Multipliers for p-Bessel Sequences in Banach Spaces

Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the continuous dependency of the multipliers on their parameters.     

Indefinite higher Riesz transforms

Stein’s higher Riesz transforms are translation invariant operators on L ²(R ⁿ ) built from multipliers whose restrictions to the unit sphere are eigenfunctions of the Laplace–Beltrami operators. In this article, generalizing Stein’s higher Riesz transforms, we construct a family of translation invariant operators by using discrete series representations for hyperboloids associated to the indefinite quadratic form of signature (p,q). We prove that these operators extend to L ^r -bounded operators for 1<r<∞ if the parameter of the discrete series representations is generic.     

L-Fourier multipliers for the Dunkl operator on the real line

We consider Fourier multipliers for L^p associated with the Dunkl operator on and establish a version of Hörmander's multiplier theorem. In applying this version, we come up with some results regarding the oscillating multipliers, partial sum operators and generalized Bessel potentials.     

Finite Section Method for a Banach Algebra of Convolution Type Operators on {L^p(\mathbb R)} with Symbols Generated by PC and SO

We prove necessary and sufficient conditions for the applicability of the finite section method to an arbitrary operator in the Banach algebra generated by the operators of multiplication by piecewise continuous functions and the convolution operators with symbols in the algebra generated by piecewise continuous and slowly oscillating Fourier multipliers on L^p(\mathbb R){L^p(\mathbb {R})}, 1 < p < ∞.     

Lambert multipliers between <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

In this paper Lambert multipliers acting between L ^p spaces are characterized by using some properties of conditional expectation operator. Also, Fredholmness of corresponding bounded operators is investigated.     

Transferring Boundedness from Conjugate Operators Associated with Jacobi, Laguerre, and Fourier–Bessel Expansions to Conjugate Operators in the Hankel Setting

In this paper we establish transference results showing that the boundedness of the conjugate operator associated with Hankel transforms on Lorentz spaces can be deduced from the corresponding boundedness of the conjugate operators defined on Laguerre, Jacobi, and Fourier–Bessel settings. Our result also allows us to characterize the power weights in order that conjugation associated with Laguerre, Jacobi, and Fourier–Bessel expansions define bounded operators between the corresponding weighted L ^p spaces. This paper is partially supported by MTM2004/05878. Third and fourth authors are also partially supported by grant PI042004/067.     

Transplanting maximal inequalities between Laguerre and Hankel multipliers

We supplement our previous paper [9] by adding a theorem that transplantsL ^p -norm maximal inequalities for Laguerre multipliers. As an immediate consequence we obtain negative results concerningL ^p -estimates of partial sum maximal operators for Laguerre expansions.Research supported in part by KBN grant No. 2 PO3A 030 09.     

Lp-Bounded Pseudodifferential Operators and Regularity for Multi-quasi-elliptic Equations

Using a classical result of Marcinkiewicz and Lizorkin about the L^p-continuity for Fourier multipliers, the authors study the action of a class of pseudodifferential operators with weighted smooth symbol on a family of weighted Sobolev spaces. Results about L^p-regularity for multi-quasi-elliptic pseudodifferential operators are also given.     

UMD Banach spaces and the maximal regularity for the square root of several operators

In this paper we prove that the maximal L ^p -regularity property on the interval (0,T), T>0, for Cauchy problems associated with the square root of Hermite, Bessel or Laguerre type operators on L ²(Ω,dμ;X), characterizes the UMD property for the Banach space X.     

BMO spaces associated with semigroups of operators

We study BMO spaces associated with semigroup of operators on noncommutative function spaces (i.e. von Neumann algebras) and apply the results to boundedness of Fourier multipliers on non-abelian discrete groups. We prove an interpolation theorem for BMO spaces and prove the boundedness of a class of Fourier multipliers on noncommutative L _p spaces for all 1 < p < ∞, with optimal constants in p.     

Estimates for some convolution operators with singularities in their kernels on a sphere with applications

We study convolution operators whose kernels have singularities on the unit sphere. For these operators we obtainH ^p -H ^q estimates, where p is less or equal q, and prove their sharpness. To this end, we develop a new method that uses special representations for the symbol of such operators as sums of certain oscillatory integrals and applies the stationary phase method and A. Miyachi results for model oscillating multipliers. Moreover, we also obtain estimates for operators from L ^p to BMO and those from BMO to BMO.     

Characterization of UMD Banach Spaces by Imaginary Powers of Hermite and Laguerre Operators

In this paper we characterize the Banach spaces with the UMD property by means of L ^p -boundedness properties for the imaginary powers of the Hermite and Laguerre operators. In order to do this we need to obtain pointwise representations for the Laplace transform type multipliers associated with Hermite and Laguerre operators.     

Uniform estimates on multi-linear operators with modulation symmetry

In a previous paper [20] in this series, we gaveL ^p estimates for multi-linear operators given by multipliers which are singular on a non-degenerate subspace of some dimensionk. In this paper, we give uniform estimates when the subspace approaches a degenerate region in the casek = 1, and when all the exponentsp are between 2 and ∞. In particular, we recover the non-endpoint uniform estimates for the bilinear Hubert transform in [12]. Dedicated to Tom Wolff     

Towards an <Emphasis Type="Italic">L</Emphasis><Superscript>p</Superscript>-potential theory for sub-Markovian semigroups: variational inequalities and balayage theory

We give a new variational approach toL ^p -potential theory for sub-Markovian semigroups. It is based on the observation that the Gâteaux-derivative of the corresponding L ^p-energy functional is a monotone operator. This allows to apply the well established theory of Browder and Minty on monotone operators to the nonlinear problems in L ^p-potential theory. In particular, using this approach it is possible to avoid any symmetry assumptions of the underlying semigroup. We prove existence of corresponding (r, p)-equilibrium potentials and obtain a complete characterization in terms of a variational inequality. Moreover we investigate associated potentials and encounter a natural interpretation of the so-called nonlinear potential operator in the context of monotone operators.     

Maximal functions associated with Fourier multipliers of Mikhlin-Hörmander type

We show that maximal operators formed by dilations of Mikhlin- Hörmander multipliers are typically not bounded on L^p(^d). We also give rather weak conditions in terms of the decay of such multipliers under which L^p boundedness of the maximal operators holds.Christ, Grafakos and Seeger were supported in part by NSF grants. Honzík was supported by 201/03/0931 Grant Agency of the Czech Republic     

On Transference of Multipliers on Matrix Weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Spaces

We consider a periodic matrix weight W defined on ℝ ^d and taking values in the N×N positive-definite matrices. For such weights, we prove transference results between multiplier operators on L _p (ℝ ^d ;W) and L_p(\mathbb T^d;W)L_{p}(\mathbb {T}^{d};W), 1<p<∞, respectively. As a specific application, we study transference results for homogeneous multipliers of degree zero.     

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L²(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L² estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.     

Applications of discrete maximal <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> regularity for finite element operators

In this paper, we present applications of discrete maximal L _p regularity for finite element operators. More precisely, we show error estimates of order h ² for linear and certain semilinear problems in various L _p (Ω)-norms. Discrete maximal regularity allows us to prove error estimates in a very easy and efficient way. Moreover, we also develop interpolation theory for (fractional powers of) finite element operators and extend the results on discrete maximal L _p regularity formerly proved by the author. The author was supported by the DFG-Graduiertenkolleg 853.     

Some Fourier-type operators for functions on unbounded intervals

In order to approximate functions defined on the real line or on the real semiaxis by polynomials, we introduce some new Fourier-type operators, connected to the Fourier sums of generalized Freud or Laguerre orthonormal systems. We prove necessary and sufficient conditions for the boundedness of these operators in suitable weighted L ^p -spaces, with 1 < p < ∞. Moreover, we give error estimates in weighted L ^p and uniform norms.     

Towards an L^p Potential Theory for Sub-Markovian Semigroups： Kernels and Capacities

We study in fairly general measure spaces （X,μ） the （non-linear） potential theory of L^p sub-Markovian semigroups which are given by kernels having a density with respect to the underlying measure. In terms of mapping properties of the operators we provide sufficient conditions for the existence （and regularity） of such densities. We give various （dual） representations for several associated capacities and, in the corresponding abstract Bessel potential spaces, we study the role of the truncation property. Examples are discussed in the case of R^n where abstract Bessel potential spaces can be identified with concrete function spaces.