By V. Kokilashvili (auth.), Alexandre Almeida, Luís Castro, Frank-Olme Speck (eds.)

ISBN-10: 3034805152

ISBN-13: 9783034805155

ISBN-10: 3034805160

ISBN-13: 9783034805162

This quantity is devoted to Professor Stefan Samko at the get together of his 70th birthday. The contributions demonstrate the diversity of his clinical pursuits in harmonic research and operator conception. specific cognizance is paid to fractional integrals and derivatives, singular, hypersingular and power operators in variable exponent areas, pseudodifferential operators in quite a few glossy functionality and distribution areas, in addition to similar purposes, to say yet a number of. such a lot contributions have been to start with provided in meetings at Lisbon and Aveiro, Portugal, in June‒July 2011.

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Those notes grew out of lectures given by way of the writer on the Institut für Angewandte Mathematik, Heidelberg collage, and on the Centre for Mathematical research, Australian nationwide Unviersity

A important goal was once to provide the elemental rules of Geometric degree conception in a mode effortlessly available to analysts. i've got attempted to maintain the notes as short as attainable, topic to the constraint of masking the relatively very important and valuable principles. There have after all been omissions; in an increased model of those notes (which i am hoping to put in writing within the close to future), themes which might evidently have a excessive precedence for inclusion are the idea of flat chains, additional purposes of G. M. T. to geometric variational difficulties, P. D. E. facets of the idea, and boundary regularity theory.

I am indebted to many mathematicians for beneficial conversations referring to those notes. specifically C. Gerhardt for his invitation to lecture in this fabric at Heidelberg, ok. Ecker (who learn completely an prior draft of the 1st few chapters), R. Hardt for plenty of worthy conversations over a couple of years. such a lot in particular i would like to thank J. Hutchinson for various positive and enlightening conversations.

As a ways as content material of those notes is anxious, i've got drawn seriously from the traditional references Federer [FH1] and Allard [AW1], even though the reader will see that the presentation and viewpoint usually differs from those references.

An define of the notes is as follows. bankruptcy 1 includes easy degree concept (from the Caratheodory point of view of outer measure). many of the effects are by means of now really classical. For a extra large therapy of a few of the subjects coated, and for a few bibliographical feedback, the reader is pointed out bankruptcy 2 of Federer's ebook [FH1], which was once as a minimum the fundamental resource used for many of the cloth of bankruptcy 1.

Chapter 2 develops extra simple preliminaries from research. In getting ready the dialogue of the world and co-area formulae we came upon Hardt's Melbourne notes [HR1] fairly worthy. there's just a brief part on BV capabilities, however it with ease suffices for the entire later functions. We stumbled on Giusti's Canberra notes [G] worthy in getting ready this fabric (especially) on the subject of the later fabric on units of in the community finite perimeter).

Chapter three is the 1st really good bankruptcy, and offers a concise therapy of crucial features of countably n-rectifiable units. There are even more basic ends up in Federer's e-book [FH1], yet confidently the reader will locate the dialogue right here appropriate for many purposes, and a great start line for any extensions which would sometimes be needed.

In Chapters four, five we improve the fundamental conception of rectifiable varifolds and turn out Allard's regularity theorem. ([AW1]. ) Our therapy here's officially even more concrete than Allard's; actually the whole argument is given within the concrete environment of rectifiable varifolds, regarded as countably n-rectifiable units outfitted with in the neighborhood Hn-integrable multiplicity functionality. confidently this can make it more uncomplicated for the reader to work out the real rules fascinated by the regularity theorem (and within the initial idea related to monotonicity formulae and so on. ).

Chapter 6 contians the fundamental idea of currents, together with integer multiplicity rectifiable currents, yet no longer together with a dialogue of flat chains. the elemental references for this bankruptcy are the unique paper of Federer and Fleming [FF] and Federer's booklet [FH1], even though in a couple of respects our therapy is a bit assorted from those references.

In bankruptcy 7 there's a dialogue of the fundamental thought of minimizing currents. the concept 36. four, the evidence of that's kind of typical, doesn't appear to seem in different places within the literature. within the final part we improve the regularity idea for condimension 1 minimizing currents. A characteristic of this part is that we deal with the case whilst the currents in query are literally codimension 1 in a few tender submanifold. (This was once in fact more often than not recognized, yet doesn't explicitly look in different places within the literature. )

Finally in bankruptcy eight we describe Allard's idea of common varifolds, which initially seemed in [AW1]. (Important points of the speculation of varifolds had past been built via Almgren [A3]. )

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Russian) Diﬀentsial’nye Uravneniya 4:2 (1968), 298–314. G. Samko, Noether’s theory for the generalized Abel integral equation (in Russian). Diﬀerencial’nye Uravneniya 4 (1968), 315–326. G. Samko, Abel’s generalized equation, Fourier transform, and convolution type equations. (Russian) Dokl. Akad. Nauk SSSR 187 (1969), 743–746. G. Samko, The general singular operator and an integral operator with an automorphic kernel. (Russian) Izv. Vysch. Uchebn. , Matematika. 1969:1 (1969), 67–77. G. Samko, The integral modulus of continuity of potentials with densities that are summable on the real line with weight.

Diﬀer. Uravn. 32:2 (1996), 275–276. V. A. G. Samko, Fractional powers of the operator −∣????∣2 Δ in ???????? -spaces. Proc. A. Razmadze Math. Inst. 124 (2000), 1–22. [3] A. Almeida, J. Hasanov, and S. Samko, Maximal and potential operators in variable exponent Morrey spaces. Georgian Math. J. 15:2 (2008), 195–208. [4] A. Almeida and S. Samko, Characterization of Riesz and Bessel potentials on variable Lebesgue spaces. J. Function Spaces and Applic. 4:2 (2006), 113–144. [5] A. Almeida and S. Samko, Pointwise inequalities in variable Sobolev spaces and applications.

K. G. Samko, Equations with involutive operators and their applications. (Russian) Rostov-na-Donu, Izdat. , 1988. 192 pages. K. G. Samko, Multidimensional integral operators with homogeneous kernels. Fract. Calculus & Applied Analysis 2:1 (1999), 67–96. K. G. Samko, On a certain approach to the investigation of equations with involutive operators. Intern. J. Math. Stat. Sci. 8:2 (1999), 66–93. K. G. Samko, On Fredholm properties of a class of Hankel operators. Math. Nachr. 217 (2000), 75–103. K.

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