lozanovskiǐ spaces E_? are presented. Points of lower local uniform monotonicity and upper local uniform monotonicity in E_? are also considered. Some sufficient conditions and necessary conditions for these properties of a given point x in S(E_?⁺) are given.     

Semismall perturbations, semi-intrinsic ultracontractivity, and integral representations of nonnegative solutions for parabolic equations

We consider nonnegative solutions of a parabolic equation in a cylinder D×I, where D is a noncompact domain of a Riemannian manifold and I=(0,T) with 0<T?∞ or I=(−∞,0). Under the assumption [SSP] (i.e., the constant function 1 is a semismall perturbation of the associated elliptic operator on D), we establish an integral representation theorem of nonnegative solutions: In the case I=(0,T), any nonnegative solution is represented uniquely by an integral on (D×{0})∪(M_∂D×[0,T)), where M_∂D is the Martin boundary of D for the elliptic operator; and in the case I=(−∞,0), any nonnegative solution is represented uniquely by the sum of an integral on M_∂D×(−∞,0) and a constant multiple of a particular solution. We also show that [SSP] implies the condition [SIU] (i.e., the associated heat kernel is semi-intrinsically ultracontractive).     

On locally compact quantum groups whose algebras are factors

In this paper we are interested in examples of locally compact quantum groups (M,Δ) such that both von Neumann algebras, M and the dual , are factors. There is a lot of known examples such that are respectively of type (I_∞,I_∞) but there is no example with factors of other types. We construct new examples of type (I_∞,II_∞), (II_∞,II_∞) and (IIIλ,IIIλ) for each λ∈[0,1]. We also show that there is no such example with M or a finite factor.     

Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation

We consider a free boundary problem modeling tumor growth in fluid-like tissue. The model equations include a diffusion equation for the nutrient concentration, and the Stokes equation with a source which represents the proliferation of tumor cells. The proliferation rate μ and the cell-to-cell adhesiveness γ which keeps the tumor intact are two parameters which characterize the “aggressiveness” of the tumor. For any positive radius R there exists a unique radially symmetric stationary solution with radius r=R. For a sequence μ/γ=Mn(R) there exist symmetry-breaking bifurcation branches of solutions with free boundary r=R+εYn,0(θ)+O(ε²) (n even ?2) for small |ε|, where Yn,0 is the spherical harmonic of mode (n,0). Furthermore, the smallest Mn(R), say Mn_∗(R), is such that n_∗=n_∗(R)→∞ as R→∞. In this paper we prove that the radially symmetric stationary solution with R=RS is linearly stable if μ/γ<N_∗(RS,γ) and linearly unstable if μ/γ>N_∗(RS,γ), where N_∗(RS,γ)?Mn_∗(RS), and we prove that strict inequality holds if γ is small or if γ is large. The biological implications of these results are discussed at the end of the paper.     

Magidor-like and radin-like forcing

We force over a model M of ZF+κ→(κ)^<γ to obtain M[G] with cf(κ)=γ. The method is reminiscent of Magidor-forcing but uses no choice. Mimicing Radin-forcing, we generalize this for strong partition cardinals κ to add a subset of κ while preserving all cardinalities, cofinalities and κ's measurability. We apply these techniques to construct models of unusual partition properties, such as ω₂→[ω₂]^ω₁ but $\text{ω}{}_2\text{?[ω}{}_2\text{]}{}^{\mathrm{\omega }}$.     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.     

On weak uniform normal structure in weighted Orlicz sequence spaces

It is proved that a weighted Orlicz sequence space ?M(w), equipped with Luxemburg or Amemiya norm has weak uniform normal structure iff ?M(w)≅hM(w) for wide class of weight sequences . An example is constructed, where M has not Δ₂-condition but by choosing a suitable weight sequence limn→∞wn=∞ we get that ?M(w) has weak uniform normal structure.     

Nonlinear random ergodic theorems for affine operators

Let (Ω,ß,μ) be a finite measure space and let (S,F,ν) be another probability measure space on which a measure preserving transformation φ is given. We introduce the so-called affine systems and prove a vector-valued nonlinear random ergodic theorem for the random affine system determined by a strongly F-measurable family of affine operators, where B is a reflexive Banach space, is a strongly F-measurable family of linear contractions on L₁(Ω,B) as well as on L_∞(Ω,B) and ξ is a function in (I−T)Lp(S×Ω,B) (1?p<∞) with the operator T defined by Tf(s,ω)=[Tsfφs](ω) which denotes the F⊗ß-measurable version of Tsfφs(ω). Moreover, some variant forms of the nonlinear random ergodic theorem are also obtained with some examples of affine systems for which the nonlinear ergodic theorems fail to hold.     

Necessary and sufficient conditions for the boundedness of rough B-fractional integral operators in the Lorentz spaces

In this paper, the necessary and sufficient conditions are found for the boundedness of the rough B-fractional integral operators from the Lorentz spaces Lp,s,γ to Lq,r,γ, 1<p<q<∞, 1?r?s?∞, and from L1,r,γ to Lq,∞,γ≡WLq,γ, 1<q<∞, 1?r?∞. As a consequence of this, the same results are given for the fractional B-maximal operator and B-Riesz potential.     

Friedrichs extensions of Schrödinger operators with singular potentials

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d²/dx²+Bx²+Ax⁻²+λx^−α (B>0, A?0) in L₂(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L₂(0,∞), namely C₀^∞(0,∞) and D(T_2,F)∩D(M_λ,α), where the latter is a subspace of the Sobolev space W_2,2(0,∞). Adjoints of these differential operators on C₀^∞(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) in L₂(0,∞) lead to the other adjoints. Further density properties C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T_2,F)∩D(M_λ,α).     

Orlicz-Sobolev空间单调性及其在最佳逼近中的应用

改进了Hudzik,Kurc关于最佳逼近中的结果,给出了赋Orlicz范数的Orlicz- Sobolev空间具有一致单调性、局部一致单调性和严格单调性的充要条件、单调系数的数值,以及在最佳逼近中的应用.     

On the calculation of the James constant of Lorentz sequence spaces

In [M. Kato, L. Maligranda, On James and Jordan-von Neumann constants of Lorentz sequence spaces, J. Math. Anal. Appl. 258 (2001) 457-465], the James constant of the 2-dimensional Lorentz sequence space d(2)(ω,q) is computed in the case where 2?q<∞. It is an open problem to compute it in the case where 1?q<2. In this paper, we completely determine the James constant of d(2)(ω,q) in the case where 1?q<2.     

Structure of boundary blow-up solutions for quasi-linear elliptic problems II: small and intermediate solutions

The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δ_pu=λf(u), u=∞ on ∂Ω, 1<p<∞, is studied in a bounded smooth domain , for a class of nonlinearities f∈C¹((0,∞)?{z₂})∩C⁰[0,∞) satisfying f(0)=f(z₁)=f(z₂)=0 with 0<z₁<z₂, f<0 in (0,z₁)∪(z₂,∞), f>0 in (z₁,z₂). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2.