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.     

Fractional integrals, potential operators and two-weight, weak type norm inequalities on spaces of homogeneous type

We prove two-weight, weak type norm inequalities for potential operators and fractional integrals defined on spaces of homogeneous type. We show that the operators in question are bounded from L^p(v) to L^q,∞(u), 1<p?q<∞, provided the pair of weights (u,v) verifies a Muckenhoupt condition with a “power-bump” on the weight u.     

Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains

For D, a bounded Lipschitz domain in Rⁿ, n ? 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L²(?D) and various subspaces of L²(?D). For 1 < p ? 2 and data in L^p(?D) with first derivatives in L^p(?D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in L^p(?D). When n = 2 the question of the invertibility of the layer potentials on every L^p(?D), 1 < p < ∞, is answered.     

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.     

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.     

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.     

Direct sums and the Szlenk index

For α an ordinal and 1<p<∞, we determine a necessary and sufficient condition for an ?p-direct sum of operators to have Szlenk index not exceeding ωα. It follows from our results that the Szlenk index of an ?p-direct sum of operators is determined in a natural way by the behaviour of the ε-Szlenk indices of its summands. Our methods give similar results for c₀-direct sums.     

The spectra of composition operators on Hp

For H^p, 1 ? p < ∞, composition operators C_?, defined by $\text{C}{}_{\mathrm{?}}\text{(?) = ? ° ?}$ for $\text{? ? H}{}^{\mathrm{p}}$, ? analytic on $\text{D = {z ¦ ¦ z ¦ < 1}}$ are considered, and their spectra determined in the case where ? is analytic on an open region containing D?.     

Restricting Hecke-Siegel operators to Jacobi modular forms

We compute the action of Hecke operators on Jacobi forms of “Siegel degree” n and m×m index M, provided 1?j?n−m. We find they are restrictions of Hecke operators on Siegel modular forms, and we compute their action on Fourier coefficients. Then we restrict the Hecke-Siegel operators T(p), Tj(p²) (n−m<j?n) to Jacobi forms of Siegel degree n, compute their action on Fourier coefficients and on indices, and produce lifts from Jacobi forms of index M to Jacobi forms of index M^′ where detM|detM^′. Finally, we present an explicit choice of matrices for the action of the Hecke operators on Siegel modular forms, and for their restrictions to Jacobi modular forms.     

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.     

Sufficiency of Favard's condition for a class of band-dominated operators on the axis

The purpose of this paper is to show that, for a large class of band-dominated operators on ?^∞(Z,U), with U being a complex Banach space, the injectivity of all limit operators of A already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of A, which, on the other hand, is often equivalent to the Fredholmness of A. As a consequence, for operators A in the Wiener algebra, we can characterize the essential spectrum of A on ?p(Z,U), regardless of p∈[1,∞], as the union of point spectra of its limit operators considered as acting on ?^∞(Z,U).     

Boundary values of holomorphic semigroups of unbounded operators and similarity of certain perturbations

We obtain sufficient conditions for a “holomorphic” semigroup of unbounded operators to possess a boundary group of bounded operators. The theorem is applied to generalize to unbounded operators results of Kantorovitz about the similarity of certain perturbations. Our theory includes a result of Fisher on the Riemann-Liouville semigroup in L^p(0, ∞) 1 < p < ∞. In this particular case we give also an alternative approach, where the boundary group is obtained as the limit of groups in the weak operator topology.     

On a Liouville-type comparison principle for solutions of quasilinear elliptic inequalities

We characterize in terms of monotonicity basic properties of quasilinear elliptic partial differential operators which make it possible to obtain a Liouville-type comparison principle for entire solutions of quasilinear elliptic partial differential inequalities of the form A(u)+|u|^q?1u?A(v)+|v|^q?1v, which belong only locally to the corresponding Sobolev spaces on $\text{R}{}^{\mathrm{n}}\text{,}\text{n?2}$. We establish that such properties are inherent for a wide class of quasilinear elliptic partial differential operators. Typical examples of such operators are the p-Laplacian and its well-known modifications for 1<p?2. To cite this article: V.V. Kurta, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

The boundedness and the fredholm property of integral operators with anisotropically homogeneous kernels of compact type and variable coefficients

In the space L _p (? ⁿ ), 1 < p < ??, we study a new wide class of integral operators with anisotropically homogeneous kernels. We obtain sufficient conditions for the boundedness of operators from this class. We consider the Banach algebra generated by operators with anisotropically homogeneous kernels of compact type and multiplicatively slowly oscillating coefficients. We establish a relationship between this algebra and multidimensional convolution operators, and construct a symbolic calculus for it. We also obtain necessary and sufficient conditions for the Fredholm property of operators from this algebra.     

Topological methods in solvability theory of multidimensional pair integral operators with homogeneous kernels of compact type

The problem of getting effective Fredholm conditions for operators with bihomogeneous kernels reduces to the question of invertibility for families of operators with homogeneous kernels and to the calculation of homotopy invariants for spaces of Fredholm and invertible operators of that type. The purpose of the present paper is to study integral operators with homogeneous kernels of compact type in L _p (? ⁿ ), 1 < p < +??. The classes of homotopy equivalence for the spaces of Fredholm and invertible operators in the C*-algebra of pair operators with homogeneous kernels of compact type are calculated by means of operator K-theory.     

The fractional Hausdorff operators on the Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

In this paper, we study the high-dimensional fractional Hausdorff operators and establish their boundedness on the real Hardy spaces H ^p (? ⁿ ) for 0 < p < 1.     

The p-approximation property in terms of density of finite rank operators

We characterize the p-approximation property (p-AP) introduced by Sinha and Karn [D.P. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of ?p, Studia Math. 150 (2002) 17-33] in terms of density of finite rank operators in the spaces of p-compact and of adjoints of p-summable operators. As application, the p-AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi-p-nuclear operators. This relates the p-AP to Saphar's approximation property APp^′. As another application, the p-AP is characterized via a trace condition, allowing to define the trace functional on certain subspaces of the space of nuclear operators.     

Fourier multipliers for Sobolev spaces on the Heisenberg group

In this paper, it is shown that the class of right Fourier multipliers for the Sobolev space W ^k,p (H ⁿ ) coincides with the class of right Fourier multipliers for L ^p (H ⁿ ) for k ∈ ?, 1 < p < ∞. Towards this end, it is shown that the operators R _j $ \bar R $ _j ?^?1 and $ \bar R $ _j R _j ?^?1 are bounded on L ^p (H ⁿ ), 1 < p < ∞, where $$ R_j = \frac{\partial } {{\partial z_j }} - \frac{i} {4}\bar z_j \frac{\partial } {{\partial t}}, \bar R_j = \frac{\partial } {{\partial \bar z_j }} + \frac{i} {4}z_j \frac{\partial } {{\partial t}} $$ and ? is the sublaplacian on H ⁿ . This proof is based on the Calderon-Zygmund theory on the Heisenberg group. It is also shown that when p = 1, the class of right multipliers for the Sobolev space W ^k,1(H ⁿ ) coincides with the dual space of the projective tensor product of two function spaces.     

Hausdorff Operators on Function Spaces

The authors mainly study the Hausdorff operators on Euclidean space Rn.They establish boundedness of the Hausdorff operators in various function spaces,such as Lebesgue spaces,Hardy spaces,local Hardy ...     

Le théorème ergodique pour les opérateurs positifs sur les espaces Lp (1<p<∞) revisitéErgodic theorem for positive operators on Lp-spaces Lp (1<p<∞) revisited

We consider positive linear operators on L_p-spaces (1<p<∞), (A(L_p⁺)?L_p⁺), satisfying the inequality A^m+n<A^m+Aⁿ for all $\text{m,n∈}\text{N}\text{.}$ We describe the structure of these operators (Theorem 1). As a consequence we obtain for all f∈L^p,Aⁿf converges a.e. The last statement contains the theorem of a.e. convergence of Cesaro averages for positive mean bounded operators. To cite this article: A. Brunel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 205–207.