Fractals and Spectra

1. Front Matter

   Pages i-viii

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2. Fractals

   • Hans Triebel

   Pages 1-32

3. ℓ _p -spaces

   • Hans Triebel

   Pages 33-48

4. Function spaces on ℝn

   • Hans Triebel

   Pages 49-123

5. Function spaces on and of fractals

   • Hans Triebel

   Pages 125-197

6. Spectra of fractal pseudodifferential operators

   • Hans Triebel

   Pages 199-259

7. Back Matter

   Pages 260-271

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n This book deals with several aspects of fractal geometry in ? which are closely connected with Fourier analysis, function spaces, and appropriate (pseudo)differ- tial operators. It emerged quite recently that some modern techniques in the theory of function spaces are intimately related to methods in fractal geometry. Special attention is paid to spectral properties of fractal (pseudo)differential operators; in particular we shall play the drum with a fractal layer. In some sense this book may be considered as the fractal twin of [ET96], where we developed adequate methods to handle spectral problems of degenerate n pseudodifferential operators in ? and in bounded domains. Besides a few special properties of function spaces we relied there on sharp estimates of entropy numbers of compact embeddings between these spaces and their relations to the distribution of eigenvalues. Some of the main assertions of the present book are based on just these techniques but now in a fractal setting. Since virtually nothing of these new methods is available in literature, a substantial part of what we have to say deals with recent developments in the theory of function spaces, also for their own sake. In this respect the book might also be considered as a continuation of [Tri83] and [Tri92].

• Eigenvalue
• Fourier analysis
• distribution
• entropy numbers
• fractal geometry
• function spaces
• functional analysis
• pseudodifferential operators

• Fak. Mathematik/Informatik, Inst. Mathematik, Universität Jena, Jena, Germany

  Hans Triebel

Hans Triebel ist Professor der Mathematik an der Friedrich-Schiller-Universität Jena.

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