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We give a detailed description of the scattering theory of solutions to the quasilinear wave equations with null conditions and small initial data in two dimensions. The scattering profile is shown to be the same as the one conjectured by Alinhac [1] (1995). Our proof is based on a framework of Fourier transform by Deng and Pusateri [6] (2018) and an expansion of oscillatory integral, together with the global well-posedness of the solutions by Alinhac [2] (2001).

We define a scale of Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, $p\in[1,\infty]$, that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for $p=1$. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of $\mathbb{R}^{n}$, and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about $L^{p}$-boundedness of Fourier integral operators, from local boundedness to global boundedness for a larger class of symbols., Comment: 59 pages. Final version before publication

Filon collocation methods are well known for their good performances in solving the second-kind highly oscillatory Volterra integral equations with weak singularities. As the frequency increases, these methods can provide much more accurate solutions for oscillatory integral equations. This paper is devoted to investigating frequency-related properties of collocation solutions of highly oscillatory Volterra equations with weakly singular kernels. Optimal convergence rates are obtained by analyzing the error equations and studying asymptotic properties of oscillatory integrals involving Bessel functions. Moreover, numerical experiments are conducted to verify the given theoretical results.

In this paper, we study efficient numerical methods for the computation of highly oscillatory Bessel transform ∫ a + ∞ f ( x ) J ν ( ω g ( x ) ) d x , g ( x ) ≠ 0 , g ′ ( x ) ≠ 0 for all x ∈ [ a , + ∞ ) . We first derive an asymptotic expansion formula for this integral by using integration by parts. Then we present an improved numerical steepest descent method, by applying complex integration theory to the remainder term of asymptotic expansion, which is also an oscillatory integral with Bessel kernel. In addition, the asymptotic orders about ω of the presented methods are given. The efficiency and accuracy of the proposed methods are investigated through theoretical results and numerical experiments.

We are concerned in this paper with the numerical computation of the spectra of highly oscillatory integrals that arise in laser simulations. Discretised using the modified Fourier basis, the spectral problem for the integral equation is converted into two independent infinite systems of linear equations whose unknowns are the coefficients of the modified Fourier functions, namely the cosine and shifted sine functions, respectively. Each ( m , n ) entry of the resulting coefficient matrices can be represented exactly by expressions involving the error function with an argument that involves the oscillatory parameter ω and the numbers m and n . Moreover, considering the behaviour of the error function for a large argument, the asymptotics for each entry are analysed for large ω or for large m and n and this enables efficient truncation of the infinite systems. Numerical experiments are provided to illustrate the effectiveness of this method.

This paper studies the existence and scattering of global small amplitude solutions to the nonlinear Boussinesq paradigm equation in R n . Firstly, with a dispersive estimate obtained by oscillatory integral and some properties of Bessel function, the existence of global small amplitude solutions is established by the method of contractive mapping principle. Then, with the help of the decay of solutions with respect to time, we construct the suitable scattering state corresponding to the small amplitude solutions.

We consider the time slicing approximations of Feynman path integrals, constructed via piecewice classical paths. A detailed study of the convergence in the norm operator topology, in the space $\mathcal{B}(L^2(\mathbb{R}^d))$ of bounded operators on $L^2$, and even in finer operator topologies, was carried on by D. Fujiwara in the case of smooth potentials with an at most quadratic growth. In the present paper we show that the result about the convergence in $\mathcal{B}(L^2(\mathbb{R}^d))$ remains valid if the potential is only assumed to have second space derivatives in the Sobolev space $H^{d+1}(\mathbb{R}^d)$ (locally and uniformly), uniformly in time. The proof is non-perturbative in nature, but relies on a precise short time analysis of the Hamiltonian flow at this Sobolev regularity and on the continuity in $L^2$ of certain oscillatory integral operators with non-smooth phase and amplitude., 28 pages

The present work focuses on formulating a numerical scheme for approximation of Volterra integral equations with highly oscillatory Bessel kernels. The application of Laplace transform reduces integral equations into algebraic equations. By application of inverse Laplace transform solution is presented as an integral along a smooth curve extending into the left half of the complex plane, which is then evaluated by quadrature. Some model problems are solved and the results are compared with other available methods. The supremacy of the present method is that the transformed problem becomes non oscillatory. Consequently such types of integral equations with highly oscillatory kernels can be approximated very effectively with large values of oscillation parameter. A small amount of work such as Clenshaw-Curtis-Filon type methods are available for efficient approximation of highly oscillatory integral equations. Keywords: Laplace transform, Numerical inverse Laplace transform, Oscillatory kernels of convolution type

The sharp range of $L^p$-estimates for the class of H\"ormander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments., Comment: Updated version incorporating minor corrections, additional clarification and an expanded discussion of applications. 95 pages. 6 figures. To appear Acta. Math

The value of a highly oscillatory integral is typically determined asymptotically by the behaviour of the integrand near a small number of critical points. These include the endpoints of the integration domain and the so-called stationary points or saddle points -- roots of the derivative of the phase of the integrand -- where the integrand is locally non-oscillatory. Modern methods for highly oscillatory quadrature exhibit numerical issues when two such saddle points coalesce. On the other hand, integrals with coalescing saddle points are a classical topic in asymptotic analysis, where they give rise to uniform asymptotic expansions in terms of the Airy function. In this paper we construct Gaussian quadrature rules that remain uniformly accurate when two saddle points coalesce. These rules are based on orthogonal polynomials in the complex plane. We analyze these polynomials, prove their existence for even degrees, and describe an accurate and efficient numerical scheme for the evaluation of oscillatory integrals with coalescing saddle points.

In this paper, we investigate the oscillation properties of solutions of a class of highly oscillatory Volterra integral equations and develop a Hermite collocation method to approximate the solution of these equations. We begin our analysis by obtaining an asymptotic expansion for the solution of these equations using their resolvent representation. We then introduce a new Gaussian radial basis function interpolation to provide a numerical solution for these equations. The convergence analysis of the proposed method is also studied, which shows that increasing the number of collocation points or the number of mesh points controls the impact of the oscillation parameter in the whole error. Some numerical examples are presented to show the accuracy of the proposed scheme.

The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions arising from the structure of the Fourier integrals.

We consider solutions in frequency bands of dispersive equations on the line defined by Fourier multipliers, these solutions being considered as wave packets. In this paper, a refinement of an existing method permitting to expand time-asymptotically the solution formulas is proposed, leading to a first term inheriting the mean position of the true solution together with a constant variance error. In particular, this first term is supported in a space-time cone whose origin position depends explicitly on the initial state, implying especially a shifted time-decay rate. This method, which takes into account both spatial and frequency information of the initial state, makes then stable some propagation features and permits a better description of the motion and the dispersion of the solutions of interest. The results are achieved firstly by making apparent the cone origin in the solution formula, secondly by applying precisely an adapted version of the stationary phase method with a new error bound, and finally by minimizing the error bound with respect to the cone origin., 32 pages, no figures

In this paper, we are concerned with the numerical approximations of one-sided Hilbert transforms with oscillatory kernel by means of the multiple integrals. This type of Hilbert transform has two computing difficulties: singularity and oscillation. To avoid the singularity, we transfer the Hilbert transform to an individual oscillatory integral which can be analytically calculated and a non-singular integral which can be well evaluated by the multiple integrals. Numerical examples are provided to illustrate the advantages of the proposed methods.

In this paper we obtain almost sharp decay estimates for $L^2$ operator norm of strongly singular oscillatory integral operators in $\mathbb{R}^{n+1}$ for $n \geq 2$; we prove some necessary condition for $L^2$ estimates. Also, we prove that the operators are bounded on $L^p$ for some $p \neq 2$ and the range of $p$ depends on the hypersingularity of the operators.

A practical and simple stable method for calculating Fourier integrals is proposed, effective both at low and at high frequencies. An approach based on the fruitful idea of Levin, which allows the use of the collocation method to approximate the slowly oscillating part of the antiderivative of the desired integral, allows reducing the calculation of the integral of a rapidly oscillating function (with a linear phase) to solving a system of linear algebraic equations with a triangular or Hermitian matrix.

In this paper we obtain some new Strichartz estimates for the wave propagator $e^{it\sqrt{-\Delta}}$ in the context of Wiener amalgam spaces. While it is well understood for the Schr\"odinger case, nothing is known about the wave propagator. This is because there is no such thing as an explicit formula for the integral kernel of the propagator unlike the Schr\"odinger case. To overcome this lack, we instead approach the kernel by rephrasing it as an oscillatory integral involving Bessel functions and then by carefully making use of cancellation in such integrals based on the asymptotic expansion of Bessel functions. Our approach can be applied to the Schr\"odinger case as well. We also obtain some corresponding retarded estimates to give applications to nonlinear wave equations where Wiener amalgam spaces as solution spaces can lead to a finer analysis of the local and global behavior of the solution., Comment: To appear in J. Funct. Anal., 24 pages

We establish sharp time decay estimates for the the Klein-Gordon equation on the cubic lattice in dimensions $d=2,3,4$. The $\ell^1\to\ell^{\infty}$ dispersive decay rate is $|t|^{-3/4}$ for $d=2$, $|t|^{-7/6}$ for $d=3$ and $|t|^{-3/2}\log|t|$ for $d=4$. These decay rates are faster than conjectured by Kevrekidis and Stefanov (2005). The proof relies on oscillatory integral estimates and proceeds by a detailed analysis of the the singularities of the associated phase function. We also prove new Strichartz estimates and discuss applications to nonlinear PDEs and spectral theory., Comment: exposition improved, some tyops corrected

Accurate algorithms are proposed for approximation of integrals involving highly oscillatory Bessel function of the first kind over finite and infinite domains. Accordingly, Bessel oscillatory integrals having high oscillatory behavior are transformed into oscillatory integrals with Fourier kernel by using complex line integration technique. The transformed integrals contain an inner non-oscillatory improper integral and an outer highly oscillatory integral. A modified meshfree collocation method with Levin approach is considered to evaluate the transformed oscillatory type integrals numerically. The inner improper complex integrals are evaluated by either Gauss–Laguerre or multi-resolution quadrature. Inherited singularity of the meshfree collocation method at x = 0 is treated by a splitting technique. Error estimates of the proposed algorithms are derived theoretically in the inverse powers of ω and verified numerically.

Highly oscillatory integrals, having amplitudes with algebraic (or logarithmic) endpoint singularities, are considered. An integral of this kind is first transformed into a regular oscillatory integral over an unbounded interval. After applying the method of finite sections, a composite modified Filon–Clenshaw–Curtis rule, recently developed by the author, is applied on it. By this strategy the original integral can be computed in a more stable manner, while the convergence orders of the composite Filon–Clenshaw–Curtis rule are preserved. By introducing the concept of an oscillation subinterval, we propose algorithms, which employ composite Filon–Clenshaw–Curtis rules on rather small intervals. The integral outside the oscillation subinterval is non-oscillatory, so it can be computed by traditional quadrature rules for regular integrals, e.g. the Gaussian ones. We present several numerical examples, which illustrate the accuracy of the algorithms.

In the paper (Greenblatt in J Funct Anal, https://doi.org/10.1016/j.jfa.2018.05.014 , 2018) the author proved $$L^p$$ Sobolev regularity results for averaging operators over hypersurfaces and connected them to associated Newton polyhedra. In this paper, we use rather different resolution of singularities techniques along with oscillatory integral methods applied to surface measure Fourier transforms to prove $$L^p$$ Sobolev regularity results for a class of averaging operators over surfaces which can be of any codimension.

We establish the mapping properties of Fourier-type transforms on rearrangement-invariant quasi-Banach function spaces. In particular, we have the mapping properties of the Laplace transform, the Hankel transforms, the Kontorovich-Lebedev transform and some oscillatory integral operators. We achieve these mapping properties by using an interpolation functor that can explicitly generate a given rearrangement-invariant quasi-Banach function space via Lebesgue spaces.

In this paper, a meshless solution procedure incorporating delaminating quadrature method for two-dimensional highly oscillatory Fredholm integral equation is put forward. The proposed method is an extension of our earlier findings of meshless methods for two-dimensional oscillatory Fredholm integral equation having kernel function free of stationary phase point(s) (Siraj-ul-Islam et al., 2015). The current method deals not only with the kernels having no stationary phase point(s) but also with the kernels having stationary phase point(s) in the context of highly oscillatory integral equations. The new method is numerically stable and computationally fast. Numerical experiments are presented to testify the robustness and efficiency of the proposed method.

In this paper, we focus on constructing the asymptotic expansion for the highly oscillatory integral including of the product of exponential and Bessel oscillations with the stationary point. Based on the exact integral representation of Bessel function, the integral is transformed into a double oscillatory integral. For the resulting inner semi-infinite integral, we present a new way of a combination of the integration by parts and the Filon-type methods to produce the asymptotic expansion. Furthermore, the original oscillatory integral can be expanded in the sum of Gaussian hypergeometric function. The corresponding asymptotic property is also analysed. With increasing the oscillatory parameter, the error of the proposed asymptotic expansions decreases very fast. Numerical experiments are provided to illustrate the effectiveness of the expansion.

In this paper, we study time-asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of phases which may have a stationary point of real order and amplitudes allowed to have an integrable singular point. The resulting estimates provide optimal decay rates which show explicitly the influence of these two particular points. Then we apply these abstract results to solution formulas of a class of dispersive equations on the line defined by Fourier multipliers. Under the hypothesis that the Fourier transform of the initial data has a compact support or an integrable singular point, we derive uniform estimates of the solutions in space-time cones, describing their motions when the time tends to infinity. The method permits also to show that symbols having a restricted growth at infinity may influence the dispersion of the solutions: we prove the existence of a cone, depending only on the symbol, in which the solution is time-asymptotically localized. This corresponds to an asymptotic version of the notion of causality for initial data without compact support., Comment: This paper is an improved and extended version of arXiv:1507.00883. Moreover this second version contains supplementary information on wave packets to motivate our results and comments on the applicability of our method to the study of certain hyperbolic equations

Green's function of the problem describing steady forward motion of bodies in an open ocean in the framework of the linear surface wave theory (the function is often referred to as Kelvin's wave source potential) is considered. Methods for numerical evaluation of the so-called ‘single integral’ (or, in other words, ‘wavelike’) term, dominating in the representation of Green's function in the far field, are developed. The difficulty in the numerical evaluation is due to integration over infinite interval of the function containing two differently oscillating factors and the presence of stationary points. This work suggests two methods to approximate the integral. First of them is based on the idea put forward by D. Levin in 1982 — evaluation of the integral is converted to finding a particular slowly oscillating solution of an ordinary differential equation. To overcome well-known numerical instability of Levin's collocation method, an alternative type of collocation is used; it is based on a barycentric Lagrange interpolation with a clustered set of nodes. The second method for evaluation of the wavelike term involves application of the steepest descent method and Clenshaw–Curtis quadrature. The methods are numerically tested and compared.

We consider long time evolution of small solutions to general multispeed Klein-Gordon systems in 3+1 dimensions. We prove that such solutions are always global and scatter to a linear flow, thus extending previous partial results. The main new ingredients of our method is an improved linear dispersion estimate exploiting the asymptotic spherical symmetry of Klein-Gordon waves, and a corresponding bilinear oscillatory integral estimate.

We revisit the multilinear Kakeya, curved Kakeya, restriction, and oscillatory integral estimates that were obtained in paper of Bennett, Carbery, and the author using a heat flow monotonicity method applied to a fractional Cartesian product, together with induction on scales arguments. Many of these estimates contained losses of the form $R^\varepsilon$ (or $\log^{O(1)} R$) for some scale factor $R$. By further developing the heat flow method, and applying it directly for the first time to the multilinear curved Kakeya and restriction settings, we are able to eliminate these losses, as long as the exponent $p$ stays away from the endpoint. In particular, we establish global multilinear restriction estimates away from the endpoint, without any curvature hypotheses on the hypersurfaces., 69 pages, no figures. This is the revised version, incorporating referee comments

In this article we prove a sharp decay estimate for certain multilinear oscillatory integral operators of a form inspired by the general framework of Christ, Li, Tao, and Thiele [6]. A key purpose of this work is to determine when such estimates are stable under smooth perturbations of both the phase and corresponding projections, which are typically only assumed to be linear. The proof is accomplished by a novel decomposition which mixes features of Gabor or windowed Fourier bases with features of wavelet or Littlewood-Paley decompositions. This decomposition very nearly diagonalizes the problem and seems likely to have useful applications to other geometrically-inspired objects in Fourier analysis., Comment: 21 pages

We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein-Gordon and fractional Schr\"odinger equations. Our estimates extend those of Frank-Sabin in the case of the wave and Klein-Gordon equations, and generalize work of Frank-Lewin-Lieb-Seiringer and Frank-Sabin for the Schr\"{o}dinger equation. Due to a certain technical barrier, except for the classical Schr\"odinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results. The main novelty of this paper is the use of estimates for weighted oscillatory integrals which we combine with an approach due to Frank and Sabin. This strategy also leads us to proving new estimates for weighted oscillatory integrals with optimal decay exponents which we believe to be of wider independent interest. Applications to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces are also provided., Comment: 55 pages, substantial change to the pervious version, especially more applications included

We consider the oscillatory hyper Hilbert transform H γ,α,β f(x) = ∫ 0 ∞ f(x - Γ(t))eit-β t-(1+α)dt; where Γ(t) = (t, γ(t)) in ℝ2 is a general curve. When γ is convex, we give a simple condition on γ such that H γ,α,β is bounded on L 2 when β > 3α, β > 0: As a corollary, under this condition, we obtain the L p -boundedness of H γ,α,β when 2β/(2β - 3α) < p < 2β/(3α). When Γ is a general nonconvex curve, we give some more complicated conditions on γ such that H γ,α,β is bounded on L 2: As an application, we construct a class of strictly convex curves along which H γ,α,β is bounded on L 2 only if β > 2α > 0.

In this paper we study singular oscillatory integrals with a nonlinear phase function. We prove estimates of \(L^2 \rightarrow L^2\) and \(L^p\rightarrow L^p\) type.

Let be the oscillatory integral operators defined by where is the unit ball in and We compare the asymptotic behaviour as of the operator norms for all We prove that, except for the dimension this asymptotic behaviour depends on the linearity or quadraticity of the phase in s only. We are led to this problem by an observation on inhomogeneous Strichartz estimates for the Schrodinger equation.

In this paper, we study the numerical methods for the highly oscillatory integral of the type ∫abf(x)(x−a)α(b−x)βeiωg(x)dx, where α>−1,β>−1, f is analytic in a sufficiently large complex region containing [a,b]. Based on substituting the original interval of integration by the paths of steepest descent, the integral can be rewritten as a sum of several line integrals, which can be efficiently computed by Gaussian quadrature rules with different weight functions. Also, we apply this method to the implementation of discontinuous Galerkin method for Volterra integral equation with the Fourier kernel. Numerical examples are used to illustrate the efficiency and accuracy of the proposed methods.

An explicit local two-dimensional resolution of singularities theorem and arguments based on the Van der Corput lemma are used to give new estimates for the decay rate of the Fourier transform of a locally defined smooth hypersurface measure in R 3, as well as to provide new proofs of some known estimates. These are then used to give Lq bounds on solutions to certain PDE problems in terms of the Lp norms of their initial data for various values of p and q.

In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals whose phases and weights are contained in a certain class of smooth functions, including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho.

Let f be a Maass cusp form for Γ0(N) with Fourier coefficients λf (n) and Laplace eigenvalue 1/4 + k. For real α 6= 0 and β > 0 consider the sum: ∑ n λf (n)e(αn )φ ( n X ) , where φ is a smooth function of compact support. We prove bounds on the second spectral moment of this sum, with the eigenvalue tending toward infinity. When the eigenvalue is sufficiently large we obtain an average bound for this sum in terms of X. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL(2)×GL(2). It contains in particular the Kuznetsov trace formula and an asymptotic expansion of a well-known oscillatory integral with an enlarged range of K ≤ L ≤ K1−e. The same bounds can be proved in an analogous way for holomorphic cusp forms. Furthermore, we prove similar bounds for ∑ n λf (n)λg(n)e(αn )φ ( n X ) , where g is a holomorphic cusp form. As a corollary, we obtain a subconvexity bound for the L-function L(s, f × g). This bound has the significant property of breaking convexity even with a trivial bound toward the Ramanujan Conjecture.

Lubich&rsquo, s convolution quadrature rule provides efficient approximations to integrals with special kernels. Particularly, when it is applied to computing highly oscillatory integrals, numerical tests show it does not suffer from fast oscillation. This paper is devoted to studying the convergence property of the convolution quadrature rule for highly oscillatory problems. With the help of operational calculus, the convergence rate of the convolution quadrature rule with respect to the frequency is derived. Furthermore, its application to highly oscillatory integral equations is also investigated. Numerical results are presented to verify the effectiveness of the convolution quadrature rule in solving highly oscillatory problems. It is found from theoretical and numerical results that the convolution quadrature rule for solving highly oscillatory problems is efficient and high-potential.

In this paper, we present a Clenshaw-Curtis-Filon-type method for the weakly singular oscillatory integral with Fourier and Hankel kernels. By interpolating the non-oscillatory and nonsingular part of the integrand at $(N+1)$ Clenshaw-Curtis points, the method can be implemented in $O(N\log N)$ operations. The method requires the accurate computation of modified moments. We first give a method for the derivation of the recurrence relation for the modified moments, which can be applied to the derivation of the recurrence relation for the modified moments corresponding to other type oscillatory integrals. By using recurrence relation, special functions and classic quadrature methods, the modified moments can be computed accurately and efficiently. Then, we present the corresponding error bound in inverse powers of frequencies $k$ and $\omega$ for the proposed method. Numerical examples are provided to support the theoretical results and show the efficiency and accuracy of the method., Comment: submitted on Dec 03, 2015

We obtain new convolutions for quadratic-phase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weight-functions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadratic-phase Fourier integral operators are also studied (including a Riemann–Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity). As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations.

We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X} \Lambda(n) \Lambda(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided that $X^{8/33+\varepsilon} \leq H \leq X^{1-\varepsilon}$, with an error term saving on average an arbitrary power of the logarithm over the trivial bound. Previous work of Mikawa, Perelli-Pintz and Baier-Browning-Marasingha-Zhao covered the range $H \geq X^{1/3+\varepsilon}$. We also obtain an analogous result for $\sum_n \Lambda(n) \Lambda(N-n)$. Our proof uses the circle method and some oscillatory integral estimates (following a paper of Zhan) to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials associated to "Type $d_3$" and "Type $d_4$" sums (as well as some other sums that are easier to treat). After applying H\"older's inequality to the Type $d_3$ sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type $d_4$ sum is treated similarly using the classical $L^2$ mean value theorem and the classical van der Corput exponential sum estimates., Comment: 80 pages, no figures. updated references

Here and in what follows, we write "a b" instead of "a ≤ Cb for some uniform constant C" for brevity; we also write ab when a b and b a .T he symbol∂j, j =1 , 2, denotes the differentiation with respect to jth variable. To be more precise, we study estimates of the scalar product of two functions in some Hilbert space (in this paper, some Sobolev space of fractional order) in terms of the product of L1- norms of some differential polynomials applied to these functions. For the author, the interest in inequalities of such type originated from the work on nonisomorphism problems for Banach spaces of smooth functions and embedding theorems used there, see the short report (5) and preprint (6). We are going to use some formalism to make our statements shorter. Let k and l be natural numbers, let α and β be real nonnegative numbers, and let σ and τ be complex nonzero numbers. The symbol BE(k, l, α, β, σ, τ) means the statement that the inequality �

In this paper, we consider the evaluation of two kinds of oscillatory integrals with a Hankel function as kernel. We first rewrite these integrals as the integrals of Fourier-type. By analytic continuation, these Fourier-type integrals can be transformed into the integrals on 0, +∞), the integrands of which are not oscillatory, and decay exponentially fast. Consequently, the transformed integrals can be efficiently computed by using the generalized Gauss-Laguerre quadrature rule. Moreover, the error analysis for the presented methods is given. The efficiency and accuracy of the methods have been demonstrated by both numerical experiments and theoretical results.

In this paper a stable and modified form of the Levin method based on Bessel radial basis functions is employed for numerical solution of highly oscillatory integrals. In the proposed technique, the multiquadric radial basis function (Levin, 1982 [1] ; Siraj-ul-Islam et al., 2013 [2] ) is replaced by Bessel radial basis functions (Fornberg et al., 2006 [3] ) and thin plate spline of order three. In this scheme the integration form is first transformed into differential form and then the numerical solution of the corresponding differential form is found. The accuracy and the algebraic stability in the form of well-conditioned coefficient matrices of the proposed methods are confirmed through numerical experiments.

The Kuznetsov formula provides a deep connection between the spectral theory in hyperbolic Riemann surfaces and some exponential sums of arithmetic nature that has been extremely fruitful in modern number theory. Unfortunately the application of the Kuznetsov formula is by no means easy in practice because it involves oscillatory integral transforms with kernels given by special functions in non-standard ranges. In this paper we introduce a new formulation of the Kuznetsov formula that rules out these complications reducing the integral transforms to something almost as simple as a composition of two Fourier transforms. This formulation admits a surprisingly short and clean proof that does not require any knowledge about special functions, solving in this way another of the disadvantages of the classical approach. Moreover the reversed formula becomes more natural and in the negative case it reduces to a direct application of Fourier inversion. We also show that our approach is more convenient in applications and gives some freedom to play with explicit test functions.

This paper presents an efficient numerical method for approximating highly oscillatory Bessel transforms. Based on analytic continuation, we transform the integrals into the problems of integrating the forms on [ 0 , + ∞ ) with the integrand that does not oscillate and decays exponentially fast, which can be efficiently computed by using Gauss–Laguerre quadrature rule. We then derive the error of the method depending on the frequency and the node number. Moreover, we apply the scheme for studying the approximations of the solutions of two kinds of highly oscillatory integral equations. Preliminary numerical results show the efficiency and accuracy of numerical approximations.

We study an operator analogue of the classical problem of finding the rate of decay of an oscillatory integral on the real line. This particular problem arose in the analysis of oscillatory Riemann–Hilbert problems associated with partial differential equations in the Ablowitz–Kaup–Newell–Segur hierarchy, but is interesting in its own right as a question in harmonic analysis and oscillatory integrals. As was the case in earlier work of the first author [9] , the approach is general and purely real-variable. The resulting estimates we achieve are strongly uniform as a function of the phase and can simultaneously accommodate phases with low regularity (as low as C 1 , α ), local singularities, and essentially arbitrary sets of stationary points that degenerate to finite or infinite order.

We prove an elementary multilinear identity for the Fourier extension operator on $\mathbb{R}^n$, generalising to higher dimensions the classical bilinear extension identity in the plane. In the particular case of the extension operator associated with the paraboloid, this provides a higher dimensional extension of a well-known identity of Ozawa and Tsutsumi for solutions to the free time-dependent Schr\"odinger equation. We conclude with a similar treatment of more general oscillatory integral operators whose phase functions collectively satisfy a natural multilinear transversality condition. The perspective we present has its origins in work of Drury., Comment: To appear in Mathematical Research Letters

In this paper we give an elementary proof for transference of local to global maximal estimates for dispersive PDEs. This is done by transferring local $L^2$ estimates for certain oscillatory integrals with rough phase functions, to the corresponding global estimates. The elementary feature of our approach is that it entirely avoids the use of the wave packet techniques which are quite common in this context, and instead is based on scalings and classical oscillatory integral estimates., Comment: 10 pages

Although the time slicing approximation of Feynman path integrals does not converge absolutely, it has a definite finite value if the potential satisfies Assumption 2.1 and if the time interval is short, because it is an oscillatory integral that satisfies Assumption 3.1. Furthermore, the time slicing approximation of Feynman path integrals converges to a limit as \(|\varDelta |\rightarrow 0\). The limit turns out to be the fundamental solution of the Schrodinger equation. The semi-classical asymptotic formula called Birkhoff’s formula is proved from the standpoint of oscillatory integrals. In this chapter, these results as well as others are explained. Proofs will be given in subsequent chapters.