Singular Integrals
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200 Umberto Neri University of Maryland, College Park, MD/USA
Singular Integrals
Spri nger-Verlag Berlin· Heidelberg· New York 1971
AMS Subject Classifications (1970): 46E 35, 47G 05
ISBN 3-540-{)5502-9 Springer-Verlag Berlin . Heidelberg . New York ISBN 0-387-05502-9 Springer-Verlag New York . Heidelberg· Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other thao private use, a fee is payable to the publisher, the amollnt of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin: Heidelberg 1971. Library of Congress Catalog Card Number 76-166077. Printed in Germany. Offsetdrud
f ~dt. x-t
I x::tl >s
This singular integral (or principal value integral) arose in the study of boundary values of analytic functions. We have two kinds of problems here. I.
Existence of f(xl.
II. Relationship between f and kept by
f?
f:
that is, what properties of fare
2
PRELIMINARIES
Let x
=
(Xl' •.. ,X ) ~ En, where En is Euclidean n-dimensional n
space, with Lebesgue measure dx
Ox , .. dx • All sets and functions n 1
=
considered will be Lebesgue measurable. If f(x) function defined a.e. sider the integral
f S
If we set f(x)
o
is a measurable
(almost everywhere) on a set SeEn, we con-
fix) dx
=
f S
f(x , ... 1
,X
n
)
dx " .. dx • 1
n
J
outside S, we may write the integral as
J
f(x)dx
=
En
f(x)dx
where the domain of integration is understood to be the entire space En.
n
(~x~
Letting Ixl
1
~
)1/2 denote the norm of x, consider the
integrals
The integrands may have trouble at 0 and at infinity, so we split the integral into two parts:
f I
j
I xl ~1
xl ~1
dx ~.
Letting r = I xl , the element of volume dx can be written in "polar coordinates" as dx the unit sphere ~
= =
n 1 r - do-dr, where do-
{x:lxl
=
is the surface element on
1}. Thus,
r
n-l r
a
1
dr C
~
r
n-a-l
dr
0, that is a < n.
(Here the constant C is the
area of the unit sphere). On the other hand,
r
a
n, then 11
With 1 LP
=
LP (En)
~
=
P < {f:
00,
12
=
r
dr
n-a-1
dr
n.
and only if n - a < 0, that is If
n-a-1
= 00.
we consider the Banach spaces
SI f (xl I Pdx < co},
wher the functions f are measurable
and complex-valued (in general), and two functions are identified if they coincide a.e.
The norm
Triangle Ineguality:
Ilf + gllp
~
Holder's Ineguality:
Ijfg dxl
~ Ilfllpllgll "
IIfllp + IIgli
p
(Minkowski). Also, we have where p > 1
p
and
p', the
conjugate exponent of p, is defined by the relation 1
P
Ifp
+
!
p'
1.
1, then Holder's inequality holds with p' Ilg
1100
=
= 00,
where we set
ess sup I g (x) I = the least number M ) 0 such that Ig(xll
> M only on a set of measure
zero
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