Norms of zonal spherical functions and Fourier series on compact symmetric spaces

New estimates are given for the p-norms of zonal spherical functions on compact symmetric spaces. These estimates are applied to give a Cohen type inequality for convolutor norms which leads to negative results on p-mean convergence of Fourier series on compact symmetric spaces of arbitrary rank.     

Chaotic dynamics of the heat semigroup on the Damek-Ricci spaces

We consider the rank one Riemannian symmetric spaces of noncompact type and their non-symmetric generalization, namely the Damek-Ricci spaces. We show that the heat semigroup generated by a certain perturbation of the Laplace-Beltrami operator of these spaces is chaotic on their L _p -spaces when p > 2. The range of p and the corresponding perturbation are sharp. A precursor to this result is due to Ji and Weber [19] where it was shown that under identical conditions the heat operator is subspace-chaotic on the Riemannian symmetric spaces, which is weaker than it being chaotic. We also extend the results to the Lorentz spaces L _p,q , which are generalizations of the Lebesgue spaces. This enables us to point out that the chaoticity degenerates to subspace-chaoticity only when q = ∞.     

Strictly singular and strictly co-singular inclusions between symmetric sequence spaces

Strict singularity and strict co-singularity of inclusions between symmetric sequence spaces are studied. Suitable conditions are provided involving the associated fundamental functions. The special case of Lorentz and Marcinkiewicz spaces is characterized. It is also proved that if E?F are symmetric sequence spaces with E≠?₁ and F≠c₀ and ?_∞ then there exist a intermediate symmetric sequence space G such that E?G?F and both inclusions are not strictly singular. As a consequence new characterizations of the spaces c₀ and ?₁ inside the class of all symmetric sequence spaces are given.     

L p -L q Estimates for Bergman Projections in Bounded Symmetric Domains of Tube Type

Let D be an irreducible bounded symmetric domain of tube type in ? ⁿ . The class of Bloch functions is well known in this context, in connection with Hankel operators or duality of Bergman spaces. Contrary to what happens in the unit ball, Bloch functions do not belong to all Lebesgue spaces L ^p (D) for p<∞ in higher rank. We give here both necessary and sufficient conditions on p for such an embedding. This question is equivalent to local boundedness properties of the Bergman projection in the tube domain over a symmetric cone that is conformally equivalent to D. We are linked to consider L ^∞–L ^q inequalities on symmetric cones, which may be of independent interest, and study more systematically estimates with loss for the Bergman projection. The proofs are based on a very precise estimate on an integral related to the Gamma function of a symmetric cone.     

Symmetric spaces with maximal projection constants

In this paper we introduce a special class of finite-dimensional symmetric subspaces of L₁, so-called regular symmetric subspaces. Using this notion, we show that for any k?2, there exist k-dimensional symmetric subspaces of L₁ which have maximal projection constant among all k-dimensional symmetric spaces. Moreover, L₁ is a maximal overspace for these spaces (see Theorems 4.4 and 4.5.) Also a new asymptotic lower bound for projection constants of symmetric spaces is obtained (see Theorem 5.3). This result answers the question posed in [12, p. 36] (see also [15, p. 38]) by H. Koenig and co-authors. The above results are presented both in real and complex cases.     

Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.     

On the exponentiality of affine symmetric spaces

An affine symmetric space G/H is said to be exponential if every two points of this space can be joined by a geodesic and weakly exponential if the union of all geodesics issuing from one point is everywhere dense in G/H. For the group space (G × G)/G _diag of a Lie group G, these properties are equivalent to the exponentiality and weak exponentiality of G, respectively. We generalize known theorems on the image of the exponential mapping in Lie groups to the case of affine symmetric spaces. We prove the weak exponentiality of the symmetric spaces of solvable Lie groups, and in the semisimple case we obtain criteria for exponentiality and weak exponentiality.     

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.     

Almost complex structures on quaternion-Kähler manifolds and inner symmetric spaces

We prove that compact quaternionic-Kähler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians Gr₂(?ⁿ⁺²). We also prove that irreducible inner symmetric spaces M ⁴ⁿ of compact type are not weakly complex, except for spheres and Hermitian symmetric spaces.     

Wavelet Characterizations for Anisotropic Besov Spaces

The goal of this paper is to provide wavelet characterizations for anisotropic Besov spaces. Depending on the anisotropy, appropriate biorthogonal tensor product bases are introduced and Jackson and Bernstein estimates are proved for two-parameter families of finite-dimensional spaces. These estimates lead to characterizations for anisotropic Besov spaces by anisotropy-dependent linear approximation spaces and lead further on to interpolation and embedding results. Finally, wavelet characterizations for anisotropic Besov spaces with respect to L_p-spaces with 0<p<∞ are derived.     

Geometric description of L 1-Spaces

We describe strongly facially symmetric spaces which are isometrically isomorphic to L ₁-space.     

On Symmetric Spaces With Convergence in Measure on Reflexive Subspaces

A closed subspace H of a symmetric space X on [0, 1] is said to be strongly embedded in X if in H the convergence in X-norm is equivalent to the convergence in measure. We study symmetric spaces X with the property that all their reflexive subspaces are strongly embedded in X. We prove that it is the case for all spaces, which satisfy an analogue of the classical Dunford–Pettis theorem on relatively weakly compact subsets in L₁. At the same time the converse assertion fails for a broad class of separableMarcinkiewicz spaces.     

Summability and estimates for polynomials and multilinear mappings

In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ?_p spaces in fact hold true for mappings on arbitrary Banach spaces.     

Équation des ondes sur les espaces symétriques riemanniens

We prove that the solutions of the homogeneous wave equation on Riemannian symmetric spaces have dispersion properties and we deduce Strichartz type estimates for these solutions. To cite this article: A. Hassani, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Hilbert polynomials and Arveson's curvature invariant

We define and study Hilbert polynomials for certain holomorphic Hilbert spaces. We obtain several estimates for these polynomials and their coefficients. Our estimates inspire us to investigate the connection between the leading coefficients of Hilbert polynomials for invariant subspaces of the symmetric Fock space and Arveson's curvature invariant for coinvariant subspaces. We are able to obtain some formulas relating the curvature invariant with other invariants. In particular, we prove that Arveson's version of the Gauss-Bonnet-Chern formula is true when the invariant subspaces are generated by any polynomials.     

Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions

We consider the nonautonomous Ornstein-Uhlenbeck operator in some weighted spaces of continuous functions in $\mathbb{R}^{N}$ . We prove sharp uniform estimates for the spatial derivatives of the associated evolution operator P _s,t , which we use to prove optimal Schauder estimates for the solution to some nonhomogeneous parabolic Cauchy problems associated with the Ornstein-Uhlenbeck operator. We also prove that, for any t>s, the evolution operator P _s,t is compact in the previous weighted spaces.     

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

In this paper we obtain generalized Clarkson–McCarthy inequalities for spaces l _q (S ^p ) of operators from Schatten ideals S ^p . We show that all Clarkson–McCarthy type inequalities are, in fact, some estimates on the norms of operators acting on the spaces l _q (S ^p ) or from one such space into another. We also extend some inequalities for partitioned operators and for Cartesian decomposition of operators.     

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 2. Linearization on a symmetric solution

The steady flow of a Navier–Stokes fluid is analysed in a two-dimensional asymmetric unbounded domain $\text{Ω}$, having two outlets to infinity, namely a half-plane K and a semi-infinite channel Π₋. Assuming that $\text{Ω}$ differs from a symmetric domain $\text{Ω}$ only by a small perturbation, we show the existence of a unique solution to the Navier–Stokes system in $\text{Ω}$. The solution is obtained as a perturbation of the symmetric solution and, at large distances in K, it takes the Jeffrey–Hamel form. Curiously, our results are valid only if the flux Φ, besides being small, is directed from the half-plane towards the semi-infinite channel, i.e. Φ is negative.The main ingredients in our proofs are estimates in weighted spaces with detached asymptotics and the study of a model problem resulting from the linearization around the symmetric solution which, for non-zero flux, leads, in contrast to the linearization around the zero solution, to the absence of compatibility conditions for the convective term and, for Φ<0, to the domination of nonlinear terms by the linear ones. We also provide some explicit examples of the domain perturbation.     

On the structure of completely flat Banach spaces

Completely flat Banach spaces (i.e. Banach spaces having a spanning centrally symmetric closed curve in the unit sphere with length 4), are characterized as spaces ‘between’ the two classical examples L₁[0,1] and C_±[0,1]. Some applications are given.