Commutators of Littlewood-Paley operators

Let b ∈ Lloc(Rn) and L denote the Littlewood-Paley operators including the LittlewoodPaley g function,Lusin area integral and gλ* function. In this paper,the authors prove that the Lp boundedness of commutators [b,L] implies that b ∈ BMO(Rn) . The authors therefore get a characterization of the Lp-boundedness of the commutators [b,L]. Notice that the condition of kernel function of L is weaker than the Lipshitz condition and the Littlewood-Paley operators L is only sublinear,so the results obtained in the p...     

Littlewood-Paley and Lusin functions on nilpotent Lie groups

In this paper, we define the Littlewood-Paley and Lusin functions associated to the sub-Laplacian operator on nilpotent Lie groups. Then we prove the Lp (1<p<∞) boundedness of Littlewood-Paley and Lusin functions.     

Parabolic Littlewood-Paley g-function with rough kernel

In this note, we give the L^p （1 〈 p 〈∞） boundedness of the parabolic Littlewood Paley g-function with rough kernel.     

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.     

Square-like Functions Generated by a Composite Wavelet Transform

The classical square functions play important role in Harmonic Analysis and have a very direct connection with L ₂-estimates and Littlewood-Paley theory. In this paper we introduce a new variant of square-like functions generated by some composite wavelet transform. We establish Calderón-type reproducing formula and then prove L ₂-boundedness of newly defined square-like functions.     

Lusin type and cotype for Laguerre <Emphasis Type="Italic">g</Emphasis>-functions

We characterize Lusin type and cotype for a Banach space in terms of the L ^p -boundedness of Littlewood-Paley g-functions associated with the Hermite and Laguerre expansions.     

Transferring strong boundedness among Laguerre orthogonal systems

Given the family of Laguerre polynomials, it is known that several orthonormal systems of Laguerre functions can be considered. In this paper we prove that an exhaustive knowledge of the boundedness in weighted L ^p of the heat and Poisson semigroups, Riesz transforms and g-functions associated to a particular Laguerre orthonormal system of functions, implies a complete knowledge of the boundedness of the corresponding operators on the other Laguerre orthonormal system of functions. As a byproduct, new weighted L ^p boundedness are obtained. The method also allows us to get new weighted estimates for operators related with Laguerre polynomials. Carlos Segovia passed away on April 3, 2007.     

Vector-valued Fourier multipliers on symmetric spaces of the noncompact type

LetX be a Riemannian symmetric space of the noncompact type. We prove the multiplier theorem for the Helgason-Fourier transform and the vector valued function spacesL _p (X, l _q ). As a consequence we get the inequalities of the Littlewood-Paley type forL _p (X) spaces.Research supported by K.B.N. Grant 210519101 (Poland).     

Littlewood-Paley functions associated to second order elliptic operators

We prove L^p-estimates for the Littlewood-Paley function associated with a second order divergence form operator L=–div A with bounded measurable complex coefficients in ⁿ.Mathematics Subject Classification (2000):42B20, 35J15The author is partially supported by NSF of China (Grant No. 10371134) and SRF for ROCS, SEM.     

Weighted sharp boundedness for multilinear commutators

We obtain sharp estimates for some multilinear commutators related to certain sublinear integral operators. These operators include the Littlewood-Paley operator and Marcinkiewicz operator. As an application, we obtain weighted L ^p (p > 1) inequalities and an L log L-type estimate for multilinear commutators. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1419–1431, October, 2007.     

L p boundedness for parabolic Littlewood-Paley operators with rough kernels belonging to block spaces

This paper is primarily concerned with the proof of the Lp boundedness of parabolic Littlewood-Paley operators with kernels belonging to certain block spaces. Some known results are essentially extended and improved.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>–dimension free boundedness for Riesz transforms associated to Hermite functions<Superscript>⋆</Superscript>

Riesz transforms associated to Hermite functions were introduced by S. Thangavelu, who proved that they are bounded operators on 1<p< . In this paper we give a different proof that allows us to show that the L^p–norms of these operators are bounded by a constant not depending on the dimension d. Moreover, we define Riesz transforms of higher order and free dimensional estimates of the L^p–bounds of these operators are obtained. In order to prove the mentioned results we give an extension of the Littlewood-Paley theory that we believe of independent interest.Mathematical Subject Classification (2000):42B20, 42B25, 42C10Partially supported by Instituto Argentino de Matemática CONICET, Convenio Universidad Autónoma de Madrid-Universidad Nacional del Litoral, UBACYT 2000-2002 and Ministerio de Ciencia y Tecnologí BFM2002-04013-C02-02     

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 ＜∞.     

An extrapolation theorem for nonlinear approximation and its applications

We prove an extrapolation theorem for the nonlinear m-term approximation with respect to a system of functions satisfying very mild conditions. This theorem allows us to prove endpoint L^p-L^q estimates in nonlinear approximation. As a consequence, some known endpoint estimates can be deduced directly and some new estimates are also obtained. Finally, applications of these new estimates are given to spherical m-widths and m-term approximation of the weighted Besov classes.     

Characterization ofL p (R n ) using Gabor frames

We characterize L^p norms of functions onR ⁿ for 1<p<∞ in terms of their Gabor coefficients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of L^p functions converge to the functions almost everywhere and in L^p for 1<p<∞. In L¹ we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L¹ in a certain Cesàro sense. Consequently, we are able to establish that a large class of Gabor families generate Banach frames for L^p (R ⁿ) when 1≤p<∞.     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

Inequalities related toH p smoothness of Sobolev type

By exploiting a class of maximal functions and Littlewood-Paley theory, a list of embedding inequalities onH ^p-Sobolev spaces andH ^p boundedness results for Riesz and Bessel potentials are obtained at one stroke.This work was supported in part by the Chung-Ang University Academic Research Special Grants, 1997.     

Walsh-Type Wavelet Packet Expansions

We consider a family of basic nonstationary wavelet packets generated using the Haar filters except for a finite number of scales where we allow the use of arbitrary filters. Such a system, which we call a system of Walsh-type wavelet packets, can be considered as a smooth generalization of the Walsh functions. We show that the basic Walsh-type wavelet packets share a number of metric properties with the Walsh system. We prove that the system constitutes a Schauder basis for L^p( ), 1<p<∞, and we construct an explicit function in L¹( ) for which the expansion fails. Then we prove that expansions of L^p( )-functions, 1<p<∞, in the Walsh-type wavelet packets converge pointwise a.e. Finally, we prove that the analogous results are true for periodic Walsh-type wavelet packets in L^p[0,1).     

Weighted estimates for the <Emphasis Type="Italic">k</Emphasis>-plane transform of radial functions on euclidean spaces

We discuss the L ^p − L ^q mapping property of k-plane transforms acting on radial functions in certain weighted L ^p spaces with power weight. We show that for all admissible power weights it is not always possible to get strong (p, q) boundedness of the k-plane transform. However, we prove the best possible estimates with respect to the Lorentz norms.     

Conical square function estimates in UMD Banach spaces and applications to H ∞-functional calculi

We study conical square function estimates for Banach-valued functions and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-M^cIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator A with certain off-diagonal bounds such that A always has a bounded H ^∞-functional calculus on these spaces. This provides a new way of proving functional calculus of A on the Bochner spaces L ^p (ℝ ⁿ ; X) by checking appropriate conical square function estimates and also a conical analogue ofBourgain’s extension of the Littlewood-Paley theory to the UMD-valued context. Even when X = ℂ, our approach gives refined p-dependent versions of known results.