Comments on a paper by Papadopoulos on the use of singular integrals in wave propagation problems

1965 ◽  
Vol 55 (2) ◽  
pp. 303-318
Author(s):  
I. M. Longman
Keyword(s):  
Wave Propagation ◽  
Point Source ◽  
Singular Integral ◽  
Singular Integrals ◽  
Sound Propagation ◽  
Elastic Solid ◽  
Special Cases

Abstract In a recent paper Papadopoulos [1]1 claims to have obtained solutions to the problem of sound propagation from an impulsive point source in a semi-infinite elastic solid which differ in certain respects from solutions previously obtained by Pekeris [2], [3], and by Pekeris and Lifson [4]. The results of Papadopoulos are based on a singular integral source representation developed by him [5]. By tests based on simple special cases we show that, in a number of cases where Papadopoulos claims a result different from that of Pekeris, the results of Papadopoulos are unacceptable whereas the results of Pekeris are acceptable. Reasons for these errors in Papadopoulos's work are suggested. A number of other errors in the paper of Papadopoulos are also noted. 1 Numbers in brackets refer to references at rear.

Multi-parameter Singular Integrals I: Examples

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Pseudodifferential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Special Cases ◽  
Stratified Lie Groups ◽  
Left And Right ◽  
Flag Kernels

This chapter discusses a few special cases where a theory of multi-parameter singular integral operators has already been developed. These include the product theory of singular integrals, convolution with flag kernels on graded groups, convolution with both the left and right invariant Calderón–Zygmund singular integral operators on stratified Lie groups, and composition of standard pseudodifferential operators with certain singular integrals corresponding to non-Euclidean geometries. The chapter outlines these examples and their applications and relates them to the trichotomy discussed in Chapter 1.

The use of singular integrals in wave propagation problems; with application to the point source in a semi-infinite elastic medium

1963 ◽  
Vol 276 (1365) ◽  
pp. 204-237 ◽  
Keyword(s):  
Free Surface ◽  
Point Source ◽  
Elastic Medium ◽  
Constant Velocity ◽  
Singular Integrals ◽  
Plane Waves ◽  
Elastic Solid ◽  
Fundamental Solutions ◽  
General Point ◽  
Set Up

This is an investigation of the field due to a general point source of energy in an isotropic, elastic solid with a free surface. The paper is divided into three parts. In part I we are concerned with the development of new plane wave representations for the fundamental solutions of elastodynamics. There are two types of situation involved; we have the simpler type involved in the case of a steady point source which moves steadily with any constant velocity in an elastic medium, this type involves superposition of plane waves with respect to a single parameter, and we have the more complicated transient problem in which a point source is set up at a given moment, and thereafter moves at constant velocity, without change of strength. In part II we make use of the new representation for the field of a steadily moving source in the calculation of fields and displacements in the presence of a free surface and in part III we do the same for the transient source. We discuss in some detail the application of the new approach to the case of a vertical load, to a horizontal load, and to a couple of arbitrary orientation, and we give a general discussion of the singularities to be expected for the general point source.

Multi-parameter Singular Integrals II: General Theory

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
General Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Main Issue

This chapter turns to a general theory which generalizes and unifies all of the examples in the preceding chapters. A main issue is that the first definition from the trichotomy does not generalize to the multi-parameter situation. To deal with this, strengthened cancellation conditions are introduced. This is done in two different ways, resulting in four total definitions for singular integral operators (the first two use the strengthened cancellation conditions, while the later two are generalizations of the later two parts of the trichotomy). Thus, we obtain four classes of singular integral operators, denoted by A1, A2, A3, and A4. The main theorem of the chapter is A1 = A2 = A3 = A4; i.e., all four of these definitions are equivalent. This leads to many nice properties of these singular integral operators.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

scholarly journals A note on singular integrals with angular integrability

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Function Spaces ◽  
Riesz Transforms ◽  
Hilbert Transforms ◽  
Rotation Method ◽  
Vector Valued Function ◽  
Vector Valued ◽  
L Log L

Abstract In this note we study the rough singular integral $$ T_{\varOmega }f(x)=\mathrm{p.v.} \int _{\mathbb{R}^{n}}f(x-y)\frac{\varOmega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, $$ T Ω f ( x ) = p . v . ∫ R n f ( x − y ) Ω ( y / | y | ) | y | n d y , where $n\geq 2$ n ≥ 2 and Ω is a function in $L\log L(\mathrm{S} ^{n-1})$ L log L ( S n − 1 ) with vanishing integral. We prove that $T_{\varOmega }$ T Ω is bounded on the mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}( \mathbb{R}^{n})$ L | x | p L θ p ˜ ( R n ) , on the vector-valued mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}(\mathbb{R}^{n},\ell ^{\tilde{p}})$ L | x | p L θ p ˜ ( R n , ℓ p ˜ ) and on the vector-valued function spaces $L^{p}(\mathbb{R}^{n}, \ell ^{\tilde{p}})$ L p ( R n , ℓ p ˜ ) if $1<\tilde{p}\leq p<\tilde{p}n/(n-1)$ 1 < p ˜ ≤ p < p ˜ n / ( n − 1 ) or $\tilde{p}n/(\tilde{p}+n-1)< p\leq \tilde{p}<\infty $ p ˜ n / ( p ˜ + n − 1 ) < p ≤ p ˜ < ∞ . The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. It should be pointed out that our proof is based on the Calderón–Zygmund’s rotation method.

scholarly journals A note on maximal singular integrals with rough kernels

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiao Zhang ◽  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Maximal Operators ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Homogeneous Mapping ◽  
Recent Developments ◽  
Maximal Singular Integrals ◽  
Maximal Singular Integral

Abstract In this note we study the maximal singular integral operators associated with a homogeneous mapping with rough kernels as well as the corresponding maximal operators. The boundedness and continuity on the Lebesgue spaces, Triebel–Lizorkin spaces, and Besov spaces are established for the above operators with rough kernels in $H^{1}({\mathrm{S}}^{n-1})$ H 1 ( S n − 1 ) , which complement some recent developments related to rough maximal singular integrals.

A criterion on oscillation and variation for the commutators of singular integral operators

2015 ◽  
Vol 27 (1) ◽  
Author(s):  
Feng Liu ◽  
Huoxiong Wu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Weighted Norm ◽  
Norm Estimates ◽  
Variation Operators

AbstractThis paper gives a criterion on the weighted norm estimates of the oscillatory and variation operators for the commutators of Calderón–Zygmund singular integrals in dimension 1. As applications, the weighted

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