Function Spaces and Potential Theory
Function spaces, especially those spaces that have become known as Sobolev spaces, and their natural extensions, are now a central concept in analysis. In particular, they play a decisive role in the modem theory of partial differential equations (PDE). P
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Editors
M. Artin S. S. Chem J. Coates J. M. Frohlich H. Hironaka F. Hirzebruch 1. Hormander C. C. Moore J. K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai J. Tits M.Waldschmidt S.Watanabe Managing Editors
M. Berger B. Eckmann S.R.S. Varadhan
Springer-Verlag Berlin Heidelberg GmbH
David R. Adams Lars Inge Hedberg
Function Spaces and Potential Theory
Springer
David R. Adams Department of Mathematics University of Kentucky Lexington, KY 40506-0027, USA e-mail: [email protected]
Lars Inge Hedberg Department of Mathematics Linkoping University 58183 Linkoping, Sweden e-mail: [email protected]
Corrected Second Print ing 1999
The Library of Congress has cataloged the original printing as follows: Adams, David R., 1941- . Function spaces and potential theory 1 David R. Adams, Lars Inge Hedberg. p. cm. (Grundlehren der mathematischen Wissenschaften; 314) Including bibliographical references and index. ISBN 978-3-642-08172-9 ISBN 978-3-662-03282-4 (eBook) DOI 10.1007/978-3-662-03282-4 1. Function spaces. 2. Potential theory (Mathematics). 1. Hedberg, Lars Inge, 1935-. II. Title. III. Series. QA323.A33 1996 515'.73-dc20 95-2.3396 CIP
Mathematics Subject Classification (1991): 46-02, 46E35, 31-02, 31C45, 31B15, 31C15, 30EI0 ISSN 0072-7830 ISBN 978-3-6 O ; Ko(r) '" log(l/r),
(1.2.20)
as r -+ O ;
(1.2.21)
Kv(r) '" (n/2r)I/2 e-r, as r -+ 00 for all v.
(1.2.22)
This again implies (1.2.14). In addition we see that GN(X) behaves at O like the logarithmic kemel, 1 GN(x) '" CN log - , as
Ixl
with cNI = 2N - 1n N / 2 r(N/2) (in particular, C2 (1.2.15) can be improved to Ga(x) '" ca(n/2)1/2Ixl(a-N-ll/2e-lxl
Ixi
=
(1.2.23)
-+ O ,
1/(2n) for N
= 2),
and that
as Ixl -+ 00 for alI a> O .
(1.2.24) In particular, (1.2.15) is true with C = 1 for a ::: N + 1. See N. Aronszajn and K. T. Smith [40] and N. Aronszajn, F. Mulla and P. Szeptycki [38]. (It could be noted that an easy estimate in (1.2.11) shows that GN(x) S A log(2/lxl) for Ixl S 1, which is enough for most purposes.) The formula (1.2.22) follows easily from the Laplace transform type representation, Kv(r)
= bvr-~e-r
1 e-ttV-~(1 + 00
1),
(1.2.25)
;rr-! dt,
-1.
with bv = (n/2)! / r(v + which is valid for r > O and v > See Watson [426], 7.3., p. 206. It also might be noted that (1.2.25) shows that Kv is an elementary function, is integer. In particular, K±! = (n/2)4r-~e-r, and thus when v -
!
GN-1(X)
= cN_I(n/2)1/2Ixl- 1e- 1xl
,
G N-3(X) = (/ - Ll)GN_1(x) = cN_3(n/2)1/2(1xl-3 + Ixr 2 )e- 1x1 ,
etc.
1.2 Sobolev Spaces and Bessel Potentials
13
Occasionally we shall aiso need estimates for the derivatives of GOI' Writing Ga(x) = Ga(r), r = Ixl, and differentiating (1.2.11), we find by (1.2.18)
(1.2.26) This implies for cx > 1, G~(r) ~ -(N -
cx)G a - 1(r),
as r ~ O ,
(1.2.27)
and G~(r) ~
_aar(a-N-l)/2e-r
~
-c Ga(r) ,
as r ~
00
(1.2.28)
with c = 2(N+a-3)/2 n (N-a-l)/2.
Remark. We would like to emphasize that the finer properties of the Bessel kernei GOI are not really important to the theory we are going to develop. What is imp
