Orthogonality Principle for Bilinear Littlewood–Paley Decompositions

We explore Littlewood–Paley like decompositions of bilinear Fourier multipliers. Grafakos and Li (Am. J. Math. 128(1):91–119 2006) showed that a bilinear symbol supported in an angle in the positive quadrant is bounded from \(L^p\times L^q\) into \(L^r\) if its restrictions to dyadic annuli are bounded bilinear multipliers in the local \(L^2\) case \(p\ge 2\) , \(q\ge 2\) , \(r= 1/(p^{-1}+q^{-1})\le 2\) . We show that this range of indices is sharp and also discuss similar results for multipliers supported near axis and negative diagonal.     

Some light on Littlewood–Paley theory

The purpose of this note is to correct an error in a paper of M. Cowling, G. Fendler and J.J.F. Fournier, and to give a counterexample to a conjecture of J.-L. Rubio de Francia. Received: 13 December 2000 / Published online: 24 September 2001     

Mixed A p -A r Inequalities for Classical Singular Integrals and Littlewood–Paley Operators

We prove mixed A _p -A _r inequalities for several basic singular integrals, Littlewood–Paley operators, and the vector-valued maximal function. Our key point is that r can be taken arbitrarily big. Hence, such inequalities are close in spirit to those obtained recently in the works by T. Hytönen and C. Pérez, and M. Lacey. On one hand, the “A _p -A _∞” constant in these works involves two independent suprema. On the other hand, the “A _p -A _r ” constant in our estimates involves a joint supremum, but of a bigger expression. We show in simple examples that both such constants are incomparable. This leads to a natural conjecture that the estimates of both types can be further improved.     

Area Littlewood–Paley Functions Associated with Hermite and Laguerre Operators

In this paper we study L ^p —boundedness properties for area Littlewood–Paley functions associated with heat semigroups for Hermite and Laguerre operators.     

Littlewood–Paley and Spectral Multipliers on Weighted L p Spaces

Let L be a non-negative self-adjoint operator acting on L ²(X), where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e ^?tL whose kernel p _t (x,y) has a Gaussian upper bound but there is no assumption on the regularity in variables x and y. In this article we study weighted L ^p -norm inequalities for spectral multipliers of L. We show that a weighted Hörmander-type spectral multiplier theorem follows from weighted L ^p -norm inequalities for the Lusin and Littlewood–Paley functions, Gaussian heat kernel bounds, and appropriate L ² estimates of the kernels of the spectral multipliers.     

Littlewood Paley g-function on the Heisenberg Group

We consider the g-function related to a class of radial functions which gives a characterization of the L^p-norm of a function on the Heisenberg group.     

Littlewood–Paley Operators on Spaces with Variable Exponent on Homogeneous Groups

In this paper,on homogeneous groups,we study the Littlewood–Paley operators in variable exponent spaces.First,we prove that the weighted Littlewood–Paley operators are controlled by the weighted Hardy–Littlewood maximal function,and obtain the vector-valued inequalities of the Littlewood–Paley operators,including the Lusin function,Littlewood–Paley g function and gλ* function.Second,we prove the boundedness of multilinear Littlewood–Paley gψ,λ* function.     

A Characterization of Two-Weight Norm Inequality for Littlewood–Paley $$g_{\lambda }^{*}$$-Function

Let \(n\ge 2\) and \(g_{\lambda }^{*}\) be the well-known high-dimensional Littlewood–Paley function which was defined and studied by E. M. Stein,

$$\begin{aligned} g_{\lambda }^{*}(f)(x) =\bigg (\iint _{\mathbb {R}^{n+1}_{+}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda } |\nabla P_tf(y,t)|^2 \frac{\mathrm{d}y \mathrm{d}t}{t^{n-1}}\bigg )^{1/2}, \ \quad \lambda > 1, \end{aligned}$$

where \(P_tf(y,t)=p_t*f(y)\), \(p_t(y)=t^{-n}p(y/t)\), and \(p(x) = (1+|x|^2)^{-(n+1)/2}\), \(\nabla =(\frac{\partial }{\partial y_1},\ldots ,\frac{\partial }{\partial y_n},\frac{\partial }{\partial t})\). In this paper, we give a characterization of two-weight norm inequality for \(g_{\lambda }^{*}\)-function. We show that \(\big \Vert g_{\lambda }^{*}(f \sigma ) \big \Vert _{L^2(w)} \lesssim \big \Vert f \big \Vert _{L^2(\sigma )}\) if and only if the two-weight Muckenhoupt \(A_2\) condition holds, and a testing condition holds:

$$\begin{aligned} \sup _{Q : \text {cubes}~\mathrm{in} \ {\mathbb {R}^n}} \frac{1}{\sigma (Q)} \int _{{\mathbb {R}^n}} \iint _{\widehat{Q}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda }|\nabla P_t(\mathbf {1}_Q \sigma )(y,t)|^2 \frac{w \mathrm{d}x \mathrm{d}t}{t^{n-1}} \mathrm{d}y < \infty , \end{aligned}$$

where \(\widehat{Q}\) is the Carleson box over Q and \((w, \sigma )\) is a pair of weights. We actually prove this characterization for \(g_{\lambda }^{*}\)-function associated with more general fractional Poisson kernel \(p^\alpha (x) = (1+|x|^2)^{-{(n+\alpha )}/{2}}\). Moreover, the corresponding results for intrinsic \(g_{\lambda }^*\)-function are also presented.

     

加权Orlicz空间上的Littlewood Paley算子

该文证明了一个与Littlewood Paley 算子有关的不等式，并由此导出Littlewood Paley 算子在加权Orlicz 空间和Morrey空间的有界特征。     

Lp estimates for non-smooth bilinear Littlewood–Paley square functions on $${\mathbb{R}}$$

In this work, we study some non-smooth bilinear analogues of linear Littlewood–Paley square functions on the real line. We prove boundedness-properties in Lebesgue spaces for them. Let us consider the functions \({\phi_{n}}\) satisfying \({\widehat{\phi_n}(\xi)={\bf 1}_{[n,n+1]}(\xi)}\) and define the bilinear operator \({S_n(f,g)(x):=\int f(x+y)g(x-y) \phi_n(y) dy}\) . These bilinear operators are closely related to the bilinear Hilbert transforms. Then for exponents \({p,q,r'\in[2,\infty)}\) satisfying \({\frac{1}{p}+\frac{1}{q}=\frac{1}{r}}\) , we prove that

$\left\| \left( \sum_{n\in \mathbb{Z}}\left|S_n(f,g) \right|^2 \right)^{1/2}\right\|_{L^{r}(\mathbb{R})}\lesssim \|f\|_{L^p(\mathbb{R})}\|g\|_{L^q(\mathbb{R})}.$

     

Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the p−1 problem

We utilize the method of Bellman functions to derive new $L^p$-estimates of Littlewood–Paley type involving $p?1$. Among the applications to singular integrals we improve the $2(p?1)$ bounds for the Ahlfors–Beurling operator on $L^p(\mathbb{C})$ when $p\to \infty$. In addition, dimensionless estimates of Riesz transforms in the classical as well as in the Ornstein–Uhlenbeck setting are attained. To cite this article: O. Dragi?evi?, A. Volberg, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

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

We apply the discrete version of Calderón??s reproducing formula and Littlewood?CPaley theory with weights to establish the $H^{p}_{w} \to H^{p}_{w}$ (0<p<??) and $H^{p}_{w}\to L^{p}_{w}$ (0<p??1) boundedness for singular integral operators and derive some explicit bounds for the operator norms of singular integrals acting on these weighted Hardy spaces when we only assume w??A _??. The bounds will be expressed in terms of the A _q constant of w if q>q _w =inf?{s:w??A _s }. Our results can be regarded as a natural extension of the results about the growth of the A _p 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?C666, 2008; Proc. Am. Math. Soc. 136(8):2829?C2833, 2008), Lerner et?al. (Int.?Math. Res. Notes 2008:rnm 126, 2008; Math. Res. Lett. 16:149?C156, 2009), Lacey et?al. (arXiv:0905.3839v2, 2009; arXiv:0906.1941, 2009), Petermichl (Am. J. Math. 129(5):1355?C1375, 2007; Proc. Am. Math. Soc. 136(4):1237?C1249, 2008), and Petermichl and Volberg (Duke Math. J. 112(2):281?C305, 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.     

Endpoint Estimates of Generalized Homogeneous Littlewood–Paley g-functions over Non-Homogeneous Metric Measure Spaces

Let(X,d,μ) be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions.Under the weak reverse doubling condition,the authors prove that the generalized homogeneous Littlewood–Paley g-function gr(r∈[2,∞)) is bounded from Hardy space H~1(μ) into L~1(μ).Moreover,the authors show that,if f∈RBMO(μ),then [gr(f)]~r is either infinite everywhere or finite almost everywhere,and in the latter case,[gr(f)]~r belongs to RBLO(μ) with the norm no more than ||f||_(RBMO(μ)) multiplied by a positive constant which is independent of f.As a corollary,the authors obtain the boundedness of gr from RBMO(μ) into RBLO(μ).The vector valued Calderón–Zygmund theory over(X,d,μ) is also established with details in this paper.     

Paley–Wiener–Schwartz type theorem for the wavelet transform

In the present paper, we discuss about extension of the wavelet transform on distribution space of compact support and develop the Paley–Wiener–Schwartz type theorem for the wavelet transform on the same. Furthermore, Paley–Wiener–Schwartz type theorem for the wavelet transform is also established using the relation between the wavelet transform and double Fourier transform.     

Skew Pieri rules for Hall–Littlewood functions

We produce skew Pieri rules for Hall–Littlewood functions in the spirit of Assaf and McNamara (J. Comb. Theory Ser. A 118(1):277–290, 2011). The first two were conjectured by the first author (Konvalinka in J. Algebraic Comb. 35(4):519–545, 2012). The key ingredients in the proofs are a q-binomial identity for skew partitions and a Hopf algebraic identity that expands products of skew elements in terms of the coproduct and the antipode.