Calderon-Zygmund capacities and operators on nonhomogeneous by Alexander Volberg

By Alexander Volberg

Singular quintessential operators play a crucial position in glossy harmonic research. least difficult examples of singular kernels are given via Calderon-Zygmund kernels. Many vital houses of singular integrals were completely studied for Calderon-Zygmund operators. within the 1980's and early 1990's, Coifman, Weiss, and Christ spotted that the idea of Calderon-Zygmund operators will be generalized from Euclidean areas to areas of homogeneous style. the aim of this e-book is to make the reader think that homogeneity (previously regarded as a cornerstone of the idea) isn't wanted. This declare is illustrated via providing harmonic research difficulties recognized for his or her hassle. the 1st challenge treats semiadditivity of analytic and Lipschitz harmonic capacities. the quantity offers the 1st self-contained and unified facts of the semiadditivity of those capacities. The ebook information Tolsa's answer of Painleve's and Vitushkin's difficulties and explains why those are difficulties of the idea of Calderon-Zygmund operators on nonhomogeneous areas. The exposition isn't really dimension-specific, which permits the writer to regard Lipschitz harmonic capability and analytic skill while. the second one challenge thought of within the quantity is a two-weight estimate for the Hilbert remodel. This challenge lately came across very important purposes in operator concept, the place it truly is in detail relating to spectral concept of small perturbations of unitary operators. The e-book offers a strategy that may be useful in overcoming quite undesirable degeneracies (i.e., exponential progress or decay) of underlying degree (volume) at the house the place the singular fundamental operator is taken into account. those occasions take place, for instance, in boundary worth difficulties for elliptic PDE's in domain names with super singular barriers. one other instance contains harmonic research at the barriers of pseudoconvex domain names that is going past the scope of Carnot-Caratheodory areas. The e-book is acceptable for graduate scholars and examine mathematicians drawn to harmonic research.

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Diesbesagtmitanderen Worten, daB die homogene Integralgleichung 2 Z (x) (83) + Iu L (x, Xl) Z (Xl) dX I = ° P linearunabhangige Losungen Zk(X) (k=l, ... ,P) hat. Wir setzen qJk (x) = Zk (x), < x < 1; Ok (x) = Zk (x + 1), < x < 1 . 32 ° Die zu (83) adjungierte Gleichung 2 T (Xl) + uI L (x, Xl) T (x) dx = ° ° hat offenbarebenfallsplinearunabhangigeLosungen T k( Xl) (k = 1, .. ,P). Setzt man "Pk(x)=Tk(x), O

RJl "'PI! y, X "'1 ••• "'I! ~l ••• ~I! YI! d Xl'" d X(!. Aus (97) und (98) ergibt sich nun, wenn wir voriibergehend ~ setzen, durch Subtraktion ; - DC = Die Glieder unter dem Integralzeichen sind von mindestens erstem Grade in bezug auf C, ; und v. It darum gewiS < 1 aus, wenn Max /v(x)/ und Max IDvl hinreichend klein sind. Es muS dann natiirlich sein, w. z. b. w. In den Anwendungen kommt der Spezialfall K OlOl = 0 identisch besonders haufig vor. Es handeIt sich also (vgl. 2 Umnp{C,DC,v}. m+n+p>l In den simuItanen Gleichungen (96), (97) fehlen dann die dritten Summanden linker Hand.

Die rechte Seite dieser Ungleichheiten konvergiert gegen Null. Man gelangt zu einem Widerspruch, wenn man M von Null verschieden annimmt. Nichtlineare Integralgleichungen im kleinen. 42 Hat, wie wir jetzt annehmen wollen, C (x) stetige Ableitung erster Ordnung, so HiBt sich der dritte Summand rechts durch teilweise Integration auf die Form x x' x - f dx' f C(x") DC(x") dx" o 0 + f C(x') C(x') dx' 0 x C(O) C(O) bringen. MaBgebend fUr die weitere Behandlung des Problems ist nunmehr die homogene Integralgleichung x x x' C(x) - f C(x') C(x') dx' + f dx' f C(x") DC(x") dx" o X 'X' - fo dx' f Cl(x") C(X") dx" 0 0 0 x' X 1 f dx' f dx" f C2 (x", Xl) C(xl ) dXl = O.

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