Littlewood-Paley Theory Research Articles

Riesz transforms and multipliers for the Grushin operator

We show that Riesz transforms associated to the Grushin operator G = -\Delta - |x|^2\partial_t^2 are bounded on L^p(R^n+1). We also establish an analogue of H\"ormander-Mihlin multiplier theorem and study Bochner-Riesz means associated to the Grushin operator. The main tools used are Littlewood-Paley theory and an operator valued Fourier multiplier theorem due to L. Weis.

On a Class of Bilinear Pseudodifferential Operators

We provide a direct proof for the boundedness of pseudodifferential operators with symbols in the bilinear Hörmander classBS1,δ0,0≤δ<1. The proof uses a reduction to bilinear elementary symbols and Littlewood-Paley theory.

Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters

Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderon’s identity. This is inspired by the work of discrete Littlewood-Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe’s covering lemma in multiparameter setting.

The maximal Hilbert transform along nonconvex curves

The Hilbert transform along curves is defined by the principal value integral. The pointwise existence of the principal value Hilbert transform can be educed from the appropriate estimates for the corresponding maximal Hilbert transform. By using the estimates of Fourier transforms and Littlewood-Paley theory, we obtain L p -boundedness for the maximal Hilbert transform associated to curves (t,P(γ(t))), where 1<p<∞, P is a real polynomial and γ is convex on [0,∞). Then, we can conclude that the Hilbert transform along curves (t,P(γ(t))) exists in pointwise sense.MSC:42B20, 42B25.

Well-posedness of the Hele–Shaw–Cahn–Hilliard system

We study the well-posedness of the Hele–Shaw–Cahn–Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. For initial data in Hs, s>d2+1, the existence and uniqueness of solution in C([0,T];Hs)∩L2(0,T;Hs+2) that is global in time in the two dimensional case (d=2) and local in time in the three dimensional case (d=3) are established. Several blow-up criterions in the three dimensional case are provided as well. One of the tools that we utilized is the Littlewood–Paley theory in order to establish certain key commutator estimates.

Calderón–Zygmund Operators on Weighted Hardy Spaces

We study the boundedness of Calderon–Zygmund operators on weighted Hardy spaces \(H^p_w\) using Littlewood-Paley theory. It is shown that if a Calderon–Zygmund operator T satisfies T *1 = 0, then T is bounded on \(H^p_w\) for \(w\in A_{p(1+\frac\varepsilon n)}\) and \(\frac n{n+\varepsilon}<p\le1\), where e is the regular exponent of the kernel of T.

Local existence and blow‐up criterion for the generalized Boussinesq equations in Besov spaces

In this paper, we consider the three‐dimensional generalized Boussinesq equations, a system of equations resulting from replacing the Laplacian − Δ in the usual Boussinesq equations by a fractional Laplacian ( − Δ)α. We prove the local existence in time and obtain a regularity criterion of solution for the generalized Boussinesq equations by means of the Littlewood–Paley theory and Bony's paradifferential calculus. The results in this paper can be regarded as an extension to the Serrin‐type criteria for Navier–Stokes equations and magnetohydrodynamics equations, respectively. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Global well-posedness for Navier–Stokes equations in critical Fourier–Herz spaces

We prove the global well-posedness for the 3D Navier–Stokes equations in critical Fourier–Herz spaces, by making use of the Fourier localization method and the Littlewood–Paley theory. The advantage of working in Fourier–Herz spaces lies in that they are more adapted than classical Besov spaces, for estimating the bilinear paraproduct of two distributions with the summation of their regularity indexes exactly zero. Our result is an improvement of a recent theorem by Lei and Lin (2011) [10].

Blow-up criterion for two-dimensional magneto-micropolar fluid equations with partial viscosity

In this paper, we derive a blow-up criterion of smooth solutions to the incompressible magneto-micropolar fluid equations with partial viscosity in two space dimensions. Our proof is based on careful Hölder estimates of heat and transport equations and the standard Littlewood–Paley theory. Copyright © 2011 John Wiley & Sons, Ltd.

On the well-posedness for Keller-Segel system with fractional diffusion

Communicated by M. Costabel In this paper, we study the Cauchy problem for the Keller–Segel system with fractional diffusion generalizing the Keller–Segel model of chemotaxis for the initial data (u0,v0) in critical Fourier-Herz spaces B˙q2−2αRn×B˙q2−2αRn with q ∈ [2, ∞], where 1 < α ≤ 2. Making use of some estimates of the linear dissipative equation in the frame of mixed time-space spaces, the Chemin ‘mono-norm method’, the Fourier localization technique and the Littlewood–Paley theory, we get a local well-posedness result and a global well-posedness result with a small initial data. In addition, ill-posedness for ‘doubly parabolic’ models is also studied. Copyright © 2011 John Wiley & Sons, Ltd.

Global Existence for the Two-Dimensional Euler Equations in a Critical Besov Space

In this paper we prove the global existence for the two-dimensional Euler equations in the critical Besov space. Making use of a new estimate of transport equation and Littlewood-Paley theory, we get the global existence result.

Bounds of Singular Integrals on Weighted Hardy Spaces and Discrete Littlewood–Paley Analysis

We apply the discrete version of Calderon’s reproducing formula and Littlewood–Paley theory with weights to establish the \(H^{p}_{w} \to H^{p}_{w}\) (0 qw=inf {s:w∈As}. Our results can be regarded as a natural extension of the results about the growth of the Ap constant of singular integral operators on classical weighted Lebesgue spaces \(L^{p}_{w}\) in Hytonen et al. (arXiv:1006.2530, 2010; arXiv:0911.0713, 2009), Lerner (Ill. J. Math. 52:653–666, 2008; Proc. Am. Math. Soc. 136(8):2829–2833, 2008), Lerner et al. (Int. Math. Res. Notes 2008:rnm 126, 2008; Math. Res. Lett. 16:149–156, 2009), Lacey et al. (arXiv:0905.3839v2, 2009; arXiv:0906.1941, 2009), Petermichl (Am. J. Math. 129(5):1355–1375, 2007; Proc. Am. Math. Soc. 136(4):1237–1249, 2008), and Petermichl and Volberg (Duke Math. J. 112(2):281–305, 2002). Our main result is stated in Theorem 1.1. Our method avoids the atomic decomposition which was usually used in proving boundedness of singular integral operators on Hardy spaces.

VMO spaces associated with divergence form elliptic operators

Consider the second order divergence form elliptic operator L with complex bounded coefficients. In this paper we introduce and develop a function space VMOL of vanishing mean oscillation associated with the operator L. Using the theory of tent spaces and the Littlewood-Paley theory, we prove that the Hardy space \({H_{L}^1}\) of Hofmann and Mayboroda introduced in Hofmann and Mayboroda (Math Ann 344:37–116, 2009) is the dual of our \({{\rm VMO}_{L^{\ast}}}\), in which L* is the adjoint operator of L. We also give an equivalent characterization of the space VMOL in the context of the theory of tent spaces.

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 L2-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 Calderon-type reproducing formula and then prove L2-boundedness of newly defined square-like functions.

Weighted estimates for strongly singular integral operators with rough kernels

The Fourier transform and the Littlewood-Paley theory are used to give the weighted boundedness of a strongly singular integral operator defined in this paper. The paper shows that the strongly singular integral operator is bounded from the Sobolev space to the Lebesgue space.

A blow-up criterion for 3D Boussinesq equations in Besov spaces

In this paper, we consider the 3D Boussinesq equations with the incompressibility condition. We obtain a regularity condition for the three-dimensional Boussinesq equations by means of the Littlewood–Paley theory and Bony’s paradifferential calculus.

Littlewood–Paley Theory for the Differential Operator $\frac{\partial^2}{\partial x_1^2}\frac{\partial^2}{\partial x_2^2}-\frac{\partial^2}{\partial x_3^2}$

Littlewood–Paley theory for the differential operator, \Delta_{\mathbf D} = \partial^2_{x_1}\partial^2_{x_2} − \partial^2_{x_3} , is developed. This study leads to the introduction of a new class of Triebel–Lizorkin spaces \dot F^{\alpha,q}_p (\mathbb D) associated with the dilation (x_1,x_2,x_3) \to (2^{\nu_1}x_1,2^{ν_2}x_2,2^{\nu_1+\nu_2}x_3) , (\nu_1,\nu_2) \in \mathbb Z^2 . The corresponding atomic and molecular decompositions are obtained. A frame generated by modulations, dilations and translations is also studied. Using this result, we show that \Delta_{\mathbb D} is a linear isomorphism from \dot F^{\alpha,q}_p to \dot F^{\alpha-2,q}_p .

Weighted Hardy spaces in the three-parameter case

In this paper, we use the idea of the discrete Littlewood–Paley theory developed by Han and Lu to carry out the three-parameter weighted Hardy spaces theory under a rather weak condition on the product weight ( w ∈ A ∞ ) and obtain the boundedness of singular integral operators on the weighted Hardy spaces.

Note on Gradient Estimate of Heat Kernel for Schr&amp;#246;dinger Operators

Let be a Schrödinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists in the Schwartz class such that the long time gradient heat kernel estimate fails.

Calderón–Zygmund operators on product Hardy spaces

Let T be a product Calderón–Zygmund singular integral introduced by Journé. Using an elegant rectangle atomic decomposition of H p ( R n × R m ) and Journé's geometric covering lemma, R. Fefferman proved the remarkable H p ( R n × R m ) − L p ( R n × R m ) boundedness of T. In this paper we apply vector-valued singular integral, Calderón's identity, Littlewood–Paley theory and the almost orthogonality together with Fefferman's rectangle atomic decomposition and Journé's covering lemma to show that T is bounded on product H p ( R n × R m ) for max { n n + ε , m m + ε } < p ⩽ 1 if and only if T 1 ∗ ( 1 ) = T 2 ∗ ( 1 ) = 0 , where ε is the regularity exponent of the kernel of T.