Weighted Estimates for Singular Integral Operators Satisfying Hörmander’s Conditions of Young Type

The following open question was implicit in the literature: Are there singular integrals whose kernels satisfy the L^r-Hörmander condition for any r > 1 but not the L^∞-Hörmander condition? We prove that the one-sided discrete square function, studied in ergodic theory, is an example of a vector-valued singular integral whose kernel satisfies the L^r-Hörmander condition for any r > 1 but not the L^∞-Hörmander condition. For a Young function A we introduce the notion of L^A-Hörmander. We prove that if an operator satisfies this condition, then one can dominate the L^p(w) norm of the operator by the L^p(w) norm of a maximal function associated to the complementary function of A, for any weight w in the A_∞ class and 0 < p < ∞. We use this result to prove that, for the one-sided discrete square function, one can dominate the L^p(w) norm of the operator by the L^p(w) norm of an iterate of the one-sided Hardy-Littlewood Maximal Operator, for any w in the A⁺ _∞ class.     

Two-parameter estimates for joint spectral projections on complex spheres

We prove sharp two-parameter estimates for the L ^p -L ² norm, 1 ≤ p ≤ 2, of the joint spectral projectors associated to the Laplace–Beltrami operator and to the Kohn Laplacian on the unit sphere S ^2n-1 in . Then, by using the notion of contraction of Lie groups, we deduce the estimates recently obtained by H. Koch and F. Ricci for joint spectral projections on the reduced Heisenberg group h ¹.      

Best Simultaneous Approximation in L(I,E)

Let G be a reflexive subspace of the Banach space E and let L^p(I,E) denote the space of all p-Bochner integrable functions on the interval I=[0,1] with values in E, 1p∞. Given any norm N( , ) on R², N nondecreasing in each coordinate on the set R²₊, we prove that L^p(I,G) is N-simultaneously proximinal in L^p(I,E). Other results are also obtained.     

Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right‐hand side in Lp (p⩾1)

Accurate modelling of heat transfer in high‐temperature situations requires accounting for the effect of heat radiation. In complex industrial applications involving dissipative heating, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L¹. In this paper, we focus on a stationary heat equation with nonlocal boundary conditions and L^p right‐hand side, with p?1 being arbitrary. Thanks to new coercivity results, we are able to produce energy estimates that involve only the L^p norm of the heat sources and to prove the existence of weak solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

Boundary Problems for Harmonic Functions and Norm Estimates for Inverses of Singular Integrals in Two Dimensions

In this paper, we establish sharp well-posedness results for tangential derivative problems for the Laplacian with data in L ^p , 1 < p < ∞, on curvilinear polygons. Furthermore, we produce norm estimates/formulas for inverses of singular integral operators relevant for the Dirichlet, Neumann, tangential derivative, and transmission boundary value problems associated with the Laplacian in a distinguished subclass of Lipschitz domains in two dimensions. Our approach relies on Calderón-Zygmund theory and Mellin transform techniques.     

Uniform estimates for spherical functions on complex semisimple Lie groups

We prove new estimates for spherical functions and their derivatives on complex semisimple Lie groups, establishing uniform polynomial decay in the spectral parameter. This improves the customary estimate arising from Harish-Chandra's series expansion, which gives only a polynomial growth estimate in the spectral parameter. In particular, for arbitrary positive-definite spherical functions on higher rank complex simple groups, we establish estimates for which are of the form in the spectral parameter and have uniform exponential decay in regular directions in the group variable a _t . Here is an explicit constant depending on G, and may be singular, for instance.?The uniformity of the estimates is the crucial ingredient needed in order to apply classical spectral methods and Littlewood—Paley—Stein square functions to the analysis of singular integrals in this context. To illustrate their utility, we prove maximal inequalities in L ^p for singular sphere averages on complex semisimple Lie groups for all p in . We use these to establish singular differentiation theorems and pointwise ergodic theorems in L ^p for the corresponding singular spherical averages on locally symmetric spaces, as well as for more general measure preserving actions. Submitted: May 2000, Revised version: October 2000.     

Littlewood–Paley and Multiplier Theorems on Weighted Spaces over Locally Compact Vilenkin Groups

We extend the Littlewood–Paley theorem toL^p_w(G), whereGis a locally compact Vilenkin group andware weights satisfying the MuckenhouptA_pcondition. As an application we obtain a mixed-norm type multiplier result onL^p_w(G) and prove the sharpness of our result. We also obtain a sufficient condition for φ L^∞(Γ) to be a multiplier on the power weightedL^p_α(G) in terms of its smoothness condition.     

On a Class of Hungarian Semigroups and the Factorization Theorem of Khinchin

Let G be a connected reductive Lie group and K be a maximal compact subgroup of G. We prove that the semigroup of all K-biinvariant probability measures on G is a strongly stable Hungarian semigroup. Combining with the result [see Rusza and Szekely⁽⁹⁾], we get that the factorization theorem of Khinchin holds for the aforementioned semigroup. We also prove that certain subsemigroups of K-biinvariant measures on G are Hungarian semigroups when G is a connected Lie group such that Ad G is almost algebraic and K is a maximal compact subgroup of G. We also prove a p-adic analogue of these results.     

L P boundedness of area function associated with operators on product spaces

Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn) with suitable upper bounds on its heat kernels. Assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper,we define the Littlewood- Paley g function associated with L on Rn × Rn, denoted by GL(f)(x1, x2), and decomposition, we prove that ‖SL(f)‖p ≈‖GL(f)‖p ≈‖f‖p for 1 ＜ p ＜∞.     

Estimating L ∞ Norms by L 2k Norms for Functions on Orbits

Abstract. Let G be a compact group acting in a real vector space V . We obtain a number of inequalities relating the L ^∞ norm of a matrix element of the representation of G with its L ^2k norm for a positive integer k . As an application, we obtain approximation algorithms to find the maximum absolute value of a given multivariate polynomial over the unit sphere (in which case G is the orthogonal group) and for the assignment problem of degree d , a hard problem of combinatorial optimization generalizing the quadratic assignment problem (in which case G is the symmetric group).     

Convolution Operators on Banach Lattices with Shift-Invariant Norms

Let G be a locally compact abelian group and let μ be a complex valued regular Borel measure on G. In this paper we consider a generalisation of a class of Banach lattices introduced in Johansson (Syst Control Lett 57:105–111, 2008). We use Laplace transform methods to show that the norm of a convolution operator with symbol μ on such a space is bounded below by the L ^∞ norm of the Fourier–Stieltjes transform of μ. We also show that for any Banach lattice of locally integrable functions on G with a shift-invariant norm, the norm of a convolution operator with symbol μ is bounded above by the total variation of μ.     

Functional Calculus for Infinitesimal Generators of Holomorphic Semigroups

We give a functional calculus formula for infinitesimal generators of holomorphic semigroups of operators on Banach spaces, which involves the Bochner–Riesz kernels of such generators. The rate of smoothness of operating functions is related to the exponent of the growth on vertical lines of the operator norm of the semigroup. The strength of the formula is tested on Poisson and Gauss semigroups inL¹(Rⁿ) andL¹(G), for a stratified Lie groupG. We give also a self-contained theory of smooth absolutely continuous functions on the half line [0, ∞).     

Espaces de Banach stables

We define the notion of “stable Banach space” by a simple condition on the norm. We prove that ifE is a stable Banach space, then every subspace ofL ^p(E) (1≦p<∞) is stable. Our main result asserts that every infinite dimensional stable Banach space containsl ^p, for somep, 1≦p<∞. This is a generalization of a theorem due to D. Aldous: every infinite dimensional subspace ofL ¹ containsl ^p, for somep in the interval [1, 2].     

The G-Ellipticity for the Sub-Laplacian on Two Step Stratified Lie Groups

This paper reveals that the sub-Laplacian L₀ on two step stratified Lie groups has a similar behavior like elliptic operators on the Euclidean space, that is, the sub - Laplacian satisfies a group-elliptic estimate, called the G- elliptic estimate (or the L^p regularity), and the general left Invariant operator L_λ has such a behavior if and only if λ is admissible.     

Hilbert Transforms on the Sphere with the Clifford Algebra Setting

Through a double-layer potential argument the inner and outer Poisson kernels, the Cauchy-type conjugate inner and outer Poisson kernels, and the kernels of the Cauchy-type inner and outer Hilbert transformations on the sphere are deduced. We also obtain Abel sum expansions of the kernels and prove the L ^p -boundedness of the inner and outer Hilbert transformations for 1<p<∞.     

Estimates for Littlewood-Paley Functions and Extrapolation

We prove certain L ^p -estimates for Littlewood-Paley functions arising from rough kernels. The estimates are useful for extrapolation to prove L ^p -boundedness of the Littlewood-Paley functions under a sharp kernel condition.      

Représentations <Emphasis Type="Italic">p</Emphasis>-adiques de GL<Subscript>2</Subscript>(L) et Catégories Dérivées

We construct some locally ℚ _p -analytic representations of GL₂(L), L a finite extension of ℚ _p , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel’d’s upper half space has a structure of rank 2 admissible filtered (φ, N)-module. Finally, we prove that this filtered module is associated, via Fontaine’s theory, to the initial Galois representation.     

Countable structures with a fixed group of automorphisms

We prove that, given a countable groupG, the set of countable structures (for a suitable languageL)U _G whose automorphism group is isomorphic toG is a complete coanalytic set and ifG ≄H thenU _G is Borel inseparable fromU _H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures forL whose automorphism group is isomorphic to ℤ _p ^ℕ ,p a prime number, is Π ₁ ¹ &σ ₁ ¹ -complete.     

Some remarks on the Fefferman-Stein inequality

We investigate the Fefferman-Stein inequality related to a function f and the sharp maximal function f ^# on a Banach function space X. It is proved that this inequality is equivalent to a certain boundedness property of the Hardy-Littlewood maximal operator M. The latter property is shown to be self-improving. We apply our results in several directions. First, we show the existence of nontrivial spaces X for which the lower operator norm of M is equal to 1. Second, in the case when X is the weighted Lebesgue space L ^p (w), we obtain a new approach to the results of Sawyer and Yabuta concerning the C _p condition.     

Optimal Extension of Fourier Multiplier Operators in L p (G)

Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier ${\psi : \Gamma \to {\mathbb C}}Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier y: G? \mathbb C{\psi : \Gamma \to {\mathbb C}} (with Γ the dual group), we study the optimal domain of the multiplier operator T^(p)_y : L^p (G) ? L^p (G){T^{(p)}_\psi : L^p (G) \to L^p (G)}. This is the largest Banach function space, denoted by L¹(m^(p)_y){L^1(m^{(p)}_\psi)}, with order continuous norm into which L ^p (G) is embedded and to which T^(p)_y{ T^{(p)}_\psi} has a continuous L ^p (G)-valued extension. Compactness conditions for the optimal extension are given, as well as criteria for those ψ for which L¹(m^(p)_y) = L^p (G){L^1(m^{(p)}_\psi) = L^p (G)} is as small as possible and also for those ψ for which L¹(m^(p)_y) = L¹ (G){L^1(m^{(p)}_\psi) = L^1 (G)} is as large as possible. Several results and examples are presented for cases when L^p (G) \subsetneqq L¹(m^(p)_y) \subsetneqq L¹ (G){L^p (G) \subsetneqq L^1(m^{(p)}_\psi) \subsetneqq L^1 (G)}.