Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the constants for some Sobolev imbeddings   We consider the imbedding inequality || f ||_{L^r(R^d)} <= S_{r,n,d} || f
||_{H^{n}(R^d)}; H^{n}(R^d) is the Sobolev space (or Bessel potential space) of
L^2 type and (integer or fractional) order n. We write down upper bounds for
the constants S_{r, n, d}, using an argument previously applied in the
literature in particular cases. We prove that the upper bounds computed in this
way are in fact the sharp constants if (r=2 or) n > d/2, r=infinity, and
exhibit the maximising functions. Furthermore, using convenient trial
functions, we derive lower bounds on S_{r,n,d} for n > d/2, 2 < r < infinity;
in many cases these are close to the previous upper bounds, as illustrated by a
number of examples, thus characterizing the sharp constants with little
uncertainty.

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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: physics