Littlewood-Paley Function Research Articles

Sharp analytic version of Fefferman's inequality

Let T be a unit circle. Assume further that f is an element of the Hardy space H1(T) and g belongs to the analytic BMO space on T. The paper contains the identification of the optimal universal constant C in the estimate|12π∫Tf(ζ)‾g(ζ)dζ|≤C‖f‖H1(T)‖g‖BMO(T). Actually, the inequality is studied in the stronger form, involving the Littlewood-Paley function on the left and the sharp maximal function of g on the right. The proof rests on the construction of an appropriate plurisuperharmonic function on a parabolic domain and the application of probabilistic techniques.

Littlewood-Paley functions without Fourier transform estimates

We prove Lp estimates for a class of Littlewood-Paley functions on homogeneous groups including the Heisenberg groups under a sharp integrability condition on the kernels. The results will be shown by methods without use of Fourier transform estimates.

Endpoint estimates for harmonic analysis operators associated with Laguerre polynomial expansions

AbstractIn this paper, we give a criterion to prove boundedness results for several operators from the Hardy‐type space to and also from to the space of functions of bounded mean oscillation , with respect to the probability measure on when is a multi‐index in . We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood–Paley functions, multipliers of the Laplace transform type, fractional integrals, and variation operators in the Laguerre setting.

Characterizations of a class of weighted 𝐵𝑀𝑂 spaces via commutators of intrinsic square functions

The intrinsic square functions including the Lusin area function, Littlewood-Paley g g -function and g λ ∗ g^\ast _\lambda -function dominate pointwisely the classical Littlewood-Paley functions and can be used to characterize the weighted Hardy spaces and more general Musielak-Orlicz Hardy spaces etc. This paper shows that for b ∈ B M O ( R n ) b\in \mathrm {BMO(\mathbb {R}^n)} , the commutators generated by these intrinsic square functions with b b are bounded from H ω p ( R n ) H^p_\omega (\mathbb {R}^n) to L ω p ( R n ) L^p_\omega (\mathbb {R}^n) for some 0 > p ≤ 1 0>p\le 1 and ω ∈ A ∞ \omega \in A_\infty if and only if b ∈ B M O ω , p ( R n ) b\in \mathcal {BMO}_{\omega ,p}(\mathbb {R}^n) , which are a class of non-trivial subspaces of B M O ( R n ) \mathrm {BMO(\mathbb {R}^n)} .

Vertical Littlewood–Paley functions related to a Schrödinger operator

Vertical Littlewood–Paley functions related to a Schrödinger operator

On the Weak Boundedness of Multilinear Littlewood–Paley Functions

On the Weak Boundedness of Multilinear Littlewood–Paley Functions

Boundedness for Littlewood-Paley functions on weighted Morrey-type spaces related to certain nonnegative potentials

Boundedness for Littlewood-Paley functions on weighted Morrey-type spaces related to certain nonnegative potentials

Anisotropic ball Campanato-type function spaces and their applications

Let A be a general expansive matrix and let X be a ball quasi-Banach function space on $${\mathbb {R}}^n$$ , whose certain power (namely its convexification) supports a Fefferman–Stein vector-valued maximal inequality and the associate space of whose other power supports the boundedness of the powered Hardy–Littlewood maximal operator. The authors first introduce some anisotropic ball Campanato-type function spaces associated with both A and X, prove that these spaces are dual spaces of anisotropic Hardy spaces $$H_X^A({\mathbb {R}}^n)$$ associated with both A and X, and obtain various anisotropic Littlewood–Paley function characterizations of $$H_X^A({\mathbb {R}}^n)$$ . Also, as applications, the authors establish several equivalent characterizations of anisotropic ball Campanato-type function spaces, which, combined with the atomic decomposition of tent spaces associated with both A and X, further induce their Carleson measure characterization. All these results have a wide range of generality and, particularly, even when they are applied to Morrey spaces and Orlicz-slice spaces, some of the obtained results are also new. The novelties of this article are reflected in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of the quasi-norm $$\Vert \cdot \Vert _X$$ under consideration, the authors embed X under consideration into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of this weighted Lebesgue space.

Maximal Operators, Littlewood–Paley Functions and Variation Operators Associated with Nonsymmetric Ornstein–Uhlenbeck Operators

In this paper, we establish L^p boundedness properties for maximal operators, Littlewood–Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein–Uhlenbeck operators. We consider the Ornstein–Uhlenbeck operators defined by the identity as the covariance matrix and having a drift given by the matrix -lambda (I+R), being lambda >0 and R a skew-adjoint matrix. The semigroups associated with these Ornstein–Uhlenbeck operators are the basic building blocks of all the normal Ornstein–Uhlenbeck semigroups.

Quantitative Weighted Bounds for Littlewood-Paley Functions Generated by Fractional Heat Semigroups Related with Schrödinger Operators

Let L = − Δ + V be a Schrödinger operator on ℝ n , where Δ denotes the Laplace operator ∑ i = 1 n ∂ 2 / ∂ x i 2 and V is a nonnegative potential belonging to a certain reverse Hölder class R H q ℝ n with q > n / 2 . In this paper, by the regularity estimate of the fractional heat kernel related with L , we establish the quantitative weighted boundedness of Littlewood-Paley functions generated by fractional heat semigroups related with the Schrödinger operators.

Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations

Abstract Let ( 𝕏 , d , μ ) {(\mathbb{X},d,\mu)} be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and let X ⁢ ( 𝕏 ) {X(\mathbb{X})} be a ball quasi-Banach function space on 𝕏 {\mathbb{X}} . In this article, the authors introduce the weak Hardy space W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} associated with X ⁢ ( 𝕏 ) {X(\mathbb{X})} via the Lusin area function. Then the authors characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the molecule, the grand maximal function, and the Littlewood–Paley g-function and g λ * {g^{*}_{\lambda}} -function. Moreover, all these results have a wide generality. Particularly, the results of this article are also new even when they are applied, respectively, to weighted Lebesgue spaces, Orlicz spaces, and variable Lebesgue spaces, which actually are new even on RD-spaces (that is, spaces of homogeneous type with additional reverse doubling condition). The main novelties of this article exist in that the authors take full advantage of the geometrical properties of 𝕏 {\mathbb{X}} expressed by both the dyadic cubes and the exponential decay of the approximations of the identity to overcome the difficulties caused by the deficiencies of both the explicit expression of the quasi-norm of X ⁢ ( 𝕏 ) {X(\mathbb{X})} and the reverse doubling condition of μ, and that the authors use the tent space on 𝕏 × ℤ {\mathbb{X}\times\mathbb{Z}} to characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the Littlewood–Paley g λ * {g^{*}_{\lambda}} -function, where the range of λ might be best possible in some cases.

Quantitative weighted estimates for Harmonic analysis operators in the Bessel setting by using sparse domination

In this paper we obtain quantitative weighted L^p-inequalities for some operators involving Bessel convolutions. We consider maximal operators, Littlewood-Paley functions and variational operators. We obtain L^p(w)-operator norms in terms of the A_p-characteristic of the weight w. In order to do this we show that the operators under consideration are dominated by a suitable family of sparse operators in the space of homogeneous type ((0,infty ),|cdot |,x^{2lambda }dx).

Littlewood-Paley-Stein Theory and Banach Spaces in the Inverse Gaussian Setting

In this paper we consider Littlewood-Paley functions defined by the semigroups associated with the operator mathcal {A}=-frac {1}{2}{Delta }-xnabla in the inverse Gaussian setting for Banach valued functions. We characterize the uniformly convex and smooth Banach spaces by using L^{p}(mathbb R^{n},gamma _{-1})- properties of the mathcal {A}-Littlewood-Paley functions. We also use Littlewood-Paley functions associated with mathcal {A} to characterize the Köthe function spaces with the UMD property.

Sublinear operators on Herz–Hardy spaces with variable exponents

AbstractIn this paper, we establish the mapping properties of sublinear operators on Herz–Hardy spaces with variable exponents. We obtain these mapping properties by extending the extrapolation theory to Herz–Hardy spaces with variable exponents. In particular, we study the mapping properties of the multiplier operators, the Littlewood–Paley functions and the parametric Marcinkiewicz integrals on the Herz–Hardy spaces with variable exponents.

Littlewood–Paley Characterization for Musielak–Orlicz–Hardy Spaces Associated with Self-Adjoint Operators

Let X , d , μ be a metric measure space endowed with a metric d and a non-negative Borel doubling measure μ . Let L be a non-negative self-adjoint operator on L 2 X . Assume that the (heat) kernel associated to the semigroup e − t L satisfies a Gaussian upper bound. In this paper, we prove that the Musielak–Orlicz–Hardy space H φ , L X associated with L in terms of the Lusin-area function and the Musielak–Orlicz–Hardy space H L , G , φ X associated with L in terms of the Littlewood–Paley function coincide and their norms are equivalent. To do this, we first establish the discrete characterization of these two spaces. It improves the known results in the literature.

Singular integrals and sublinear operators on amalgam spaces and Hardy-amalgam spaces

In this paper, we establish the extrapolation theory for the amalgam spaces and the Hardy-amalgam spaces. By using the extrapolation theory, we obtain the mapping properties for the Calderón-Zygmund operators and its commutator, the Carleson operators and establish the Rubio de Francia inequalities for Littlewood-Paley functions of arbitrary intervals to the amalgam spaces. We also obtain the boundedness of the Calder{ó}n-Zygmund operators and the intrinsic square function on the Hardy-amalgam spaces.

Extrapolation to mixed norm spaces and applications

This paper establishes extrapolation theory to mixed norm spaces. By applying this extrapolation theory, we obtain the mapping properties of the Rubio de Francia Littlewood-Paley functions and the geometrical maximal functions on mixed norm spaces. As special cases of these results, we have the mapping properties on the mixed norm Lebesgue spaces with variable exponents and the mixed norm Lorentz spaces.

Singular integral operators and sublinear operators on Hardy local Morrey spaces with variable exponents

This paper establishes the mapping properties of the singular integral operators, the Littlewood-Paley functions and the maximal Bochner-Riesz means on the Hardy local Morrey spaces with variable exponents by using extrapolation theory.

Littlewood-Paley-Stein Estimates for Non-Local Dirichlet Forms

We obtain the boundedness in Lp spaces for all 1 < p < ∞ of the so-called vertical Littlewood-Paley functions for non-local Dirichlet forms in the metric measure space under some mild assumptions. For 1 < p ⩽ 2, the pseudo-gradient is introduced to overcome the difficulty that chain rules are not available for non-local operators, and then the Mosco convergence is used to pave the way from the uniformly bounded jumping kernel case to the general case, while for 2 ⩽ p ⩽ ∞, the Burkholder-Davis-Gundy inequality is effectively applied. The former method is analytic and the latter one is probabilistic. The results extend those for pure jump symmetric Lévy processes in Euclidean spaces.

Real‐variable characterizations of local Hardy spaces on spaces of homogeneous type

AbstractLet be a space of homogeneous type, with upper dimension μ, in the sense of R. R. Coifman and G. Weiss. Let η be the Hölder regularity index of wavelets constructed by P. Auscher and T. Hytönen. In this article, the authors introduce the local Hardy space via local grand maximal functions and also characterize via local radial maximal functions, local non‐tangential maximal functions, local atoms and local Littlewood–Paley functions. Furthermore, the authors establish the relationship between the global and the local Hardy spaces. Finally, the authors also obtain the finite atomic characterizations of . As an application, the authors give the dual spaces of when , which further completes the result of G. Dafni and H. Yue on the dual space of . This article also answers the question of R. R. Coifman and G. Weiss on the nonnecessity of any additional geometric assumptions except the doubling condition for the radial maximal function characterization of when .