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Quantitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory (American Mathematical Society Translations Series 2): M. Sh. Birman

The lectures in this book are devoted to quantitative analysis in connection with Sobolev imbedding theorems (see [1]) for the function spaces W^alpha_p which Sobolev introduced. The fundamental significance of the Sobolev imbedding theorems in analysis and mathematical physics is well known. Usually by imbedding theorems we mean "qualitative" assertions concerning continuity or compactness of an imbedding of one class W^alpha_p into another. Our principal goal is a "quantitative" investigation of the corresponding imbedding operators, which usually turn out to be compact. The conventional quantitative characteristics of compact imbeddings are epsilon-entropy and n-widths of the corresponding unit sphere. Meaningful information on these quantities can be obtained by means of approximation theory if a suitable approximation device is found and investigated. For the subject matter which is of interest to us such an approximation device was presented by the authors in [2], namely, a special method of piecewise-polynomial approximation with an "irregular" grid. These piecewise- polynomial approximations will be discussed in detail in the following lectures. A nonlinear version of this approximation method enabled us, in particular, to overcome the difficulties in estimating the entropy of the imbedding of W^alpha_p into L_q for q > p. Another area where we will make an application is the spectral theory of integral and differential operators. We will deal with estimates for and the asymptotic behavior of singular numbers and eigenvalues in problems with a discrete spectrum. From theorems on piecewise-polynomial approximation we deduce very precise estimates on the spectrum. In some cases these estimates enable us to clarify completely the limits of applicability for classical formulas describing the asymptotic behavior of the spectra of boundary value problems. In particular, we can justify asymptotic formulas for problems with "bad" (discontinuous) leading coefficients. We can also include problems in infinite domains, problems where ellipticity degenerates "weakly" or "strongly" and so forth. For integral operators we obtain estimates on singular numbers which turn out to be uniform with respect to a broad class of weighted spaces. Due to the duality between the classes of nuclear and bounded operators, such uniform estimates lead to useful results concerning pseudodifferential operators and some other problems of the multiplier problem type. We feel that our method of piecewise-polynomial approximation may be useful for approximate computations.

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