Estimates for the commutator of bilinear Fourier multiplier

Let b ₁, b ₂ ∈ BMO(? ⁿ ) and T _σ be a bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that $\mathop {\sup }\limits_{\kappa \in \mathbb{Z}} \left\| {\sigma _\kappa } \right\|_{W^{s_1 ,s_2 } \left( {\mathbb{R}^{2n} } \right)} < \infty$ for some s ₁, s ₂ ∈ (n/2, n]. In this paper, the behavior on $L^{p_1 } \left( {\mathbb{R}^n } \right) \times L^{p_2 } \left( {\mathbb{R}^n } \right)\left( {p_1 ,p_2 \in \left( {1,\infty } \right)} \right)$ , on H ¹(? ⁿ ) × L ^p2 (? ⁿ ) (p ₂ ∈ [2,∞)), and on H ¹(? ⁿ ) × H ¹(? ⁿ ), is considered for the commutator $T_{\sigma ,\vec b}$ defined by $${T_{\sigma ,\vec b}}({f_1},{f_2})(x) = {b_1}(x){T_\sigma }({f_1},{f_2})(x) - {T_\sigma }({b_1}{f_1},{f_2})(x) + {b_2}(x){T_\sigma }({f_1},{f_2})(x) - {T_\sigma }({f_1},{b_2}{f_2})(x)$$ . By kernel estimates of the bilinear Fourier multiplier operators and employing some techniques in the theory of bilinear singular integral operators, it is proved that these mapping properties are very similar to those of the bilinear Fourier multiplier operator which were established by Miyachi and Tomita.     

Multiple ergodic theorems

Given measure preserving transformationsT ₁,T ₂,...,T _s of a probability space (X,B, μ) we are interested in the asymptotic behaviour of ergodic averages of the form $$\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {T_1^n f_1 \cdot T_2^n f_2 } \cdot \cdots \cdot T_s^n f_s $$ wheref ₁,f ₂,...,f _s ?L ^∞(X,B,μ). In the general case we study, mainly for commuting transformations, conditions under which the limit of (1) inL ²-norm is ∫ _x f ₁ dμ·∫ _x f ₂ dμ...∫ _x f _s dμ for anyf ₁,f ₂...,f _s ?L ^∞(X,B,μ). If the transformations are commuting epimorphisms of a compact abelian group, then this limit exists almost everywhere. A few results are also obtained for some classes of non-commuting epimorphisms of compact abelian groups, and for commuting epimorphisms of arbitrary compact groups.     

Eine Bemerkung zur lokalen Werteverteilung der Lösungen linearer Differentialgleichungen

We consider the homogeneous linear differential equation (D) $$w^{(n)} + \sum\limits_{j = 1}^n {f_{n - j} } (z)w^{(n - j)} = 0$$ with coefficients f_n-j holomorphic in the disc |z-ζ|n-j about z=ζ.     

Chebyshev's inequalities for functions monotone in the mean

LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ?,n} →R. Choosew ∈ F withw _i = w(i) > 0. PutW _i = w₁ + ? + w_i. Givenf ∈ F, define \(\bar f\) ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if \(\bar f\) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$      

A general class of inequalities with mixed means

Suppose (T, Σ, μ) is a space with positive measure,f: ? → ? is a strictly monotone continuous function, and &;(T) is the set of real μ-measurable functions onT. Letx(·) ∈ &;(T) andf ○x)(·) ∈L ₁(T,μ). Comparison theorems are proved for the means $\mathfrak{M}_{(T,{\mathbf{ }}\mu ,{\mathbf{ }}f)} (x( \cdot ))$ and the mixed means $\mathfrak{M}_{(T_1 ,{\mathbf{ }}\mu _1 ,{\mathbf{ }}f_1 )} (\mathfrak{M}_{(T_2 ,{\mathbf{ }}\mu _2 ,{\mathbf{ }}f_2 )} (x( \cdot )))$ these inequalities imply analogs and generalizations of some classical inequalities, namely those of Hölder, Minkowski, Bellman, Pearson, Godunova and Levin, Steffensen, Marshall and Olkin, and others. These results are a continuation of the author's studies.     

A moment problem associated to rational Szegő functions

Leta ₁,...,a _p be distinct points in the finite complex plane ?, such that |a _j|>1,j=1,..., p and let \(b_j = 1/\bar \alpha _j ,\) j=1,..., p. Let μ₀, μ _π ^(j) , ν _π ^(j) j=1,..., p;n=1, 2,... be given complex numbers. We consider the following moment problem. Find a distribution ψ on [?π, π], with infinitely many points of increase, such that $$\begin{array}{l} \int_{ - \pi }^\pi {d\psi (\theta ) = \mu _0 ,} \\ \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - a_j )^n }} = \mu _n^{(j)} ,} \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - b_j )^n }} = v_n^{(j)} ,} j = 1,...,p;n = 1,2,.... \\ \end{array}$$ It will be shown that this problem has a unique solution if the moments generate a positive-definite Hermitian inner product on the linear space of rational functions with no poles in the extended complex plane ?^* outside {a ₁,...,a _p,b ₁,...,b _p}.     

The individual ergodic theorem forp-sequences

Let (Ω,f,P) be a probability space and letT be a measure-preserving weak mixing transformation. We define a large class of sequences of integers calledp-sequences, such that iff∈L ₁ there exists a set Ω′⊂Ω of probability one and for ω∈Ω′ we have for everyp-sequence {k_n}.     

Orlicz norm estimates for eigenvalues of matrices

Let φ be a supermultiplicative Orlicz function such that the function $t \mapsto \varphi \left( {\sqrt t } \right)$ is equivalent to a convex function. Then each complexn×n matrixT=(τ _ij ) _{i, j} satisfies the following eigenvalue estimate: $\left\| {\left( {\lambda _i \left( T \right)} \right)_{i = 1}^n } \right\|_{\ell _\varphi } \leqslant C\left\| ( \right\|\left( {\tau _{ij} } \right)_{i = 1}^n \left\| {_{_{\ell _{\varphi *} } } )_{j = 1}^n } \right\|\ell _{\bar \varphi } $ . Here, ?_* stands for Young’s conjugate function of φ, ?, $\bar \varphi $ is the minimal submultiplicative function dominating φ andC>0 a constant depending only on φ. For the power function φ(t)=t ^p ,p≥2 this is a celebrated result of Johnson, König, Maurey and Retherford from 1979. In this paper we prove the above result within a more general theory of related estimates.     

The “Full Müntz Theorem” inL p[0, 1] for 0<p<∞

Denote by span {f ₁,f ₂, …} the collection of all finite linear combinations of the functionsf ₁,f ₂, … over ?. The principal result of the paper is the following. Theorem (Full Müntz Theorem in L_p(A) for p ∈ (0, ∞) and for compact sets A ? [0, 1] with positive lower density at 0). Let A ? [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ _j ) _j=1 ^∞ is a sequence of distinct real numbers greater than ?(1/p). Then span {x ^λ1,x ^λ2,…} is dense in L_p(A) if and only if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ . Moreover, if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ , then every function from the L_p(A) closure of {x ^λ1,x ^λ2,…} can be represented as an analytic function on {z ∈ ? \ (?∞,0] : |z| < r_A} restricted to A ∩ (0, r_A) where $r_A : = \sup \left\{ {y \in \mathbb{R}:\backslash ( - \infty ,0]:\left| z \right|< r_A } \right\}$ (m(·) denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Müntz, Szász, Clarkson, Erdös, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of {x ^λ1,x ^λ2,…} are also considered.     

Generalized Jacobi Identities and Ball-Box Theorem for Horizontally Regular Vector Fields

Consider a family $\mathcal{H}:= \{X_{j} =: f_{j}\cdot\nabla: j=1,\ldots , m\}$ of C ¹ vector fields in ? ⁿ and let s∈?. We assume that for all p∈{1,…,s} and j ₁,…,j _p ∈{1,…,m} the horizontal derivatives $X_{j_{1}}X_{j_{2}}\cdots X_{j_{p-1}}f_{j_{p}}$ exist and are Lipschitz continuous with respect to the control distance defined by  $\mathcal{H}$ . Then we show that different notions of commutator agree. This involves an accurate analysis of some algebraic identities involving nested commutators which seem to have an independent interest. Our principal applications are a ball-box theorem, the doubling property, and the Poincaré inequality for Hörmander vector fields under an intrinsic “horizontal regularity” assumption on their coefficients.     

Multiple recurrence and almost sure convergence for weakly mixing dynamical systems

We prove the following: Let (X, β, μ,T) be a weakly mixing dynamical system such that the restriction ofT to its Pinsker algebra has singular spectrum, then for all positive integersH, for allf _i ∈L ^∞, 1≤i≤H, the averages

Optimal methods for the approximate calculation of functional on classesW r L ∞

Пусть T_n(f)={L₁(f), ..., L_n(f)} — набор линейных функционал ов, заданных на простран стве \(C_{(r - 1)} (\parallel f\parallel _{C_{(r - 1)} } = \mathop {\max }\limits_{0 \leqq i \leqq r - 1} \parallel f^{(i)} \parallel _C );A_{n,r}\) — множество всех так их наборов функцио налов; С_{2n, 2} — множество всех н аборов из 2n функциона лов вида $$T_{2n} (f) = \{ f(x_1 ), \ldots ,f(x_n ),f'(x_1 ), \ldots ,f'(x_n )\}$$ и s: Е_n→Е₁. Доказано, что е слиW _∞ ^r множество всех 2π-периодических функ цийfεW∞0, 2π_r, то приr=1,2,3,... ирε(1, ∞) и $$\begin{gathered} \mathop {\inf }\limits_{T_{2n} \in A_{2n,r} } \parallel \mathop {\inf }\limits_s \mathop {\sup }\limits_{f \in W_\infty ^r } |f( \cdot ) - s(T_{2n} ,f, \cdot )|\parallel _p = \parallel \varphi _{n,r} \parallel _p \hfill \\ \mathop {\inf }\limits_{T_{2n} \in C_{2n,2} } \parallel \mathop {\inf }\limits_s \mathop {\sup }\limits_{f \in W_\infty ^r } |f( \cdot ) - s(T_{2n} ,f, \cdot )|\parallel _p = \parallel \parallel \varphi _{n,r} \parallel _\infty - \varphi _{n,r} \parallel _p , \hfill \\ \end{gathered}$$ где ?_n,r —r-й периодичес кий интеграл, в средне м равный нулю на периоде, от фун кции ?_n, 0t=sign sinnt. При этом указан ы оптимальные методы приближенного вычис ления.     

Exact simulation of final, minimal and maximal values of Brownian motion and jump-diffusions with applications to option pricing

We introduce a method for generating (W_x,T^(m,s),m_x,T^(m,s),M_x,T^(m,s))(W_{x,T}^{(\mu,\sigma)},m_{x,T}^{(\mu,\sigma)},M_{x,T}^{(\mu,\sigma)}) , where W_x,T^(m,s)W_{x,T}^{(\mu,\sigma)} denotes the final value of a Brownian motion starting in x with drift μ and volatility σ at some final time T, m_x,T^(m,s) = inf_{0 ￡ t ￡ T}W_x,t^(m,s)m_{x,T}^{(\mu,\sigma)} = {\rm inf}_{0\leq t \leq T}W_{x,t}^{(\mu,\sigma)} and M_x,T^(m,s) = sup_{0 ￡ t ￡ T} W_x,t^(m,s)M_{x,T}^{(\mu,\sigma)} = {\rm sup}_{0\leq t \leq T} W_{x,t}^{(\mu,\sigma)} . By using the trivariate distribution of (W_x,T^(m,s),m_x,T^(m,s),M_x,T^(m,s))(W_{x,T}^{(\mu,\sigma)},m_{x,T}^{(\mu,\sigma)},M_{x,T}^{(\mu,\sigma)}) , we obtain a fast method which is unaffected by the well-known random walk approximation errors. The method is extended to jump-diffusion models. As sample applications we include Monte Carlo pricing methods for European double barrier knock-out calls with continuous reset conditions under both models. The proposed methods feature simple importance sampling techniques for variance reduction.     

Some results on non-linear recurrence

We study the behavior of measure-preserving systems with continuous time along sequences of the form {n ^α}n∈#x2115;} where α is a positive real number¹. Let {S ^t } _t∈? be an ergodic continuous measure preserving flow on a probability Lebesgue space (X, β, μ). Among other results we show that:

1. For all but countably many α (in particular, for all α∈???) one can find anL ^∞-functionf for which the averagesA _N (f)(1/N)=Σ _n=1 ^N f(S ^nα x) fail to converge almost everywhere (the convergence in norm holds for any α!).
2. For any non-integer and pairwise distinct numbers α₁, α₂,..., α _k ∈(0, 1) and anyL ^∞-functionsf ₁,f ₂, ...,f _k , one has $$\mathop {lim}\limits_{N \to \infty } \left\| {\frac{1}{N}\sum\limits_{n - 1}^N {\prod\limits_{i - 1}^k {f_i (S^{n^{\alpha _i } } x) - \prod\limits_{i - 1}^k {\int_X {f_i d\mu } } } } } \right\|_{L^2 } = 0$$

We also show that Furstenberg’s correspondence principle fails for ?-actions by demonstrating that for all but a countably many α>0 there exists a setE?? having densityd(E)=1/2 such that, for alln∈?, $$d(E \cap (E - n^\alpha )) = 0$$ .     

Estimation of the remainder of a cubature formula on a Chebyshev grid

Let C(Q) denote the space of continuous functions f(x, y) in the square Q = [?1, 1] × [?1, 1] with the norm $\begin{gathered} \left\| f \right\| = \max \left| {f(x,y)} \right|, \hfill \\ (x,y) \in Q. \hfill \\ \end{gathered} $ On a Chebyshev grid, a cubature formula of the form $\int\limits_{ - 1}^1 {\int\limits_{ - 1}^1 {\frac{1} {{\sqrt {(1 - x^2 )(1 - y^2 )} }}f(x,y)dxdy = \frac{{\pi ^2 }} {{mn}}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {f\left( {\cos \frac{{2i - 1}} {{2n}}\pi ,\cos \frac{{2j - 1}} {{2m}}\pi } \right)} + R_{m,n} (f)} } } $ is considered in some class H(r ₁, r ₂) of functions f ?? C(Q) defined by a generalized shift operator. The remainder R _{m, n} (f) is proved to satisfy the estimate $\mathop {\sup }\limits_{f \in H(r_1 ,r_2 )} \left| {R_{m,n} (f)} \right| = O(n^{ - r_1 + 1} + m^{ - r_2 + 1} ), $ where r ₁, r ₂ > 1; ??^?1 ?? n/m ?? ?? with ?? > 0; and the constant in O(1) depends on ??.     

The maximal variation of a bounded martingale

Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦X_i≦1;i=0, …,n. For 0≦p≦1 denote by ? _p ⁿ the set of all such martingales satisfying alsoE(X₀)=p. Thevariation of a martingale χ ₀ ⁿ is denoted byV ₀ ⁿ and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ ₀ ⁿ ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

Non-Uniformity and Generalised Sacks Splitting

We show that there do not exist computable functions f ₁(e, i), f ₂(e, i), g ₁(e, i), g ₂(e, i) such that for all e, i ∈ ω, (1) $ {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)} \leqslant _{{\rm T}} {\left( {W_{e} - W_{i} } \right)}; $ (2) $ {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)} \leqslant _{{\rm T}} {\left( {W_{e} - W_{i} } \right)}; $ (3) $ {\left( {W_{e} - W_{i} } \right)} \not\leqslant _{{\rm T}} {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)} \oplus {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)}; $ (4) $ {\left( {W_{e} - W_{i} } \right)} \not\leqslant _{{\rm T}} {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)}{\text{unless}}{\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\emptyset};{\text{and}} $ (5) $ {\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)}{\text{unless}}{\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\emptyset}. $ It follows that the splitting theorems of Sacks and Cooper cannot be combined uniformly.     

A remark on the existence of entire large and bounded solutions to a (<Emphasis Type="Italic">k</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">k</Emphasis><Subscript>2</Subscript>)-Hessian system with gradient term

In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system

$$\left\{ {\begin{array}{*{20}c}{S_{k_1 } \left( {\lambda \left( {D^2 u_1 } \right)} \right) + a_1 \left( {\left| x \right|} \right)\left| {\nabla u_1 } \right|^{k_1 } = p_1 \left( {\left| x \right|} \right)f_1 \left( {u_2 } \right)} & {for x \in \mathbb{R}^N ,} \\{S_{k_2 } \left( {\lambda \left( {D^2 u_2 } \right)} \right) + a_2 \left( {\left| x \right|} \right)\left| {\nabla u_2 } \right|^{k_2 } = p_2 \left( {\left| x \right|} \right)f_2 \left( {u_1 } \right)} & {for x \in \mathbb{R}^N .} \\\end{array} } \right.$$

Here \({S_{{k_i}}}\left( {\lambda \left( {{D^2}{u_i}} \right)} \right)\) is the k _i -Hessian operator, a ₁, p ₁, f ₁, a ₂, p ₂ and f ₂ are continuous functions.

     

Improved error bounds for near-minimax approximations

Givenf εC ⁽ⁿ⁺¹⁾[?1, 1], a polynomialp _n, of degree ≤n, is said to be near-minimax if (*) $$\left\| {f - p_n } \right\|_\infty = 2^{ - n} |f^{(n + 1)} (\xi )|/(n + 1)!,$$ for some ζ ε (?1,1). For three sets of near-minimax approximations, by considering the form of the error ∥f ?p _n∥_∞ in terms of divided differences, it is shown that better upper and lower bounds can be found than those given by (*).     

Mixing via the extended family

In this paper,the relationship between the extended family and several mixing properties in measuretheoretical dynamical systems is investigated.The extended family eF related to a given family F can be regarded as the collection of all sets obtained as"piecewise shifted"members of F.For a measure preserving transformation T on a Lebesgue space(X,B,μ),the sets of"accurate intersections of order k"defined below are studied,Nε(A0,A1,...,Ak)=n∈Z+:μk i=0T inAiμ(A0)μ(A1)μ(Ak)ε,for k∈N,A0,A1,...,Ak∈B and ε0.It is shown that if T is weakly mixing(mildly mixing)then for any k∈N,all the sets Nε(A0,A1,...,Ak)have Banach density 1(are in(eFip),i.e.,the dual of the extended family related to IP-sets).