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We prove l p ( Z d ) bounds, for p ∈ ( 1 , ∞ ) , of discrete maximal functions corresponding to averaging operators and truncated singular integrals of Radon type, and their applications to pointwise ergodic theory. Our new approach is based on a unified analysis of both types of operators, and also yields an extension to the vector-valued form of these results.

This is a survey article about recent developments in dimension-free estimates for maximal functions corresponding to the Hardy–Littlewood averaging operators associated with convex symmetric bodies in \(\mathbb R^d\) and \(\mathbb Z^d\).

We survey recent work and announce new results concerning two singular integral operators whose kernels are holomorphic functions of the output variable, specifically the Cauchy–Leray integral and the Cauchy–Szegő projection associated to various classes of bounded domains in \(\mathbb C^n\) with n ≥ 2.

This chapter describes some results concerning oscillatory integrals and, in particular, their application to Radon-like transforms. It briefly discusses three classes of oscillatory integrals. The first class consists of maximal oscillatory integrals. The second is made up of the oscillatory integrals arising in restrictions theorems and Bochner–Riesz summability. The third class contains the oscillatory integrals related to the Radon-like transforms, which is most closely related to Fourier integral operators. The chapter shows that the study of Radon-like transforms has an impact on the estimates for averaging operators involving integration over lower dimensional manifolds and connections with hyperbolic equations, relations of Radon transforms with Fourier integral operators, the initial study of singular Radon transforms as “Hilbert transforms along curves,” and the relevance to several complex variables.

We study dimension-free L p inequalities for r-variations of the Hardy–Littlewood averaging operators defined over symmetric convex bodies in $${\mathbb{R}^d}$$ .

The purpose of this paper is to complement the results in [LS-1] by showing the dense definability of the Cauchy-Leray transform for the domains that give the counterexamples of [LS-1], where $L^p$-boundedness is shown to fail when either the "near" $C^2$ boundary regularity, or the strong $\mathbb C$-linear convexity assumption is dropped., Comment: 19 pages

We show that the discrete Hardy–Littlewood maximal functions associated with the Euclidean balls in \(\mathbb Z^d\) with dyadic radii have bounds independent of the dimension on \(\ell ^p(\mathbb Z^d)\) for p ∈ [2, ∞].

We prove \(\ell ^p\left( {\mathbb {Z}}^d\right) \) bounds for \(p\in (1, \infty )\), of r-variations \(r\in (2, \infty )\), for discrete averaging operators and truncated singular integrals of Radon type. We shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of p and r which coincide with the ranges of their continuous counterparts.

We prove strong jump inequalities for a large class of operators of Radon type in the discrete and ergodic theoretical settings. These inequalities are the $r=2$ endpoints of the $r$-variational estimates studied in arXiv:1512.07523., v2: 39 pages, revised following referee report

The aim of this paper is to present an abstract and general approach to jump inequalities in harmonic analysis. Our principal conclusion is the refinement of $r$-variational estimates, previously known for $r>2$, to end-point results for the jump quasi-seminorm corresponding to $r=2$. This is applied to the dimension-free results recently obtained by the first two authors in collaboration with Bourgain and Wr��bel (arXiv:1708.04639 and arXiv:1804.07679), and also to operators of Radon type treated by Jones, Seeger, and Wright., 25 pages, small corrections

Jump inequalities are the $r=2$ endpoint of L\'epingle's inequality for $r$-variation of martingales. Extending earlier work by Pisier and Xu we interpret these inequalities in terms of Banach spaces which are real interpolation spaces. This interpretation is used to prove endpoint jump estimates for vector-valued martingales and doubly stochastic operators as well as to pass via sampling from $\mathbb{R}^{d}$ to $\mathbb{Z}^{d}$ for jump estimates for Fourier multipliers., Comment: v3: corrections following referee report

We prove L p ( b D ) -regularity of the Cauchy–Leray integral for bounded domains D ⊂ C n whose boundary satisfies the minimal regularity condition of class C 1 , 1 , together with a naturally occurring notion of convexity.

The purpose of this paper is to study the L2 boundedness of operators of the form f ↦ ψ(x) ∫ f (γt(x))K(t)dt, where γt(x) is a C∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝN × ℝn, satisfying γ0(x) ≡ x, ψ is a C∞ cut-off function supported on a small neighborhood of 0 ∈ ℝn, and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝN. The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L2. The case when K is a Calderon-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderon- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of Lp boundedness, while the third paper deals with the special case when γ is real analytic.

We present the theory of Cauchy-Fantappi\'e integral operators, with emphasis on the situation when the domain of integration, $D$, has minimal boundary regularity. Among these operators we focus on those that are more closely related to the classical Cauchy integral for a planar domain, whose kernel is a holomorphic function of the parameter $z\in D$. The goal is to prove $L^p$ estimates for these operators and, as a consequence, to obtain $L^p$ estimates for the canonical Cauchy-Szeg\"o and Bergman projection operators (which are not of Cauchy-Fantappi\'e type)., Comment: This is an electronic reprint of the original article published in the Bulletin of the Mathematical Sciences, vol. 3 (2) (2013) 241-285. This reprint differs from the original in pagination and typographical detail

We prove the optimality of the hypotheses guaranteeing the $L^p$-boundedness for the Cauchy-Leray integral in $\mathbb C^n$, $n\geq 2$, obtained in [LS-4]. Two domains, both elementary in nature, show that the geometric requirement of strong $\mathbb C$-linear convexity, together with regularity of order 2, are both necessary.

We prove the Lp(bD)-regularity of the Cauchy–Szegő projection (also known as the Szegő projection) for bounded domains D⊂Cn which are strongly pseudoconvex and whose boundary satisfies the minimal regularity condition of class C2.

We study the phenomena that arise when we combine the standard pseudodifferential operators with those operators that appear in the study of some sub-elliptic estimates, and on strongly pseudoconvex domains. The algebra of operators we introduce is geometrically invariant, and is adapted to a smooth distribution of tangent subspaces of constant rank. We isolate certain ideals in the algebra whose analysis is of particular interest., Comment: typos fixed

Let $\mathcal K$ be a flag kernel on a homogeneous nilpotent Lie group $G$. We prove that operators $T$ of the form $T(f)= f*\mathcal K$ form an algebra under composition, and that such operators are bounded on $L^{p}(G)$ for $1, Comment: 92 pages

The purpose of this announcement is to describe a development given in a series of forthcoming papers by the authors that concern operators of the form \[ f\mapsto \psi(x) \int f(\gamma_t(x)) K(t)\: dt, \] where $\gamma_t(x)=\gamma(t,x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \mathbb{R}^N\times \mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $K(t)$ is a "multi-parameter singular kernel" supported near $t=0$, and $\psi$ is a cutoff function supported near $x=0$. This note concerns the case when $K$ is a "product kernel". The goal is to give conditions on $\gamma$ such that the above operator is bounded on $L^p$ for $1, Comment: 18 pages; an announcement for a three part series of papers; final version to appear in Math. Res. Lett

We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of a geometric Brownian motion and the density of the stock price process in the Hull–White model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the price of an Asian option are obtained.

We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Stein-Stein and the Heston model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the implied volatility in the Stein-Stein and the Heston model are obtained.

We study the implied volatility KI(K) in the Hull-White model of option pricing, and obtain asymptotic formulas for this function as the strike price K tends to infinity or zero. We also prove that the function I is convex near zero and concave near infinity, and characterize the behavior of the first two derivatives of this function.

We prove various representations and density results for Hardy spaces of holomorphic functions for two classes of bounded domains in \(\mathbb{C}^{n}\), whose boundaries satisfy minimal regularity conditions (namely the classes C2 and C1, 1, respectively) together with naturally occurring notions of convexity.

We prove Lp boundedness of certain non-translation-invariant discrete maximal Radon transforms and discrete singular Radon transforms. We also prove maximal, pointwise, and Lp ergodic theorems for certain families of non-commuting operators.

In the present Note, we study the asymptotic behavior of the distribution density of the stock price process in the Hull–White model. The leading terms in the asymptotic expansions at zero and infinity are found for such a density and the corresponding error estimates are given. Similar problems are solved for time averages of the volatility process, which are also of interest in the study of Asian options. To cite this article: A. Gulisashvili, E.M. Stein, C. R. Acad. Sci. Paris, Ser. I 343 (2006).