Writing $(T_R^{\lambda }f)\widehat{}\left(\xi \right)=(1-|\xi {|}^2/R^2{)}_{+}^{\lambda }\widehat{f}\left(\xi \right)$. E. Stein conjectured$$\parallel {\left(\sum _j|T_{R_j}^{\lambda }{f}_i{|}^2\right)}^{1/2}{\parallel }_p\le C\parallel {\left(\sum _j|f_j{|}^2\right)}^{1/2}{\parallel }_p$$for $\lambda \>0$, $\frac{4}{3}\le p\le 4$ and $C=C_{\lambda ,p}$. We prove this conjecture. We prove also $f\left(x\right)={lim}_{j\to \infty }{T}_{2^j}^{\lambda }f\left(x\right)$ a.e. We only assume $\frac{4}{3+2\lambda }\<p\<\frac{4}{1-2\lambda }$.

We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, $L^p$ and BMO of classical ℒ-square functions.

We establish L2 and Lp bounds for a class of square functions which arises in the study of singular integrals and boundary value problems in non-smooth domains. As an application, we present a simplified treatment of a class of parabolic smoothing operators which includes the caloric single layer potential on the boundary of certain minimally smooth, non-cylindrical domains.

A number of recent results in Euclidean harmonic analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen...

We discuss recent progress on issues surrounding the Brascamp–Lieb inequalities.

In this paper we study the Cauchy problem for viscous shallow water equations. We work in the Sobolev spaces of index s > 2 to obtain local solutions for any initial data, and global solutions for small initial data.

In this work we give sufficient and necessary conditions for the boundedness of the fractional integral operator acting between weighted Orlicz spaces and suitable $BM{O}_{\phi }$ spaces, in the general setting of spaces of homogeneous type. This result generalizes those contained in [P1] and [P2] about the boundedness of the same operator acting between weighted $L^p$ and Lipschitz integral spaces on ${ℝ}^n$. We also give some properties of the classes of pairs of weights appearing in connection with this boundedness.

The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator $L:=d^2/d{x}^2-2xd/dx$, x ∈ ℝ, need not be of weak type (1,1). A function in $L^1\left(d\gamma \right)$, where dγ is the Gaussian measure, is given such that the distribution function of the higher order Riesz transform decays more slowly than C/λ.

In this paper we study the Hilbert transform and maximal function related to a curve in R2.

A version of the John-Nirenberg inequality suitable for the functions $b\in \mathrm{BMO}$ with $b^{-}\in L^{\infty }$ is established. Then, equivalent definitions of this space via the norm of weighted Lebesgue space are given. As an application, some characterizations of this function space are given by the weighted boundedness of the commutator with the Hardy-Littlewood maximal operator.