By Elias M. Stein, Guido Weiss
This booklet offers with the extension of genuine and complicated equipment in harmonic research to the many-dimensional case. So, its pre-requisites are a powerful historical past in actual and intricate research and a few acquaintance with common harmonic research, that's, this e-book is meant for graduate scholars and dealing mathematicians. possibly a few complicated undergraduates may disguise sure elements of the material.
This e-book is one part of the Stein trilogy on harmonic research (together with "Singular Integrals and Differentiability homes of capabilities" and "Harmonic Analysis", either additionally reviewed by way of myself), and as such it has to be considered as an authoritative reference at the topic because Elias Stein and Guido Weiss are of the major specialists within the box, and the fabric they chose used to be taken from their instructing and learn experience.
The contents of the ebook are: The Fourier rework; Boundary Values of Harmonic services; the idea of H^p areas on Tubes; Symmetry homes of the Fourier remodel; Interpolation of Operators; Singular Integrals and structures of Conjugate Harmonic services; a number of Fourier Series.
Includes motivation and whole factors for every subject, excercises for every bankruptcy, referred to as "further results", and large references. extraordinary printing caliber and great clothbound.
These 3 volumes may be found in each analyst's library.
Please have a look to the remainder of my stories (just click my identify above).
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Additional info for Introduction to Fourier Analysis on Euclidean Spaces. (PMS-32)
1), provided that f(t, u) is bounded. In the second paragraph, we study systems with asymmetric nonlinearities. These are non-selfadjoint problems. e. 2). The benefit of this assumption is that it gives some localization of the solution in the sense that there exists t 0 E [0, 1r] such that f3(t 0 ) < u(t0 ) < a(t 0 ). Such a property can be used to obtain multiplicity results. The last paragraph extends some of the ideas on monotone iterations. 1). 1 Assume the function f : [0, 21r] x lR --+ lR is continuous and bounded.
Assume further (i) for every R > 0, there exists hR E A such that, for a. e. t E ]0, 1r[ and all u E [a(t), R], if(t, u)i ~ hR(t); (ii) there exist p > 0 and b, c E A, b > 0 on [0, 1r] such that • for a. e. 1 of u +:Abu= 0, u(O) = 0, u(1r) = 0, is such that >. 1 < 1. +) nC 2 (]0, 1r[, IRci). A similar result can be obtained from a resonance condition. 3 Let f: ]0, 1r[ xiRt --+ lR be a continuous function. Assume there exist a, E C([O, 1r]), respectively a W 2 •1 -lower solution and a strict W 2 •1 -upper solution, such that /3(0) > 0, j3(1r) > 0, and for all t E ]0, 1r[ f3 0 < a(t) :S j3(t).
2 Let f: ]0, 1r[ xJRci -+JR. be a continuous function. Assume there exist a, (3 E C([O, 1r]), respectively a W 2•1 -lower solution and a strict W 2•1 -upper solution, such that (3(0) > 0, (3(1r) > 0, and for all t E ]0, 1r[ 0 < a(t) :::; (J(t). Assume further (i) for every R > 0, there exists hR E A such that, for a. e. t E ]0, 1r[ and all u E [a(t), R], if(t, u)i ~ hR(t); (ii) there exist p > 0 and b, c E A, b > 0 on [0, 1r] such that • for a. e. 1 of u +:Abu= 0, u(O) = 0, u(1r) = 0, is such that >.