Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the Hormander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k≥ 3 using the multiparameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k≥ 3:■where x =(x_1,x_2,x_3)∈R~(n_1)×R~(n_2)×R~(n_3) and ξ =(ξ_1,ξ_2,ξ_3)∈R~(n_1)×R~(n_2)×R~(n_3). One of our main results is the following:Assume that m(ξ) is a function on R~(n_1+n_2+n_3) satisfying ■ with s_i n_i(1/p-1/2) for 1≤i≤3. Then T_m is bounded from H~p(R~(n_1)×R~(n_2)×R~(n_3) to H~p(R~(n_1)×R~(n_2)×R~(n_3)for all 0 p≤1 and ■ Moreover, the smoothness assumption on s_i for 1≤i≤3 is optimal. Here we have used the notations m_(j,k,l)(ξ)=m(2~jξ_1,2~kξ_2,2~lξ_3)Ψ(ξ_1)Ψ(ξ_2)Ψ(ξ_3) and Ψ(ξ_i) is a suitable cut-off function on R~(n_i) for1≤i≤3, and W~(s_1,s_2,s_3) is a three-parameter Sobolev space on R~(n_1)×R~(n_2)× R~(n_3).Because the Fefferman criterion breaks down in three parameters or more, we consider the L~p boundedness of the Littlewood-Paley square function of T_mf to establish its boundedness on the multi-parameter Hardy spaces.     

A Supplement to Hölder’s Inequality. The Resonance Case. I

Suppose that m ≥ 2, numbers p₁, …, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ... + \frac{1}{{{p_m}}} < 1\), and functions γ₁ ∈ \({L^{{p_1}}}\)(?¹), …, γ _m ∈ \({L^{{p_m}}}\)(?¹) are given. It is proved that if the set of “resonance points” of each of these functions is nonempty and the so-called “resonance condition” holds, then there are arbitrarily small (in norm) perturbations Δγ_k ∈ \({L^{{p_k}}}\)(?¹) under which the resonance set of each function γ_k + Δγ_k coincides with that of γ_k for 1 ≤ k ≤ m, but \({\left\| {\int\limits_0^t {\prod\limits_{k = 0}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]d\tau } } } \right\|_{{L^\infty }\left( {{\mathbb{R}^1}} \right)}} = \infty \). The notion of a resonance point and the resonance condition for functions in the spaces L ^p (?¹), p ∈ (1, +∞], were introduced by the author in his previous papers.     

Embeddings between weighted Copson and Cesàro function spaces

In this paper, characterizations of the embeddings between weighted Copson function spaces \(Co{p_{{p_1},{q_1}}}\left( {{u_1},{v_1}} \right)\) and weighted Cesàro function spaces \(Ce{s_{{p_2},{q_2}}}\left( {{u_2},{v_2}} \right)\) are given. In particular, two-sided estimates of the optimal constant c in the inequality

$${\left( {\int_0^\infty {{{\left( {\int_0^t {f{{\left( \tau \right)}^{{p_2}}}{v_2}\left( \tau \right)d\tau } } \right)}^{{q_2}/{p_2}}}{u_2}\left( t \right)dt} } \right)^{1/{q_2}}} \leqslant c{\left( {\int_0^\infty {{{\left( {\int_t^\infty {f{{\left( \tau \right)}^{{p_1}}}{v_1}\left( \tau \right)d\tau } } \right)}^{{q_1}/{p_1}}}{u_1}\left( t \right)dt} } \right)^{1/{q_1}}},$$

where p₁, p₂, q₁, q₂ ∈ (0,∞), p₂ ≤ q₂ and u₁, u₂, v₁, v₂ are weights on (0,∞), are obtained. The most innovative part consists of the fact that possibly different parameters p₁ and p₂ and possibly different inner weights v₁ and v₂ are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.

     

On the towers of torsion Bertrandias and Payan modules

For an odd prime p, let K/k be a Galois p-extension and S be a set of primes of k containing the primes lying over p. For the p ^r th roots \({\mu _{{p^r}}}\left( K \right)\) of unity in K, we describe the so-called Sha group Sha _S (G(K/k), \({\mu _{{p^r}}}\left( K \right)\)) in terms of the Galois groups of certain subfields of K corresponding to S. As an application, we investigate a tower of extension fields \({\left\{ {{k_{{T^i}}}} \right\}_i} \geqslant 0\) where \({k_{{T^{i + 1}}}}\) is defined as the fixed field of a free part of the Galois group of the Bertrandias and Payan extension of \({k_{{T^i}}}\) over \({k_{{T^i}}}\). This is called a tower of torsion parts of the Bertrandias and Payan extensions over k. We find a relation between the degrees \({\left\{ {\left[ {{k_{{T^{i + 1}}}}:{k_{{T^i}}}} \right]} \right\}_{i \geqslant 0}}\) over the towers. Using this formula we investigate whether the towers are stationary or not.     

Goldbach’s conjecture in arithmetic progressions: number and size of exceptional prime moduli

Set \({T=N^{\frac{1}{3}-\epsilon}}\). It is proved that for all but \({\ll TL^{-H},\,H > 0}\), exceptional prime numbers \({k\leq T}\) and almost all integers b ₁, b ₂ co-prime to k, almost all integers \({n\sim N}\) satisfying \({n\equiv b_{1}+b_{2}(mod\,k)}\) can be written as the sum of two primes p ₁ and p ₂ satisfying \({p_{i}\equiv b_{i}(mod\,k),\,i=1,2}\). For prime numbers \({k\leq N^{\frac{5}{24}-\epsilon}}\), this result is even true for all but \({\ll (\log\,N)^{D}}\) primes k and all integers b ₁, b ₂ co-prime to k.     

On a class of strongly contractive quadratic recurrent systems

We consider a class of nonlinear recurrent systems of the form \( {\Lambda_p} = \frac{1}{p}\sum\limits_{{p_1} = 1}^{p - 1} {f\left( {\frac{{{p_1}}}{p}} \right){\Lambda_{{p_1}}}{\Lambda_{p - {p_1}}}} \), p > 1, where f is a given function on the interval [0, 1] and Λ₁ = x is an adjustable real-valued parameter. Under some suitable assumptions on the function f, we show that there exists an initial value x ^* for which Λ _p = Λ _p (x ^* ) → const as p → ∞. More precise asymptotics of Λ _p is also derived.     

On the Gosper’s <Emphasis Type="Italic">q</Emphasis>-constant Π<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>

Gosper introduced the functions sin_qz and cos_qz as q-analogues for the trigonometric functions sin z and cos z respectively. He stated a variety of identities involving these two q-trigonometric functions along with certain constants denoted by \({\Pi _{{q^n}}}\) (n ∈ N). Gosper noticed that all his formulas on these constants have more than two of the \({\Pi _{{q^n}}}\). So, it is natural to raise the question of establishing identities involving only two of the \({\Pi _{{q^n}}}\). In this paper, our main goal is to give examples of such formulas in only two \({\Pi _{{q^n}}}\).     

Some remarks on Wente’s inequality and the Lorentz-Sobolev space

In this note we consider Wente's type inequality on the Lorentz-Sobolev space.If▽f∈L~p1,q1(R~n),G ∈ L~(p2,q2)(R~n) and div G≡0 in the sense of distribution where(1/p1)+(1/P2)=(1/q1)+(1/q2)=1,1P1,P2∞,it is known that G·▽f belongs to the Hardy space H~1 and furthermore‖G·▽f‖H~1≤C‖▽f‖L~(p1,q1)(R~2)‖G‖L~(p2,q2)(R~2).Reader can see[9]Section 4.Here we give a new proof of this result.Our proof depends on an estimate of a maximal operator on the Lorentz space which is of some independent interest.Finally,we use this inequality to get a generalisation of Bethuel's inequality.     

A continuous model for systems of complexity 2 on simple abelian groups

It is known that if p is a sufficiently large prime, then, for every function f: Z_p → [0, 1], there exists a continuous function f′: T → [0, 1] on the circle such that the averages of f and f′ across any prescribed system of linear forms of complexity 1 differ by at most ∈. This result follows from work of Sisask, building on Fourier-analytic arguments of Croot that answered a question of Green. We generalize this result to systems of complexity at most 2, replacing T with the torus T² equipped with a specific filtration. To this end, we use a notion of modelling for filtered nilmanifolds, that we define in terms of equidistributed maps and combine this notion with tools of quadratic Fourier analysis. Our results yield expressions on the torus for limits of combinatorial quantities involving systems of complexity 2 on Z_p. For instance, let m₄(α, Z_p) denote the minimum, over all sets A ? Z_p of cardinality at least αp, of the density of 4-term arithmetic progressions inside A. We show that limp→∞ m₄(α, Z_p) is equal to the infimum, over all continuous functions f: T₂ →[0, 1] with \({\smallint _{{T^2}}}f \geqslant a\), of the integral

$$\int_{{T^5}} {f\left( {\begin{array}{*{20}{c}}{{x_1}} \\ {{y_1}} \end{array}} \right)} f\left( {\begin{array}{*{20}{c}}{{x_1} + {x_2}} \\ {{y_1} + {y_2}} \end{array}} \right)f\left( {\begin{array}{*{20}{c}}{{x_1} + 2{x_2}} \\ {{y_1} + 2{y_2} + {y_3}} \end{array}} \right).f\left( {\begin{array}{*{20}{c}}{{x_1} + 3{x_2}} \\ {{y_1} + 3{y_2} = 3{y_3}} \end{array}} \right)d{\mu _{{T^5}}}({x_1},{x_2},{y_1},{y_2},{y_3})$$

     

<Emphasis Type="Italic">p</Emphasis>-adic valuation of exponential sums associated to trinomials and some consequences

In this paper, we compute the p-adic valuation of exponential sums associated to trinomials \(F\left( X \right) = a{X^{{d_1}}} + b{X^{{d_2}}} + c{X^{{d_3}}}\) over Fp. As a byproduct of our results, we obtain restrictions for permutation polynomials of type \(a{X^{{d_1}}} + b{X^{{d_2}}} + c{X^{{d_3}}}\) over F_p.     

On one complement to the Hölder inequality: I

Let m ≥ 2, the numbers p ₁,…, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ...\frac{1}{{{p_m}}} < 1\), and γ₁ ∈ L ^p1(?¹), …, γ _m ∈ \({L^{{p_m}}}\)(?¹). We prove that, if the set of “resonance” points of each of these functions is nonempty and the “nonresonance” condition holds (both concepts have been introduced by the author for functions of spaces L ^p (?¹), p ∈ (1, +∞]), we have the inequality \(\mathop {\sup }\limits_{a,b \in {R^1}} \left| {\int\limits_a^b {\prod\limits_{k = 1}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]} d\tau } } \right| \leqslant C{\prod\limits_{k = 1}^m {\left\| {{\gamma _k} + \Delta {\gamma _k}} \right\|} _{L_{{a_k}}^{{p_k}}}}\left( {{\mathbb{R}^1}} \right)\), where the constant C > 0 is independent of functions \(\Delta {\gamma _k} \in L_{{a_k}}^{{p_k}}\left( {{\mathbb{R}^1}} \right)\) and \(L_{{a_k}}^{{p_k}}\left( {{\mathbb{R}^1}} \right) \subset {L^{{p_k}}}\left( {{\mathbb{R}^1}} \right)\), 1 ≤ k ≤ m are some specially constructed normed spaces. In addition, we give a boundedness condition for the integral of the product of functions over a subset of ?¹.     

A remark on the values of binary linear forms at prime arguments

Let λ₁, λ₂ be positive real numbers such that \({\frac{{\lambda_1}}{{\lambda_2}}}\) is irrational and algebraic. For any (C, c) well-spaced sequence \({\mathcal {V} = \{{v_i}\}_{i = 1}^\infty}\) and δ > 0 let \({E( {\mathcal {V},X,\delta})}\) denote the number of elements \({v \in \mathcal {V}, v \le X}\) for which the inequality

$| {\lambda_1 p_1 + \lambda_2 p_2 - v} | < X^{- \delta}$

is not solvable in primes p ₁, p ₂. In this paper it is proved that

$E( {\mathcal {V},X,\delta}) \ll X^{\frac{4}{5} + \delta + \varepsilon}$

for any \({\varepsilon > 0}\). This result constitutes an improvement upon that of Brüdern, Cook, and Perelli for the range \({\frac{2}{{15}} < \delta < \frac{1}{5}}\).

     

Some estimates for the Schrödinger type operators with nonnegative potentials

In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).     

The Representation of a C0-Semigroup of Linear Operators in a Banach Space on the Set of Entire Vectors of its Generator

For a C₀-semigroup \({\{U(t)\}_{t \geq 0}}\) of linear operators in a Banach space \({{\mathfrak{B}}}\) with generator A, we describe the set of elements \({x \in {\mathfrak{B}}}\) whose orbits U(t)x can be extended to entire \({{\mathfrak{B}}}\)-valued functions of a finite order and a finite type, and establish the conditions under which this set is dense in \({{\mathfrak{B}}}\). The Hille problem of finding vectors \({x \in {\mathfrak{B}}}\) such that there exists the limit \({\lim\limits_{n \to \infty}\left(I + \frac{tA}{n}\right)^{n}x}\) is also solved in the paper. We prove that this limit exists if and only if x is an entire vector of the operator A, and if this is the case, then it coincides with U(t)x.     

Subgraph Transversal of Graphs

Given graphs F and G, denote by \({\tau_F}(G)\) the cardinality of a smallest subset \(T {\subseteq}V(G)\) that meets every maximal F-free subgraph of G. Erdös, Gallai and Tuza [9] considered the question of bounding \(\tau_{\overline{K_2}}(G)\) by a constant fraction of |G|. In this paper, we will give a complete answer to the following question: for which F, is τ _F (G) bounded by a constant fraction of |G|?In addition, for those graphs F for which \({\tau_F}(G)\) is not bounded by any fraction of |G|, we prove that \(\tau_F(G)\le|G|-\frac{1}{2}\sqrt{|G|}+\frac{1}{2}\), provided F is not K _k or \(\overline{K_k}\).     

The statistics of the number of rational points in the hyperelliptic ensemble over a finite field

We study the distribution of the numbers of \({F_{{q^r}}}\)-rational points of hyperelliptic curves over a finite field F_q in odd characteristic. This extends the result of Kurlberg and Rudnick [4]. We also study the distribution of the number of \({F_{{q^r}}}\)-rational points and the trace of high powers of the Frobenius class of real hyperelliptic curves over a finite field F_q in even characteristic.     

The <Emphasis Type="Italic">p</Emphasis>-adic order of some fibonomial coefficients whose entries are powers of <Emphasis Type="Italic">p</Emphasis>

Let (F _n ) _n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

$${\left[ {\begin{array}{*{20}{c}} m \\ k \end{array}} \right]_F} = \frac{{{F_{m - k + 1}} \cdots {F_{m - 1}}{F_m}}}{{{F_1} \cdots {F_k}}}$$

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±2 (mod 5), then \(p{\left| {\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]} \right._F}\) for all integers a ≥ 1. In 2015, Marques and Trojovský worked on the p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all a ≥ 1 when p ≠ 5. In this paper, we shall provide the exact p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a, b ≥ 1 and for all prime number p.

     

On a hybrid of bilinear Hilbert transform and paraproduct

We introduce a class of tri-linear operators that combine features of the bilinear Hilbert transform and paraproduct. For two instances of these operators, we prove boundedness property in a large range D = {(p1,p2,p_3) ∈ R~3 : 1 p1,p2 ∞,1/(p1)+ 1/(p2)3/2,1 p3 ∞}.     

2-Arc-transitive cyclic covers of $$K_{n,n}$$

In this paper, a complete classification is achieved of all the regular covers of the complete bipartite graphs \(K_{n,n}\) with cyclic covering transformation group, whose fibre-preserving automorphism group acts 2-arc-transitively. All these covers consist of one threefold covers of \(K_{6,6}\), one twofold cover of \(K_{12, 12}\) and one infinite family X(r, p) of p-fold covers of \(K_{p^r,p^r}\) with p a prime and r an integer such that \(p^r\ge 3\). This infinite family X(r, p) can be derived by a very simple and nice voltage assignment f as follows: \(X(r, p)=K_{p^r, p^r}\times _f \mathbb {Z}_p\), where \(K_{p^r, p^r}\) is a complete bipartite graph with the bipartition \(V=\{ \alpha \bigm |\alpha \in V(r,p)\}\cup \{ \alpha '\bigm |\alpha \in V(r,p)\}\) for the r-dimensional vector space V(r, p) over the field of order p and \(f_{\alpha ,\beta '}=\sum _{i=1}^ra_ib_i,\,\, \mathrm{for\,\,all}\,\,\alpha =(a_i)_r, \beta =(b_i)_r\in V(r,p)\).     

Quantization of the Kadomtsev–Petviashvili equation

We propose a quantization of the Kadomtsev–Petviashvili equation on a cylinder equivalent to an infinite system of nonrelativistic one-dimensional bosons with the masses m = 1, 2,.... The Hamiltonian is Galilei-invariant and includes the split and merge terms \(\Psi _{{m_1}}^\dag \Psi _{{m_2}}^\dag {\Psi _{{m_1} + {m_2}}}\) and \(\Psi _{{m_1} + {m_2}}^\dag {\Psi _{{m_1}}}{\Psi _{{m_2}}}\) for all combinations of particles with masses m ₁, m ₂, and m ₁ + m ₂ for a special choice of coupling constants. We construct the Bethe eigenfunctions for the model and verify the consistency of the coordinate Bethe ansatz and hence the quantum integrability of the model up to the mass M=8 sector.