Page 20 - Textos de Matemática Vol. 34

10 DAVID E. EDMUNDS
(iv) While exact formulae have been given for the approximation numbers of the
0
embeddings W1(I) → L (I) and W1 (I) → L (I), I have been silent about the
p p p,C p,C
basic embedding E : Wp1(I) → Lp(I). There is a very good reason for this: in the two cases dealt with the Sobolev space is normed by ∥f′ |Lp(I)∥, but this is not possible for Wp1(I) and precise estimates are accordingly much harder to obtain. Can an exact formula be obtained for the approximation numbers of E?
(v) The operator Ha which appears in Section 2 is but a very special case of weighted Hardy integral operators of the form
x
a
where u and v are given real-valued functions satisfying some integrability con- ditions. In [7] a detailed analysis is provided of the properties of such opera- tors. Here I draw attention to the result given in [8] that if u,v ∈ L2(I) and u′,v′ ∈ L2/3(I)∩C(I), then the approximation numbers am(T) of T as a map from L2(I) to itself satisfy
1b
am(T)= mπ
Can this be sharpened, under suitable conditions on u and v, to give a genuine second term in the expansion plus a remainder term decaying more quickly? It is tempting to conjecture that given sufficiently pleasant u and v there would be an asymptotic expansion for am(T) containing as many terms as desired, a situation somewhat reminiscent of that known to hold for the eigenvalues of the Dirichlet Laplacian on suitable domains in Rn.
References
[1] M. S. Birman and M. Z. Solomyak, Piecewise polynomial approximations of functions of the class Wpα, Mat. Sb. (N.S.) 73 (115) (1967), 331-355 (Russian). Engl. transl. in Math. USSR Sb. (1967), 295-317.
[2] C. Bennewitz and Y. Saito¯, An embedding norm and the Lindqvist trignometric functions, Electron. J. of Differential Equations 86 (2002), 1-6.
[3] A. M. Caetano, About the approximation numbers in function spaces, J. Approx. Theory 94 (1998), 383-395.
[4] B. Carl and I. Stephani, Entropy, compactness and the approximation of operators, Cam- bridge University Press, Cambridge, 1990.
[5] P. Drabek and R. Mana´sevich, On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian, Differential Integral Equations 12 (1999), no. 6, 773-788.
[6] D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford University Press, Oxford, 1987.
[7] , Hardy operators, function spaces and embeddings, Springer-Verlag, Berlin, 2004.
[8] D. E. Edmunds, R. Kerman and R. Lang, Remainder estimates for the approximation num-
bers of weighted Hardy operators acting on L2, J. Anal. Math. 85 (2001), 225-243.
(Tf)(x) = v(x)
u(t)f(t)dt, x ∈ I,
|u(t)v(t)|dt+O(m−3/2).
a