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We study a distance graph Γ n that is isomorphic to the 1 -skeleton of an n -dimensional unit hypercube. We show that every measurable set of positive upper Banach density in the plane contains all sufficiently large dilates of Γ n . This provides the first examples of distance graphs other than the trees for which a dimensionally sharp embedding in positive density sets is known. [ABSTRACT FROM AUTHOR]

The Burton-Miller method is widely used to solve the problem of solution non-uniqueness in a conventional boundary element method (CBEM). Although this method is very robust, the hypersingular integral kernel contained in its normal derivative equation increases the computational complexity and reduces the computational accuracy. In this paper, a coupled double boundary Burton-Miller method with unique solutions at full wave numbers is proposed. The aforementioned process is done by replacing the normal integral equation of the Burton-Miller method with the virtual indirect boundary element method (VIBEM) integral equation of combined layer potential and using the equivalent relationship between their coefficient matrices. The proposed method inherits the high-precision advantages of VIBEM and avoids the hypersingular integrals of traditional Burton-Miller methods. In particular, no singular integral is present when the plane element is used for discretization. The numerical results of acoustic radiation and scattering show that the calculation accuracy of the coupled double boundary Burton-Miller method is higher than that of CBEM and the conventional Burton-Miller method. Moreover, the condition number of the coefficient matrix is much lower than that of VIBEM. Lastly, the computation time is less than that of the conventional Burton-Miller method. • Avoiding the calculation of hypersingular integrals in the Burton-Miller method. • Significantly improved the computational accuracy of the Burton-Miller method. • The non-uniqueness of the solution of the boundary integral equation is overcome. • The matrix condition number of the virtual boundary element method is reduced. [ABSTRACT FROM AUTHOR]

In this article we present the L p , p ≥ 1 , approximation properties of activated singular integral operators over the real line. We establish their approximation to the unit operator with rates. The kernels here come from neural network activation functions and we employ the related density functions. The derived inequalities use the high order L p modulus of smoothness. [ABSTRACT FROM AUTHOR]

As a consequence of a general result, we prove that in the case of singular integrals the set of convergence consists only of the two functions 1 and cos . We prove also a multivariate version of this result and apply it to find the necessary and sufficient conditions for the convergence of the sequences of positive linear operators associated to the rectangular and triangular summation. [ABSTRACT FROM AUTHOR]

The use of boundary elements in two-dimensional acoustic analysis is presented in this study, along with a detailed explanation of how to derive the final discrete equations from the fundamental fluctuation equations. In order to overcome the fictitious eigenfrequency problem that might arise during the examination of the external sound field, this work employs the Burton-Miller approach. Additionally, this work uses the Taylor expansion to extract the frequency-dependent component from the BEM function, which speeds up the computation and removes the frequency dependency of the system coefficient matrix. The effect of the radiated acoustic field generated by underwater structures' on thin-walled structures such as submarines and ships is inspected in this work. Numerical examples verify the accuracy of the proposed method and the efficiency improvement. [ABSTRACT FROM AUTHOR]

In this article, we study the approximation properties of activated singular integral operators over the real line. We establish their convergence to the unit operator with rates. The kernels here derive from neural network activation functions and their corresponding density functions. The estimates are mostly sharp, and they are pointwise and uniform. The derived inequalities involve the higher order modulus of smoothness. [ABSTRACT FROM AUTHOR]

In this article we continue our research in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19), about the asymptotic expansion of thick distributions. We compute more examples of asymptotic expansion of integral transforms using the techniques developed in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19). Besides, we derive a new “Laplace Formula” for the situation in which a point singularity is allowed. [ABSTRACT FROM AUTHOR]

The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. It is also the main part of the theory of singular integral equations on the real line. Therefore, approximations of Hilbert transform are of great interest. Many papers have dealt with the numerical approximation of singular integrals in case of bounded intervals. On the other hand, the literature concerning the numerical integration on unbounded intervals is much sparser than the one on bounded intervals. There is very little literature concerning the case of Hilbert transform. This article is dedicated to the approximation of Hilbert transform in Hölder spaces by the operators introduced by V.R.Kress and E.Mortensen to approximate the Hilbert transform of analytic functions in a strip. [ABSTRACT FROM AUTHOR]

For an integer d ≥ 2 , t ∈ R , and a 0-homogeneous function Φ ∈ C ∞ (R d \ { 0 } , R) , we consider the family of Fourier multiplier operators T Φ t associated with symbols ξ ↦ exp (i t Φ (ξ)) and prove that for a generic phase function Φ , one has the estimate ‖ T Φ t ‖ L p → L p ≳ d , p , Φ ⟨ t ⟩ d | 1 p - 1 2 | . That is the maximal possible order of growth in t → ± ∞ , according to the previous work by V. Kovač and the author and the result shows that the two special examples of functions Φ that induce the maximal growth, given by V. Kovač and the author and independently by D. Stolyarov, to disprove a conjecture of Maz'ya actually exhibit the same general phenomenon. [ABSTRACT FROM AUTHOR]

We present a simple yet accurate method to compute the adjoint double layer potential, which is used to solve the Neumann boundary value problem for Laplace's equation in three dimensions. An expansion in curvilinear coordinates leads us to modify the expression for the adjoint double layer so that the singularity is reduced when evaluating the integral on the surface. Then, to regularize the integral, we multiply the Green's function by a radial function with length parameter δ chosen so that the product is smooth. We show that a natural regularization has error O (δ 3) , and a simple modification improves the error to O (δ 5) . The integral is evaluated numerically without the need of special coordinates. We use this treatment of the adjoint double layer to solve the classical integral equation for the interior Neumann problem, altered to account for the solvability condition, and evaluate the solution on the boundary. Choosing δ = c h 4 / 5 , we find about O (h 4) convergence in our examples, where h is the spacing in a background grid. [ABSTRACT FROM AUTHOR]

AbstractThe moving electric-magnetic polarization saturation (EMPS) model is studied for two equal collinear interface cracks in mode-III dissimilar magneto-electro-elastic (MEE) bimaterial under semipermeable crack face conditions. The closed-form solutions of all three possible cases of the proposed EMPS model are derived using the distributed dislocation technique (DDT) and singular integral method. An iterative scheme and explicit expressions of fracture parameters are used to implement the semipermeable crack-face conditions. The study of local energy release rate based on constant electric and magnetic breakdown strength-based saturated conditions concludes that increasing the mismatch of the MEE bimaterial can improve its fracture toughness. [ABSTRACT FROM AUTHOR]

The singular integral equation method is applied to present analytical solutions for modified electric-magnetic-polarization saturation (EMPS) models subjected to semipermeable center cracked magneto-electro-elastic (MEE) materials. A generalized methodology is presented to explicitly solve the EMPS model under any arbitrary saturated electric and magnetic conditions using the distributed dislocation method (DDM) and finding the analytical solutions for developed singular integral equations. The explicit expressions are derived for distributed dislocation densities, crack opening displacement (COD), crack opening potential (COP), crack opening induction (COI), saturated zone lengths, and local stress intensity factor (LSIF) for particular cases of the generalized EMPS model such as constant, linear, quadratic and cubic polynomial varying saturated electric and magnetic conditions. Using an iterative scheme and the explicit solutions of COD, COP, and COI, the exact semipermeable crack-face conditions are evaluated at the center of the crack, and the same applies throughout the crack-surface for the study of semipermeable modified EMPS models. A comparative analysis of the fracture parameters has also been carried out between the results of exact semipermeable crack-face conditions and the results of semipermeable crack-face conditions of the center crack problem to analyze the impact of polynomial varying saturated conditions on the crack-face conditions. Numerical studies performed under different electromagnetic-mechanical loadings, volume fractions, and crack-face conditions demonstrate the increasing effects in the fracture parameters' values with the degree of the polynomial varying saturated conditions. • We present closed-form solutions of variable saturated conditions based EMPS models in MEE materials. • A generalized methodology using DDT and singular integral method is derived. • Explicit expressions for saturated zone lengths and fracture parameters for each modified EMPS model are given. • Fracture parameters based on exact semipermeable and semipermeable conditions of the centre crack problem are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of singular integrals which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation ∂ t ⁡ θ + u ⋅ ∇ ⁡ θ + κ ⁢ Λ 2 ⁢ β ⁢ θ = 0 , (x , t) ∈ ℝ 2 × ℝ + , κ > 0 , where u = - ∇ ⊥ ⁡ Λ - 2 + 2 ⁢ α ⁢ θ , α ∈ [ 0 , 1 2 ] and β ∈ (0 , 1 ] . First, we give a relationship between this kind of singular integrals and Calderón–Zygmund singular integral operators and obtain a uniform Besov estimates. As an application, we give the well-posedness of the generalized 2D dissipative quasi-geostrophic (QG) in the critical Besov space. [ABSTRACT FROM AUTHOR]

The use of boundary elements in two-dimensional acoustic analysis is presented in this study, along with a detailed explanation of how to derive the final discrete equations from the fundamental fluctuation equations. In order to overcome the fictitious eigenfrequency problem that might arise during the examination of the external sound field, this work employs the Burton-Miller approach. Additionally, this work uses the Taylor expansion to extract the frequency-dependent component from the BEM function, which speeds up the computation and removes the frequency dependency of the system coefficient matrix. The effect of the radiated acoustic field generated by underwater structures’ on thin-walled structures such as submarines and ships is inspected in this work. Numerical examples verify the accuracy of the proposed method and the efficiency improvement.

We prove boundedness and compactness for the iterated commutators of the $ \theta $-type Calderón-Zygmund singular integral and its fractional variant on the weighed Morrey spaces.

This paper is devoted to the real analysis methods for the problem of construction of Markov processes with boundary conditions in probability. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called Ventcel' (Wentzell) boundary condition, on the boundary of the domain. Probabilistically, a Markovian particle moves both by continuous paths and by jumps in the state space and it obeys the Ventcel' boundary condition that consists of six terms corresponding to a diffusion along the boundary, an absorption phenomenon, a reflection phenomenon, a sticking (or viscosity) phenomenon and a jump phenomenon on the boundary and an inward jump phenomenon from the boundary. More precisely, we study a class of first order Ventcel' boundary value problems for second order elliptic Waldenfels integro-differential operators. By using the Calderón–Zygmund theory of singular integrals in real analysis, we prove existence and uniqueness theorems in the framework of Sobolev and Besov spaces, which extend earlier theorems due to Bony–Courrège–Priouret to the VMO (vanishing mean oscillation) case. Our proof is based on various maximum principles for second order elliptic Waldenfels operators with discontinuous coefficients in the framework of Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of singular integrals T λ f (x) = p.v. ∫ R n Ω (y) | y | n − λ f (x − y) d y for any f ∈ L q (R n) , 1 < q < ∞ and 0 < λ < n , which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation (0.1) ∂ t θ + u ⋅ ∇ θ + κ Λ 2 β θ = 0 , (x , t) ∈ R 2 × R + , κ > 0 , where u = − ∇ ⊥ Λ − 2 + 2 α θ , α ∈ [ 0 , 1 2 ] and β ∈ (0 , 1 ]. Firstly, we give a uniform sparse domination for this kind of singular integral operators. Secondly, we obtain the uniform quantitative weighted bounds for the operator T λ with rough kernel. As an application, we obtain the uniform quantitative weighted bounds for the commutator [ b , T λ ] with rough kernel and study solutions to the generalized 2D dissipative quasi-geostrophic (QG) equation. [ABSTRACT FROM AUTHOR]

The article is devoted to uniform approximations of the solution of a periodic strongly singular integral equation of the first kind. We note that an integral equation of the first kind is an ill-posed problem. However, the presence of certain features in the kernels makes it possible to establish a correct formulation of the problem on specially selected function spaces. After this selection, various approximation methods can be applied to solve the equation under consideration. In the present article, the authors use this approach, previously applied in the periodic and non-periodic cases to integral equations of the first kind with a logarithmic singularity in the kernel, to integral equations with the Cauchy kernel on a segment of the real axis, and to some singular integro-differential equations. The spaces of the required elements and right-hand sides are taken to be certain narrowings of the space of -periodic continuous functions with a certain norm. The boundedness of the operator, inverse to the characteristic one on a pair of selected spaces, is established. Next, projection methods for solving the characteristic and complete equations are considered. Constructive estimates of the approximation of a function by partial sums of its Fourier series and Lagrange interpolation polynomials in the proposed spaces make it possible to establish uniform approximations of solutions to the equation using the Galerkin and collocation methods. [ABSTRACT FROM AUTHOR]

The present article proposes a time‐domain boundary element method (TDBEM) for the three‐dimensional (3D) dissipative wave equation (DWE). Although the fundamental ingredients such as the Green's function for the 3D DWE have been known for a long time, the details of formulation and implementation for such a 3D TDBEM have been unreported yet to the author's best knowledge. The present formulation is performed truly in time domain on the basis of the time‐dependent Green's function and results in a marching‐on‐in‐time fashion. The main concern is in the evaluation of the boundary integrals. For this regard, weakly‐ and removable‐singularities are carefully treated. The proposed TDBEM is checked through the numerical examples whose solutions can be obtained semi‐analytically by means of the inverse Laplace transform. The results of the present TDBEM are satisfactory to validate its formulation and implementation for Neumann problems. On the other hand, the present formulation based on the ordinary boundary integral equation (BIE) is unstable for Dirichlet problems. The numerical analyses for the non‐dissipative case imply that the instability issue can be partially resolved by using the Burton–Miller BIE even in the dissipative case. [ABSTRACT FROM AUTHOR]

The approach to "locally" aggrandize Lebesgue spaces, previously suggested by the authors and based on the notion of "aggrandizer", is combined with the usual "global" aggrandization. We study properties of such spaces including embeddings, dependence of the choice of the aggrandizer and, in particular, we discuss the question when these spaces are not new, coinciding with globally aggrandized spaces, and when they proved to be new. We study the boundedness of the maximal, singular, and maximal singular operators in the introduced spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the maximal truncated singular integral operators with rough kernel along certain compound curves, which contain many classical model examples. We prove the Lp boundedness of such maximal singular integral operators under very weak conditions on the integral kernels both on the unit sphere and the radial direction. The main results essentially improve and extend certain previous results. [ABSTRACT FROM AUTHOR]

In this article we present the Lp, p≥1, approximation properties of activated singular integral operators over the real line. We establish their approximation to the unit operator with rates. The kernels here come from neural network activation functions and we employ the related density functions. The derived inequalities use the high order Lp modulus of smoothness.

In this article, we study the approximation properties of activated singular integral operators over the real line. We establish their convergence to the unit operator with rates. The kernels here derive from neural network activation functions and their corresponding density functions. The estimates are mostly sharp, and they are pointwise and uniform. The derived inequalities involve the higher order modulus of smoothness.

The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. The Hilbert transform is the main part of the singular integral equations on the real line. Therefore, approximations of the Hilbert transform are of great interest. Many papers have dealt with the numerical approximation of the singular integrals in the case of bounded intervals. On the other hand, the literature concerning the numerical integration on unbounded intervals is by far poorer than the one on bounded intervals. The case of the Hilbert Transform has been considered very little. This article is devoted to the approximation of the Hilbert transform in Lebesgue spaces by operators which introduced by V.R.Kress and E.Mortensen to approximate the Hilbert transform of analytic functions in a strip. In this paper, we prove that the approximating operators are bounded maps in Lebesgue spaces and strongly converges to the Hilbert transform in these spaces.

In Electromagnetics, the field scattered by an ensemble of particles—of arbitrary size, shape, and material—can be obtained by solving the Lippmann–Schwinger equation. This singular vectorial integral equation is generally formulated in the direct space R n (typically n = 2 or n = 3 ). In the article, we rigorously computed the Fourier transform of the vectorial Lippmann–Schwinger equation in the space of tempered distributions, S ′ (R 3) , splitting it in a singular and a regular contribution. One eventually obtains a simple equation for the scattered field in the Fourier space. This permits to draw an explicit link between the shape of the scatterer and the field through the Fourier Transform of the body indicator function. We compare our results with accurate calculations based on the T-matrix method and find a good agreement. [ABSTRACT FROM AUTHOR]

Let p (·) : R n → (1 , ∞) be a variable exponent, such that the Hardy–Littlewood maximal operator is bounded on the variable exponent Lebesgue space L p (·) (R n) , and ϕ : R n × (0 , ∞) → (0 , ∞) be a function satisfying some conditions. In this article, we give some properties of variable Campanato spaces L p (·) , ϕ , d (R n) , with a non-negative integer d, and variable Morrey spaces L p (·) , ϕ (R n) , and then establish their predual spaces. As an application of duality obtained in this article, we consider the boundedness of singular integral operators on variable Morrey spaces and their predual spaces. [ABSTRACT FROM AUTHOR]

A complete hypersingular integral equation of the second kind was obtained as a boundary integral equation for the diffraction and scattering problem of electromagnetic waves in space separated by the periodically placed non-perfectly conducting strips. The equation includes a singular integral that distinguishes it from the studied second-kind hypersingular equation. Our motivation is the need to have a numerical method for the equation, its applicability borders, and guaranteed convergence. The numerical method has the type of Nyström. The justification of the method envelops a proof of the theorem of existence and uniqueness of the solution and an estimate of the convergence rate of sequence of the approximate solutions to an exact solution. [ABSTRACT FROM AUTHOR]

A complete hypersingular integral equation of the second kind was obtained as a boundary integral equation for the diffraction and scattering problem of electromagnetic waves in space separated by the periodically placed non-perfectly conducting strips. The equation includes a singular integral that distinguishes it from the studied second-kind hypersingular equation. Our motivation is the need to have a numerical method for the equation, its applicability borders, and guaranteed convergence. The numerical method has the type of Nyström. The justification of the method envelops a proof of the theorem of existence and uniqueness of the solution and an estimate of the convergence rate of sequence of the approximate solutions to an exact solution.

In this paper, we prove that for n 2 - 1 2 < α < n 2 , α ∉ ℕ , the convolution operator S α ⁢ f ⁢ (x) = ∫ | y | ≥ 1 f ⁢ (x - y) ⁢ ( | y | 2 - 1) - α ⁢ 푑 y is bounded from L p to L q 1 + L q 2 for certain values of p and q 1 , q 2 . [ABSTRACT FROM AUTHOR]

We study Fourier multiplier operators associated with symbols \xi \mapsto \exp (\mathbb {i}\lambda \phi (\xi /|\xi |)), where \lambda is a real number and \phi is a real-valued \mathrm {C}^\infty function on the standard unit sphere \mathbb {S}^{n-1}\subset \mathbb {R}^n. For 1

In this paper, we establish a weak‐type (1,1) boundedness criterion for vector‐valued singular integral operators with rough kernels. As applications, we obtain weak‐type (1,1) bounds for the convolution singular integral operator taking value in the Banach space Y with a rough kernel, the maximal operator taking vector value in Y with a rough kernel and several square functions with rough kernels. Here, Y=[H,X]θ$Y=[H,X]_\theta$ is a complex interpolation space between a Hilbert space H and a UMD space X. [ABSTRACT FROM AUTHOR]

We investigate the maximal singular integral with Riesz potentials and give a short proof of its estimates through a result from Duoandikoetxea and Rubio de Francia in dimension n > 1 and the estimates of Fourier transforms of the measures in dimension n = 1 . In particular, this enables us to drop the Dini condition when n > 1 . In the appendix, we provide a proof of a counterexample for Corollary 4.1 in [Invent. Math. (1986)] by Duoandikoetxea and Rubio de Francia in dimension n = 1 . [ABSTRACT FROM AUTHOR]

Let V q K , q ∈ (2 , ∞) , be the variation operator of a family truncated operators of a singular integral K on an RD-space (X , d , μ) with additional layer decay property. Under the assumption that the kernel of K satisfies a Dini condition and V q K is bounded on L p 0 (μ) for some p 0 ∈ (1 , ∞) , the authors obtain the boundedness of V q K on L p (w) for w ∈ A p and from L 1 (w) to L 1 , ∞ (w) for w ∈ A 1 . These results also hold for oscillation operators of K. As applications, weighted boundedness of variations and oscillations for some known singular integrals are provided. [ABSTRACT FROM AUTHOR]

This paper is concerned with the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation. For the generalized singular integral operator, we obtain uniform L p − L q estimates with respect to a parameter β. From this one can cover the L p -boundedness of the Calderón-Zygmund operator with rough kernel by letting β → 0. We applied this estimate to study the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation. Local well-posedness in the Besov space B p , q s and some limit behaviour of the solutions are obtained. Our results improve the previous ones by Yu-Zheng-Jiu in 2019 and by Yu-Jiu-Li in 2021. [ABSTRACT FROM AUTHOR]

This study aims to bring together the ultimate formation of the integro-differential equations involving the functional delay and the singular integral with constant delay on the real line, presenting an exclusive spectral computational approach made up of quadratic orthoexponential polynomials. Frankly, the method makes possible to evaluate the singular and delayed integral part by way of the matrix expansion of quadratic property of orthoexponential polynomials, as well as transforming the unknown terms into the matrix relations. Hence, the stability of the integral part is materialized. An error improvement technique is also performed via a residual function. Several integral and integro-differential equations are considered by a devised programme, which immediately returns the method of solution. The outcomes are, thus, exposed to be compared sensitively in tables and figures. Having investigated the comparisons, one can admit that the method accomplishes exclusive and efficient approximation to the equations in question. [ABSTRACT FROM AUTHOR]

We find asymptotic expansions as for integrals of the form , where sufficiently smooth functions and satisfy natural assumptions on their behavior at infinity and all critical points of in the set are nondegenerate. These asymptotic expansions play a crucial role in analyzing stochastic models for nonlinear waves systems. We generalize a result of Kuksin that a similar asymptotic expansion occurs in a particular case where is a nondegenerate quadratic form of signature with even . [ABSTRACT FROM AUTHOR]

In this paper we prove uniform oscillation estimates on L p , with p ∈ (1 , ∞) , for truncated singular integrals of the Radon type associated with the Calderón–Zygmund kernel, both in continuous and discrete settings. In the discrete case we use the Ionescu–Wainger multiplier theorem and the Rademacher–Menshov inequality to handle the number-theoretic nature of the discrete singular integral. The result we obtained in the continuous setting can be seen as a generalisation of the results of Campbell, Jones, Reinhold and Wierdl for the continuous singular integrals of the Calderón–Zygmund type. [ABSTRACT FROM AUTHOR]

Let T be a linear operator that, for some p1∈(1,∞), is bounded on Lp1(w~) for all w~∈Ap1(Rd) and in addition compact on Lp1(w1) for some w1∈Ap1(Rd). Then T is bounded and compact on Lp(w) for all p∈(1,∞) and all w∈Ap(Rd). This "compact version" of Rubio de Francia's celebrated weighted extrapolation theorem follows from a combination of results in the interpolation and extrapolation theory of weighted spaces on the one hand, and of compact operators on abstract spaces on the other hand. Moreover, generalizations of this extrapolation of compactness are obtained for operators that are bounded from one space to a different one ("off-diagonal estimates") or only in a limited range of the Lp scale. As applications, we easily recover several recent results on the weighted compactness of commutators of singular integral operators, fractional integrals and pseudo-differential operators, and obtain new results about the weighted compactness of commutators of Bochner-Riesz multipliers. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of the well-posedness of the problem of solving some classes of integral and integro-differential equations of the first kind. The authors consider integral equations with a logarithmic singularity in periodic and non-periodic cases, a weakly singular equations with indeterminate parameters, integral equations with the Cauchy kernel on the segment of the real axis, a singular integro-differential equation. In accordance with the theory of ill-posed problem, integral equations of the first kind are one of the well-known example of such problem. However, it is possible to establish the well-posedness of the problem in some particular cases, for example, for singular or weakly singular kernels. In this cases function spaces are selected in a special way. The authors propose a certain technique to select a pair of spaces of the desired elements and the right-hand sides. The approach is based on the narrowing the space of continuous functions and the properties of singular integrals. For each class the authors propose a pair of appropriate function spaces. The norms in them are chosen in a certain way. The results obtained by the authors allow us to solve the equations under consideration numerically and obtain uniform estimates of the errors of approximate solutions. [ABSTRACT FROM AUTHOR]

We prove a range of L p bounds for singular Brascamp–Lieb forms with cubical structure. We pass through sparse and local bounds, the latter proved by an iteration of Fourier expansion, telescoping, and the Cauchy–Schwarz inequality. We allow 2 m - 1 < p ≤ ∞ with m the dimension of the cube, extending an earlier result that required p = 2 m . The threshold 2 m - 1 is sharp in our theorems. [ABSTRACT FROM AUTHOR]

In Electromagnetics, the field scattered by an ensemble of particles—of arbitrary size, shape, and material—can be obtained by solving the Lippmann–Schwinger equation. This singular vectorial integral equation is generally formulated in the direct space Rn (typically n=2 or n=3). In the article, we rigorously computed the Fourier transform of the vectorial Lippmann–Schwinger equation in the space of tempered distributions, S′(R3), splitting it in a singular and a regular contribution. One eventually obtains a simple equation for the scattered field in the Fourier space. This permits to draw an explicit link between the shape of the scatterer and the field through the Fourier Transform of the body indicator function. We compare our results with accurate calculations based on the T-matrix method and find a good agreement.

Abstract In this paper, the authors establish the boundedness of singular integral operators associated to polynomial curves as well as the related maximal operators with rough kernels Ω ∈ H 1 ( S n − 1 ) $\Omega \in H^{1}({\mathrm{S}}^{n-1})$ and h ∈ Δ γ ( R + ) $h\in \Delta _{\gamma }(\mathbb{R}_{+})$ for some γ > 1 $\gamma >1$ on the Triebel–Lizorkin spaces. It should be pointed out that the bounds are independent of the coefficients of the polynomials in the definition of the operators. The main results of this paper not only improve and generalize essentially some known results but also complement some recent boundedness results.

In this paper, we present some explicit exponents in the estimates for the volumes of sub-level sets of polynomials on bounded sets and applications to the decay of oscillatory integrals and the convergence of singular integrals.

This article concerns an analytic and numerical analysis of a class of weighted singular Cauchy integrals with exponential weights w : = exp (− Q) with finite moments and with smooth external fields Q : R → [ 0 , ∞) , with varying smooth convex rate of increase for large argument. Our analysis relies in part on weighted polynomial interpolation at the zeros of orthonormal polynomials with respect to w2. We also study bounds for the first derivatives of a class of functions of the second kind for w2. [ABSTRACT FROM AUTHOR]