Estimates for the maximal multilinear singular integral operators

In this paper, some mapping properties are considered for the maximal multilinear singular integral operator whose kernel satisfies certain minimum regularity condition. It is proved that certain uniform local estimate for doubly truncated operators implies the Lp(Rn) (1 < p < ∞) boundedness and a weak type L log L estimate for the corresponding maximal operator.     

Pseudodifferential Operators with Rough Symbols

In this work, we develop L ^p boundedness theory for pseudodifferential operators with rough (not even continuous in general) symbols in the x variable. Moreover, the B(L ^p ) operator norms are estimated explicitly in terms of scale invariant quantities involving the symbols. All the estimates are shown to be sharp with respect to the required smoothness in the ξ variable. As a corollary, we obtain L ^p bounds for (smoothed out versions of) the maximal directional Hilbert transform and the Carleson operator.     

A remark on a maximal function over a Cantor set of directions

LetM _e ⁰ be the maximal operator over segments of length 1 with directions belonging to a Cantor set. It has been conjectured that this operator is bounded onL ². We consider a sequence of operators over finite sets of directions converging toM _e ⁰ . We improve the previous estimate for the (L ²,L ²)-norm of these particular operators. We also prove thatM _e ⁰ is bounded from some subsets ofL ² toL ². These subsets are composed of positive functions whose Fourier transforms have a very weak decay or are supported in a vertical strip. Partially supported by Spanish DGICYT grant no. PB90-0187.     

The Spectral Measure of Certain Elements of the Complex Group Ring of a Wreath Product

We use elementary methods to compute the L ²-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk and Zuk, and later used by them, together with Linnell and Schick, to produce a counterexample to a strong version of the Atiyah conjecture about the range of L ²-Betti numbers. We use our results to construct manifolds with certain L ²-Betti numbers (given as convergent infinite sums of rational numbers) which are not obviously rational, but we have been unable to determine whether any of them are irrational.     

A note on the sunouchi operator with respect to the walsh—kaczmarz system

An analogue of the so—called Sunouchi operator with respect to the Walsh—Kaczmarz system will be investigated. We show the boundedness of this operator if we take it as a map from the dyadic Hardy space H ^p to L ^p for all 0<p≤1.. For the proof we consider a multiplier operator and prove its (H ^p H ^p)—boundedness for 0<p≤1. Since the multiplier is obviously bounded from L ² to L ², a theorem on interpolation of operators can be applied to show that our multiplier is of weak type (1,1) and of type (q q) for all 1<q<∞. The same statements follow also for the Sunouchi operator.     

Spectral functions of self-adjoint and symmetric multiplication operators inL 2(X, μ)-spaces

Explicit formulas are derived for the spectral function of double multiplication operator containing a multiplicative evolution inL ²(X, μ)-space and a convolution-type operator inL ²(ℝ ⁿ )-spaces. Symmetric convolution and multiplication operators are considered inL ²(X, μ) andL ²(ℝ ⁿ )-spaces. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 803–810, June, 2000.     

Decoupling of Modes for the Elastic Wave Equation in Media of Limited Smoothness

We establish a decoupling result for the P and S waves of linear, isotropic elasticity, in the setting of twice-differentiable Lamé parameters. Precisely, we show that the P?S components of the wave propagation operator are regularizing of order one on L ² data, by establishing the diagonalization of the elastic system modulo a L ²-bounded operator. Effecting the diagonalization in the setting of twice-differentiable coefficients depends upon the symbol of the conjugation operator having a particular structure.     

The Vector-Valued Big <Emphasis Type="Italic">q</Emphasis>-Jacobi Transform

Big q-Jacobi functions are eigenfunctions of a second-order q-difference operator L. We study L as an unbounded self-adjoint operator on an L ²-space of functions on ℝ with a discrete measure. We describe explicitly the spectral decomposition of L using an integral transform ℱ with two different big q-Jacobi functions as a kernel, and we construct the inverse of ℱ.      

Ergodic theory and iterative solution of linear operator equations

In this paper we consider the problem of approximating solutions of linear operator equations of the type u-Tu=f. The main tools are Dotson's extension of the Eberlein ergodic theorem to affine mappings and the DeMoivre-Laplace theorem of probability theory. The main results are applied to obtain theorems on the iterative approximation of solutions of linear operator equations in Hilbert space and the approximation in L ^ρ norm of solutions of a certain functional equation in the space L ^∞     

Directional maximal operators with smooth densities

We study directional maximal operators on ?ⁿ with smooth densities. We prove that if the classical directional maximal operator in a given set of directions is weak type (1, 1), then the corresponding smooth‐density maximal operator in that set of directions will be bounded on L^q for q suitably large, depending on the order of the stationary points of the density function. In contrast to the classical case, if q is too small, the smooth density operator need not be bounded on L^q. Improving upon previously known results, we also establish that if the density function has only finitely many extreme points, each of finite order, then any maximal operator in a finite sum of diadic directions is bounded on all L^q for q > 1 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weighted Inequalities for the Generalized Maximal Operator in Martingale Spaces

The generalized maximal operator M in martingale spaces is considered. For 1 < p ≤ q < ∞, the authors give a necessary and sufficient condition on the pair ([^(m)]\hat \mu , v) for M to be a bounded operator from martingale space L ^p (v) into L ^q ([^(m)]\hat \mu ) or weak-L ^q ([^(m)]\hat \mu ), where [^(m)]\hat \mu is a measure on Ω × ℕ and v a weight on Ω. Moreover, the similar inequalities for usual maximal operator are discussed.     

Non-supercyclicity of Volterra Convolution and Related Operators

We show that if V ^α (α > 0) is the Riemann-Liouville fractional integration operator and T is an invertible operator on L ²(0, 1) which commutes with V , then TV ^α is not supercyclic on L ²(0, 1); in particular, many Volterra convolution operators are not supercyclic. The technique is based on an argument used by Gallardo-Gutiérrez and Montes-Rodríguez to show that V is not supercyclic.     

The higher order Riesz transform and BMO type space associated to Schrödinger operators

Let L = ?Δ + V be a Schrödinger operator on $\mathbb {R}^nLet L = ?Δ + V be a Schrödinger operator on $\mathbb {R}^n$ (n ≥ 3), where $V \not\equiv 0$ is a nonnegative potential belonging to certain reverse Hölder class B_s for $s \ge \frac{n}{2}$. In this article, we prove the boundedness of some integral operators related to L, such as L^?1?², L^?1V and L^?1( ? Δ) on the space $BMO_L(\mathbb {R}^n)$.     

On the Riemann Summability of Fourier Integrals and Real Hardy Spaces

We consider the Riemann means of single and multiple Fourier integrals of functions belonging to L¹ or the real Hardy spaces defined on ℝⁿ, where n ≥ 1 is an integer. We prove that the maximal Riemann operator is bounded both from H¹(ℝ) into L¹(ℝ) and from L¹(ℝ) into weak –L¹(ℝ). We also prove that the double maximal Riemann operator is bounded from the hybrid Hardy spaces H^(1,0)(ℝIsup2), H^(0,1)(ℝ²⁾ into weak –L¹(ℝ²). Hence pointwise Riemann summability of Fourier integrals of functions in H^(1,0) ∪ H^(0,1)(ℝ²) follows almost everywhere.The maximal conjugate Riemann operators as well as the pointwise convergence of the conjugate Riemann means are also dealt with.     

The Heat Equation for the Generalized Hermite and the Generalized Landau Operators

We give a formula for the one-parameter strongly continuous semigroups ${e^{-tL^{\lambda}}}We give a formula for the one-parameter strongly continuous semigroups e^-tLl{e^{-tL^{\lambda}}} and e^{-t [(A)\tilde]}{e^{-t \tilde{A}}}, t > 0 generated by the generalized Hermite operator L^l, l ? R\{0}{L^{\lambda}, \lambda \in {\bf R}\backslash \{0\}} respectively by the generalized Landau operator ?. These formula are derived by means of pseudo-differential operators of the Weyl type, i.e. Weyl transforms, Fourier-Wigner transforms and Wigner transforms of some orthonormal basis for L ²(R ²ⁿ ) which consist of the eigenfunctions of the generalized Hermite operator and of the generalized Landau operator. Applications to an L ² estimate for the solutions of initial value problems for the heat equations governed by L ^λ respectively ?, in terms of L ^p norm, 1 ≤ p ≤ ∞ of the initial data are given.     

Dual frames in connected with generalized sampling in shift-invariant spaces

The aim of this article is to derive stable generalized sampling in a shift-invariant space by using some special dual frames in L²(0,1). These sampling formulas involve samples of filtered versions of the functions in the shift-invariant space. The involved samples are expressed as the frame coefficients of an appropriate function in L²(0,1) with respect to some particular frame in L²(0,1). Since any shift-invariant space with stable generator is the image of L²(0,1) by means of a bounded invertible operator, our generalized sampling is derived from some dual frame expansions in L²(0,1).     

Walsh Multipliers for Dyadic Hardy Spaces

In this article we are interested in conditions on the coefficients of a Walsh multiplier operator that imply the operator is bounded on certain dyadic Hardy spaces H ^p , 0 < p < ∞. In particular, we consider two classical coefficient conditions, originally introduced for the trigonometric case, the Marcinkiewicz and the Hörmander–Mihlin conditions. They are known to be sufficient in the spaces L ^p , 1 < p < ∞. Here we study the corresponding problem on dyadic Hardy spaces, and find the values of p for which these conditions are sufficient. Then, we consider the cases of H ¹ and L ¹ which are of special interest. Finally, based on a recent integrability condition for Walsh series, a new condition is provided that implies that the multiplier operator is bounded from L ¹ to L ¹, and from H ¹ to H ¹. We note that existing multiplier theorems for Hardy spaces give growth conditions on the dyadic blocks of the Walsh series of the kernel, but these growth are not computable directly in terms of the coefficients.     

Spherical means and CR functions on the Heisenberg group

The injectivity of the spherical mean value operator on the Heisenberg group is studied. Whenf ∈L ^P (Hⁿ), 1 ≤p < ∞ it is proved that the spherical mean value operator is injective. When 1 ≤p ≤ 2,f(z, ·) ∈L ^P (ℝ) the same is proved under much weaker conditions in the z-variable. Some extensions of recent results of Agranovskyet al. regardingCR functions on the Heisenberg group are also obtained.     

Extended Best Polynomial Approximation Operator in Orlicz Spaces

In this article we consider the best polynomial approximation operator, defined in an Orlicz space L ^Φ(B), and its extension to L ^?(B) where ? is the derivative function of Φ. A characterization of these operators and several properties are obtained.     

On Iterations of Non–Negative Operators and Their Applications to Elliptic Systems

Let H₀, H₁ be Hilbert spaces and L : H₀ → H₁ be a linear bounded operator with ∥L∥ ≤ 1. Then L*L is a bounded linear self–adjoint non–negative operator in the Hilbert space H₀ and one can use the Neumann series Σ^∞_v=0(I — L*L)^v L*f in order to stud solvabilit of the operator equation Lu = f. In particular, applying this method to the ill–posed Cauch problem for solutions to an elliptic system Pu = 0 of linear PDE's of order p with smoothcoefficients we obtain solvabilit conditions and representation formulae for solutions of the problem in Hardy spaces whenever these solutions exist. For the Cauch–Riemann system in ℂ the summands of the Neumann series are iterations of the Cauch type integral.