Stein Square Research Articles

Littlewood–Paley–Stein square functions for the fractional discrete Laplacian on $$\mathbb {Z}$$

Littlewood–Paley–Stein square functions for the fractional discrete Laplacian on $$\mathbb {Z}$$

$$L^p$$-Boundedness of Stein’s Square Functions Associated with Fourier–Bessel Expansions

In this paper we prove \(L^p\) estimates for Stein’s square functions associated with Fourier–Bessel expansions. Furthermore, we prove transference results for square functions from Fourier–Bessel series to Hankel transforms. Actually, these are transference results for vector-valued multipliers from discrete to continuous in the Bessel setting. As a consequence, we deduce the sharpness of the range of p for the \(L^p\)-boundedness of Fourier–Bessel Stein’s square functions from the corresponding property for Hankel–Stein square functions. Finally, we deduce \(L^p\) estimates for Fourier–Bessel multipliers from that ones we have got for our Stein square functions.

Hörmander multiplier theorem with regularity in modulation spaces

In this paper, we mainly establish the Hörmander type multiplier theorem related to the modulation spaces M2,ps(Rn) for all 0<p≤1. This extends the result of Tomita (2009) when p=1 to all the cases 0<p≤1. We use the Littlewood–Paley–Stein square functions and atomic decompositions to establish our main result. Furthermore, our method is substantially different from that used in Tomita (2009) when p=1.

Mixed estimates for singular integrals on weighted Hardy spaces

In this paper we give quantitative bounds for the norms of different kinds of singular integral operators on weighted Hardy spaces $$H_w^p$$ , where $$0<p\le 1$$ and w is a weight in the Muckenhoupt $$A_{\infty }$$ class. We deal with Fourier multiplier operators, Calderon–Zygmund operators of homogeneous type which are particular cases of the first ones, and, more generally, we study singular integrals of convolution type. In order to prove mixed estimates in the setting of weighted Hardy spaces, we need to introduce several characterizations of weighted Hardy spaces by means of square functions, Littlewood–Paley functions and the grand maximal function. We also establish explicit quantitative bounds depending on the weight w when switching between the $$H^p_w$$ -norms defined by the Littlewood–Paley–Stein square function and its discrete version, and also by applying the mixed bound $$A_q-A_\infty $$ result for the vector-valued extension of the Hardy–Littlewood maximal operator given in Buckley (Trans Am Math Soc 340(1):253–272, 1993).

Discrete Littlewood–Paley–Stein characterization of multi-parameter local Hardy spaces

Abstract In our recent work [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380], the multi-parameter local Hardy space h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} has been introduced by using the continuous inhomogeneous Littlewood–Paley–Stein square functions. In this paper, we will first establish the new discrete multi-parameter local Calderón’s identity. Based on this identity, we will define the local multi-parameter Hardy space h dis p ⁢ ( ℝ n 1 × ℝ n 2 ) {h_{\mathrm{dis}}^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} by using the discrete inhomogeneous Littlewood–Paley–Stein square functions. Then we prove that these two multi-parameter local Hardy spaces are actually the same. Moreover, the norms of the multi-parameter local Hardy spaces under the continuous and discrete Littlewood–Paley–Stein square functions are equivalent. This discrete version of the multi-parameter local Hardy space is also critical in establishing the duality theory of the multi-parameter local Hardy spaces.

Hörmander type theorem on bi-parameter Hardy spaces for bi-parameter Fourier multipliers with optimal smoothness

The main purpose of this paper is to establish, using the bi-parameter Littlewood–Paley–Stein theory (in particular, the bi-parameter Littlewood–Paley–Stein square functions), a Hörmander–Mihlin type theorem for the following bi-parameter Fourier multipliers on bi-parameter Hardy spaces H^p(\mathbb{R}^{n_1}\!\times \mathbb{R}^{n_2}) ( 0 &lt; p\le 1 ) with optimal smoothness: T_mf(x_1, x_2)=\frac{1}{(2\pi)^{n_1+n_2}}\int_{\mathbb{R}^n\times \mathbb{R}^m}m(\xi, \eta)\hat{f}(\xi, \eta)e^{2\pi(x_1\xi+x_2\eta)}\,d\xi \,d\eta. One of our results (Theorem 1.7) is the following: assume that m(\xi,\eta) is a function on \mathbb{R}^{n_1}\!\times \mathbb{R}^{n_1} satisfying \sup_{j,k\in \mathbb{Z}}\|m_{j,k}\|_{W^{(s_1,s_2)}}&lt;\infty, with s_1 &gt; n_1({1}/{p}-{1}/{2}) , s_2 &gt; n_2({1}/{p}-{1}/{2}) . Then T_m is bounded from H^p(\mathbb{R}^{n_1}\!\times \mathbb{R}^{n_2}) to H^p(\mathbb{R}^{n_1}\!\times \mathbb{R}^{n_2}) for all 0 &lt; p\le 1 , and \|T_m\|_{H^p\rightarrow H^p}\lesssim \sup_{j,k\in \mathbb{Z}}\|m_{j,k}\|_{W^{(s_1,s_2)}}. Moreover, the smoothness assumption on s_1 and s_2 is optimal. Here, m_{j,k}(\xi,\eta)=m(2^j\xi,2^k\eta)\Psi(\xi)\Psi(\eta) , where \Psi(\xi) and \Psi(\eta) are suitable cut-off functions on \mathbb{R}^{n_1} and \mathbb{R}^{n_2} , respectively, and W^{(s_1, s_2)} is a two-parameter Sobolev space on \mathbb{R}^{n_1}\!\times \mathbb{R}^{n_2} . We also establish that under the same smoothness assumption on the multiplier m , \|T_m\|_{H^p\rightarrow L^p}\!\lesssim\! \sup_{j,k\in \mathbb{Z}}\|m_{j,k}\|_{W^{(s_1,s_2)}} and \|T_m\|_{{\rm CMO}_p\rightarrow {\rm {CMO}}_p}\lesssim \sup_{j,k\in \mathbb{Z}}\|m_{j,k}\|_{W^{(s_1,s_2)}} for all 0 &lt; p\le 1 . Moreover, \|T_m\|_{L^p\rightarrow L^p}\lesssim \sup_{j,k\in \mathbb{Z}}\|m_{j,k}\|_{W^{(s_1,s_2)}} for all 1 &lt; p &lt; \infty under the assumption s_1 &gt; n_1/2 and s_2 &gt; n_2/2 .

Square function estimates for the Bochner–Riesz means

We consider the square function (known as Stein's square function) estimate associated with the Bochner-Riesz means. The previously known range of sharp estimate is improved. Our results are based on vector valued extensions of Bennett-Carbery-Tao's multilinear (adjoint) restriction estimate and adaption of induction argument due to Bourgain-Guth. Unlike the previous work by Bourgain-Guth on $L^p$ boundedness of Bochner-Riesz means in which oscillatory operators associated to the kernel had been studied, we take more direct approach by working on the Fourier transform side. This enables us to obtain the correct order of smoothing which is essential for obtaining the sharp estimate for the square function.

Hardy space estimates for limited ranges of Muckenhoupt weights

In this article, we give a full necessary and sufficient set of conditions for a Calderón–Zygmund operator to be bounded on weighted Hardy spaces Hwp where w is an Muckenhoupt weight and 0<p<∞. In fact, this result is new even when 1<p<∞ since it allows for Hwp boundedness of an operator when 1<p<q<∞ and w∈Aq, where it is possible that Hwp≠Lwp. These singular integral results are achieved by proving Littlewood–Paley–Stein square function type estimates from Hwp into Lwp for 0<p<∞ and a Muckenhoupt weight w, which are interesting results in their own right. New techniques involving A∞ weight invariant spaces are also used to prove the weighted estimates for Calderón–Zygmund operators. More precisely, we prove the following BMO type weight invariance properties: for a fixed s≥0, the weighted Sobolev-BMO spaces Is(BMOw) coincide for all w∈A∞, the weighted p=∞ type Triebel–Lizorkin spaces F˙∞,ws,2 coincide for all w∈A∞, and these two classes of spaces coincide with each other as well, all of which have comparable norms up to constants depending on an Ap character of the weight w∈A∞.

Hardy Space Estimates for Littlewood–Paley–Stein Square Functions and Calderón–Zygmund Operators

In this work, we give new sufficient conditions for Littlewood–Paley–Stein square function operators and necessary and sufficient conditions for Calderon–Zygmund operators to be bounded on Hardy spaces $$H^p$$ with indices smaller than 1. New Carleson measure type conditions are defined for Littlewood–Paley–Stein operators, and the authors show that they are sufficient for the associated square function to be bounded from $$H^p$$ into $$L^p$$ . New polynomial growth $$BMO$$ conditions are also introduced for Calderon–Zygmund operators. These results are applied to prove that Bony paraproducts can be constructed such that they are bounded on Hardy spaces with exponents ranging all the way down to zero.

$H^{\infty}$ functional calculus and square function estimates for Ritt operators

A Ritt operator T\colon X\to X on a Banach space is a power bounded operator satisfying an estimate n\| T^{n}-T^{n-1}\| \leq C . When X=L^p(\Omega) for some 1\leq p \leq \infty , we study the validity of square functions estimates \| (\sum_k k|T^{k}(x) - T^{k-1}(x)|^2)^{1/2}\|_{L^p}\lesssim\ \|x\|_{L^p} for such operators. We show that T and T^* both satisfy such estimates if and only if T admits a bounded functional calculus with respect to a Stolz domain. This is a single operator analogue of the famous Cowling–Doust–McIntosh–Yagi characterization of bounded H^\infty -calculus on L^p -spaces by the boundedness of certain Littlewood–Paley–Stein square functions. We also prove a similar result for Hilbert spaces. Then we extend the above to more general Banach spaces, where square functions have to be defined in terms of certain Rademacher averages. We focus on noncommutative L^p -spaces, where square functions are quite explicit, and we give applications, examples, and illustrations on such spaces, as well as on classical L^p .

Boundedness of stein's square functions associated to operators on hardy spaces

Let (X, d, μ) be a metric measure space endowed with a metric d and a nonnegative Borel doubling measure μ. Let L be a second order non-negative self-adjoint operator on L2(X). Assume that the semigroup e-tL generated by L satisfies the Davies-Gaffney estimates. Also, assume that L satisfies Plancherel type estimate. Under these conditions, we show that Stein's square function Gδ(L) arising from Bochner-Riesz means associated to L is bounded from the Hardy spaces Hp(X) to Lp(X) for all 0 <p ≤ 1.

$L^p$-bounds for Stein's square functions associated to operators and applications to spectral multipliers

Let $(X, d, \mu)$ be a metric measure space endowed with a metric $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $X$. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables $x$ and $y$. Also assume that $L$ satisfies a Plancherel type estimate. Under these conditions, we show the $L^p$ bounds for Stein's square functions arising from Bochner-Riesz means associated to the operator $L$. We then use the $L^p$ estimates on Stein's square functions to obtain a Hormander-type criterion for spectral multipliers of $L$. These results are applicable for large classes of operators including sub-Laplacians acting on Lie groups of polynomial growth and Schrodinger operators with rough potentials.

A $$T(1)$$ -Theorem for non-integral operators

Let \(X\) be a space of homogeneous type and let \(L\) be a sectorial operator with bounded holomorphic functional calculus on \(L^2(X)\). We assume that the semigroup \(\{e^{-tL}\}_{t>0}\) satisfies Davies–Gaffney estimates. Associated with \(L\) are certain approximations of the identity. We call an operator \(T\) a non-integral operator if compositions involving \(T\) and these approximations satisfy certain weighted norm estimates. The Davies–Gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on \(T\) in Calderon–Zygmund theory. In this paper, we show, under the additional assumption that a vertical Littlewood–Paley–Stein square function associated with \(L\) is bounded on \(L^2(X)\), that a non-integral operator \(T\) is bounded on \(L^2(X)\) if and only if \(T(1) \in BMO_L(X)\) and \(T^{*}(1) \in BMO_{L^{*}}(X)\). Here, \(BMO_L(X)\) and \(BMO_{L^{*}}(X)\) denote the recently defined \(BMO(X)\) spaces associated with \(L\) that generalize the space \(BMO(X)\) of John and Nirenberg. Generalizing a recent result due to F. Bernicot, we show a second version of a \(T(1)\)-Theorem under weaker off-diagonal estimates, which gives a positive answer to a question raised by him. As an application, we prove \(L^2(X)\)-boundedness of a paraproduct operator associated with \(L\). We moreover study criterions for a \(T(b)\)-Theorem to be valid.

Harmonic analysis operators related to symmetrized Jacobi expansions

Following a symmetrization procedure proposed recently by Nowak and Stempak, we consider the setting of symmetrized Jacobi expansions. In this framework we investigate mapping properties of several fundamental harmonic analysis operators, including Riesz transforms, Poisson semigroup maximal operator, Littlewood–Paley–Stein square functions and multipliers of Laplace and Laplace–Stieltjes transform type. Our paper delivers also some new results in the original setting of classical Jacobi expansions.

Improved bounds for Stein's square functions

We prove a weighted norm inequality for the maximal Bochner--Riesz operator and the associated square-function. This yields new $L^p(R^d)$ bounds on classes of radial Fourier multipliers for $p\ge 2+4/d$ with $d\ge 2$, as well as space-time regularity results for the wave and Schr\"odinger equations.

Calderón–Zygmund operators in the Bessel setting

We study several fundamental operators in harmonic analysis related to Bessel operators, including maximal operators related to heat and Poisson semigroups, Littlewood–Paley–Stein square functions, multipliers of Laplace transform type and Riesz transforms. We show that these are (vector-valued) Calderon–Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory.

Weighted Norm Inequalities on Graphs

Let (Γ,μ) be an infinite graph endowed with a reversible Markov kernel p and let P be the corresponding operator. We also consider the associated discrete gradient ∇. We assume that μ is doubling, a uniform lower bound for p(x,y) when p(x,y)>0, and gaussian upper estimates for the iterates of p. Under these conditions (and in some cases assuming further some Poincare inequality) we study the comparability of (I−P)1/2 f and ∇f in Lebesgue spaces with Muckenhoupt weights. Also, we establish weighted norm inequalities for a Littlewood–Paley–Stein square function, its formal adjoint, and commutators of the Riesz transform with bounded mean oscillation functions.

Uniform estimates for spherical functions on complex semisimple Lie groups

We prove new estimates for spherical functions and their derivatives on complex semisimple Lie groups, establishing uniform polynomial decay in the spectral parameter. This improves the customary estimate arising from Harish-Chandra's series expansion, which gives only a polynomial growth estimate in the spectral parameter. In particular, for arbitrary positive-definite spherical functions \( \varphi_\lambda \) on higher rank complex simple groups, we establish estimates for \( |d^k\varphi_\lambda(a_t)/dt^k| \) which are of the form \( (1\,+ \parallel \lambda \parallel)^{k-\gamma} \) in the spectral parameter \( \lambda \) and have uniform exponential decay in regular directions in the group variable a t . Here \( \gamma \) is an explicit constant depending on G, and \( \lambda \) may be singular, for instance.¶The uniformity of the estimates is the crucial ingredient needed in order to apply classical spectral methods and Littlewood—Paley—Stein square functions to the analysis of singular integrals in this context. To illustrate their utility, we prove maximal inequalities in L p for singular sphere averages on complex semisimple Lie groups for all p in \( (1\,+ {1 \over 2\gamma}, \infty) \). We use these to establish singular differentiation theorems and pointwise ergodic theorems in L p for the corresponding singular spherical averages on locally symmetric spaces, as well as for more general measure preserving actions.