Chapter 1. One-dimensional singular integrals

1983 ◽  
Vol 6 (3) ◽  
pp. 567-587 ◽  
Author(s):  
P. S. Theocaris

A numerical technique, first reported in 1979 in refs.[1] and [2], for the numerical evaluation of two-dimensional Cauchy-type principal-value integrals, is extended in this paper to include several cubature formlas of the Radau and Lobatto types. For the construction of such a cubature formula the 2-D singular integral is considered as an iterated one, and the second-order pole involved in this integral analyzed into a pair of complex poles. Based on this procedure, the methods of numerical integration, valid for one-dimensional singular integrals, are extanded to the case of two-dimensional singular integrals. The cubature formulas of the Lobatto- and Radau-type are now formulated to include the cases where some of the desired abscissas may be chosen accordins to any appropriate criterion.Moreover, the theory developed is enlarged to include the case of a 2-D principal-value integral, containing a logarithmic singularity. The validity of the results is illustrated by considering certain numerical examples. Furthermore, a complete analysis of the convergence and the construction of error estimates is also presented.


Author(s):  
Elena Prestini

AbstractIt is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.


Author(s):  
Brian Street

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.


2006 ◽  
Vol 58 (1) ◽  
pp. 154-179 ◽  
Author(s):  
Elena Prestini

AbstractWe prove Lp(T2) boundedness, 1 < p ≤ 2, of variable coefficients singular integrals that generalize the double Hilbert transform and present two phases that may be of very rough nature. These operators are involved in problems of a.e. convergence of double Fourier series, likely in the role played by the Hilbert transform in the proofs of a.e. convergence of one dimensional Fourier series. The proof due to C.Fefferman provides a basis for our method.


Author(s):  
Jessica Vantine Birkenholtz

Chapter 1 introduces Nepal’s popular Svasthānī tradition: the goddess Svasthānī, the Svasthānīvratakathā text that Nepali Hindus recite annually, and the Svasthānī vrat (ritual vow) that is described in the text and performed annually to honor the goddess. Both Nepal’s Newar Hindus and high-caste (Brahman and Chetri) hill Hindus, Parbatiyās, participate in these devotional practices, and have influenced in different ways the many stories that the Svasthānīvratakathā contains within its pages. The chapter also enumerates the theoretical concerns that fuel the book, such as the tensions between local (Newar) and translocal (Brahmanical Hindu) influences, and the methodology that underpins it. Finally, the chapter maps out in very broad strokes a general political history of Nepal that subsequent chapters in this book reinvigorate with a focused discussion of concurrent religious, sociocultural, literary, and linguistic developments that round out Nepal’s often one-dimensional master political narrative.


Author(s):  
Francesco Di Plinio ◽  
Kangwei Li ◽  
Henri Martikainen ◽  
Emil Vuorinen

Abstract We prove that the class of trilinear multiplier forms with singularity over a one-dimensional subspace, including the bilinear Hilbert transform, admits bounded $L^p$-extension to triples of intermediate $\operatorname{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\operatorname{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\textrm{UMD}$-valued setting. This is then employed to obtain appropriate single-tree estimates by appealing to the $\textrm{UMD}$-valued bound for bilinear Calderón–Zygmund operators recently obtained by the same authors.


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