Fonctions lipschitziennes et espaces de Sobolev fractionnaires

We consider a semigroup of Markovian and symmetric operators to which we associate fractional Sobolev spaces D^α_p (0 < α < 1 and 1 < p < ∞) defined as domains of fractional powers (−A_p)^α/2, where A_p is the generator of the semigroup in L^p. We show under rather general assumptions that Lipschitz continuous functions operate by composition on D^α_p if p ≥ 2. This holds in particular in the case of the Ornstein-Uhlenbeck semigroup on an abstract Wiener space.     

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c₀ and ?_p for 1?p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E?(⊕?₂ⁿ)_c0, that is, E is the c₀-direct sum of the finite-dimensional Hilbert spaces ?₂¹,?₂²,…,?₂ⁿ,… .     

Process convergence of self-normalized sums of i.i.d. random variables coming from domain of attraction of stable distributions

In this paper we show that the continuous version of the self-normalized process Y _n,p (t)?=?S _n (t)/V _n,p ?+?(nt???[nt])X _[nt]?+?1/V _n,p ,0?<?t?≤?1; p?>?0 where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}=(\sum_{i=1}^{n}|X_i|^p)^{1/p}$ and X _i i.i.d. random variables belong to DA(α), has a non-trivial distribution iff p?=?α?=?2. The case for 2?>?p?>?α and p?≤?α?<?2 is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörg? et al. who showed Donsker’s theorem for Y _n,2(·), i.e., for p?=?2, holds iff α?=?2 and identified the limiting process as a standard Brownian motion in sup norm.     

Banach Spaces with Property (M) and their Szlenk Indices

We show that a separable Banach space with property (M*) has a Szlenk index equal to ω₀, and a norm with an optimal modulus of asymptotic uniform smoothness. From this we derive a condition on the Szlenk functions of the space and its dual which characterizes embeddability into c ₀ or an ℓ _p -sum of finite dimensional spaces. We also prove that two Lipschitz-isomorphic Orlicz sequence spaces contain the same ℓ _p -spaces.      

Multidimensional integral operators with homogeneous kernels of compact type and multiplicatively weakly oscillating coefficients

In the space L _p (? ⁿ ), 1 < p < +∞, we consider a new class of integral operators with kernels homogeneous of degree ?n, which includes the class of operators with homogeneous SO(n)-invariant kernels; we study the Banach algebra generated by such operators with multiplicatively weakly oscillating coefficients. For operators from this algebra, we define a symbol in terms of which we formulate a Fredholm property criterion and derive a formula for calculating the index. An important stage in obtaining these results is the establishment of the relationship between the operators of the class under study and the operators of one-dimensional convolution with weakly oscillating compact coefficients.     

Existence of multiple positive solutions for one-dimensional p-Laplacian operator

In this paper, we consider the multipoint boundary value problem for one-dimensional p-Laplacian $$(\phi_{p}(u'))'+f(t,u,u')=0,\quad t\in [0,1],$$ subject to the boundary value conditions: $$u'(0)=\sum_{i=1}^{n-2}\alpha_{i}u'(\xi_{i}),\qquad u(1)=\sum_{i=1}^{n-2}\beta_{i}u(\xi_{i}),$$ where φ _p (s)=|s| ^p?2?s,p>1;ξ _i ∈(0,1) with 0<ξ ₁<ξ ₂<???<ξ _n?2<1 and α _i ,β _i satisfy α _i ,β _i ∈[0,∞),0≤∑ _i=1 ^n?2 α _i <1 and 0≤∑ _i=1 ^n?2 β _i <1. Using a fixed point theorem for operators in a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.     

Some weighted estimates for Littlewood-Paley functions and radial multipliers

We prove some weighted estimates for certain Littlewood-Paley operators on the weighted Hardy spaces H_w^p (0<p?1) and on the weighted L^p spaces. We also prove some weighted estimates for the Bochner-Riesz operators and the spherical means.     

Martingale Transforms and Their Projection Operators on Manifolds

We prove the boundedness on L ^p , 1?<?p?<?∞, of operators on manifolds which arise by taking conditional expectation of transformations of stochastic integrals. These operators include various classical operators such as second order Riesz transforms and operators of Laplace transform-type.     

Weighted Weak Behaviour of Fourier-Jacobi Series

Let w(x) = (1 - x)^α (1 + x)^β be a Jacobi weight on the interval [-1, 1] and 1 < p < ∞. If either α > ?1/2 or β > ?1/2 and p is an endpoint of the interval of mean convergence of the associated Fourier-Jacobi series, we show that the partial sum operators S_n are uniformly bounded from L^p,1 to L^p,∞, thus extending a previous result for the case that both α, β > ?1/2. For α, β > ?1/2, we study the weak and restricted weak (p, p)-type of the weighted operators f→uS_n(u^?1f), where u is also Jacobi weight.     

Boundedness of the fractional maximal operator in generalized Morrey space on the Heisenberg group

In this paper we study the fractional maximal operator M _α , 0 ≤ α < Q on the Heisenberg group ? _n in the generalized Morrey spaces M _{p, ?}(? _n ), where Q = 2n + 2 is the homogeneous dimension of ? _n . We find the conditions on the pair (? ₁, ? ₂) which ensures the boundedness of the operator M _α from one generalized Morrey space M _{p, ?1}(? _n ) to another M _{q, ?2}(? _n ), 1 < p < q < ∞, 1/p?1/q = α/Q, and from the space M _{1, ?1}(? _n ) to the weak space WM _{q, ?2}(? _n ), 1 < q < ∞, 1 ? 1/q = α/Q. We also find conditions on the φ which ensure the Adams type boundedness of M _α from $M_{p,\phi ^{\tfrac{1} {p}} } \left( {\mathbb{H}_n } \right)$ to $M_{q,\phi ^{\tfrac{1} {q}} } \left( {\mathbb{H}_n } \right)$ for 1 < p < q < ∞ and from M _{1, ?}(? _n ) to $WM_{q,\phi ^{\tfrac{1} {q}} } \left( {\mathbb{H}_n } \right)$ for 1 < q < ∞. As applications we establish the boundedness of some Schrödinger type operators on generalized Morrey spaces related to certain nonnegative potentials V belonging to the reverse Hölder class B _∞(” _n ).     

Uniqueness of positive radial solutions for Dirichlet problems on annular domains

This paper deals with the uniqueness of positive radial solutions to Dirichlet problems on annular domains in Rn, n?3. As an application we can obtain the results to equation Δu+up−α₁u−α₀=0, where p>1, α₁?0, α₀?0 and α₁+α₀>0.     

Gaps of operators

We construct examples which distinguish clearly the classes of p-hyponormal operators for 0<p?∞. In addition, we show that those examples classify the classes of w-hyponormal, absolute-p-paranormal, and normaloid operators on the complex Hilbert space. Only a few examples of p-hyponormal operators have been examined. Our technique can provide many examples related to the above operators.     

Vector-valued multiparameter singular integrals and pseudodifferential operators

We consider multiparameter singular integrals and pseudodifferential operators acting on mixed-norm Bochner spaces Lp₁,…,pN(Rn₁×?×RnN;X) where X is a UMD Banach space satisfying Pisier's property (α). These geometric conditions are shown to be necessary. We obtain a vector-valued version of a result by R. Fefferman and Stein, also providing a new, inductive proof of the original scalar-valued theorem. Then we extend a result of Bourgain on singular integrals in UMD spaces with an unconditional basis to a multiparameter situation. Finally we carry over a result of Yamazaki on pseudodifferential operators to the Bochner space setting, improving the known vector-valued results even in the one-parameter case.     

Existence and iteration of positive solutions for a three-point boundary value problem of second order integro-differential equation with p-Laplace operator

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for the following three-point boundary value problem $$\left\{\begin{array}{l}(\phi_p(u'))'(t)+q(t)f\left(u(t),u'(t),Tu(t),Su(t)\right)=0,\quad0 < t< 1,\\u'(0)=\alpha u'(\eta),\quad u(1)=g(u'(1)),\end{array}\right.$$ where ? _p (s)=|s| ^p?2 s,p>1,α∈[0,1),η∈(0,1), T and S are all linear operators, g(t) is continuous and nonincreasing on (?∞,0]. The main tools are monotone iterative technique and numerical simulation. We illustrate our results by one example, and give its numerical results by iterative scheme.     

Sharp weighted bounds without testing or extrapolation

We give a short proof of the sharp weighted bound for sparse operators that holds for all p,1?<?p?< ??. By recent developments this implies the bounds hold for any Calderón?CZygmund operator. The novelty of our approach is that we avoid two techniques that are present in other proofs: two weight inequalities and extrapolation. Our techniques are applicable to fractional integral operators as well.     

Sparsity and non-Euclidean embeddings

We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables one to construct embeddings of ? _p ⁿ , p > 0, into various types of Banach or quasi-Banach spaces. In particular, for 0 < r < p < 2 with r ≤ 1, we construct a family of operators that embed ? _p ⁿ into $\ell _r^{(1 + \eta )n}$ , with sharp polynomial bounds in η > 0.     

A note on geometrical properties of Banach spaces using ψ-direct sums

Let X be a Banach space and ψ a continuous convex function on [0,1] satisfying certain conditions. Let Xψ_⊕X be the ψ-direct sum of X. In this note, we characterize the strict convexity, uniform convexity and uniformly non-squareness of Banach spaces using ψ-direct sums, which extends the well-known characterization of these spaces.     

Homogenization of monotone operators under conditions of coercitivity and growth of variable order

We obtain a homogenization procedure for the Dirichlet boundary-value problem for an elliptic equation of monotone type in the domain Ω ? ? ^d . The operator of the problem satisfies the conditions of coercitivity and of growth with variable order p _? (x) = p(x/?); furthermore, p(y) is periodic, measurable, and satisfies the estimate 1 < α ≤ p(y) ≤ β < ∞, while the parameter ? > 0 tends to zero. Here α and β are arbitrary constants. Taking Lavrent’ev’s phenomenon into account, we consider solutions of two types: H- and W-solutions. Each of the solution types calls for a distinct homogenization procedure. Its justification is carried out by using the corresponding version of the lemma on compensated compactness, which is proved in the paper.     

Non Integral Regularizing Operators on Lp- Spaces

We present bounded positivity preserving operators from L_p(?) to L_q (?), for 1 < p < ∞, 1/p-1/q < 1/2, which are not integral operators.