ANNALES SOCIETATIS MATHEMATIOAE POLONAE Series I: COMMENTATIONES MATHEMATIOAE X IX (1976) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE X IX (1976)
R. T
a b e r sk i(Poznan)
O n double singular integrals
In this note the suitable analogues of Theorems 4.2.1 and 4.2.2 of [1], p. 255, 262 are given.
1. Preliminaries. Let f{x, y) be a real function defined on the two- dimensional interval (rectangle) R = [a, b; c, d] and let R tj = [жг-_1?
.x{ ; yj_lt уД be the rectangles such th a t
a = x 0< æ1< ... < xm = b, с = y 0 < у г < ... < yn = d.
Denote by 1г, I 2 the intervals <a, b> and (c, d}, respectively, and set A ( / ; Щ) = Уз-1) Уз) Уз-1) + / K - , Vj) •
Then
m n
V (/; B) = s u P y , 2 ' | z l ( / ; В„)I i=1j = l
is called the variation of f over the rectangle It, and
Ш
V(f(-,y)-, II) = s n p ^ !/(«<_!, y) - f ( ®i , y)\,
i ~ l n
•); 1
2) = supJ^!/(a>, ^ _ i ) - / ( ® , %•)!
3=1
the partial variations of this function over the intervals I 1} I 2 (with para meters y, x).
If V( f ) R) < 00 , the function / is said to be of bounded variation in the sense of Vitali over the rectangle R = I x x l 2- In this case, for all (æ, y) eR,
f {æ, y) = - f ( a , c ) + f ( x , c ) + f ( a , y ) + P ( x , y ) - N ( x , y ) ,
where P( x, y ) and N( x , y ) are non-negative and non-decreasing in each variable, separately, and
Zl(P; R y ) > 0 , A( N; R {j) > 0 for all R tj ([3], p. 470-72).
If, moreover, F(/(-,.c)j I x) < oo and V(f (a, •); I 2j < 00 , we say th a t /
is of bounded variation in the sense of Hardy over R. The class of all these
functions will be^ denoted by H(R).
I t is easy to see th a t if f e H(B), then
v ( n - , y ) ; h ) <У( Д- , о) - , h ) + r(f-, в ) < « and
V(f(œ, •); 1 г ) < y ( f ( a, •); h ) + V ( f ; В) < oo for arbitrary {x, y) eB. Consequently, / is bounded in B.
Let q ( x , y ) be a weight-function in the rectangle В = [a, b\ c, d]T i.e. q is non-negative in В and has a positive Lebesgue integral over this two-dimensional interval. Denote by L e(B) the class of all measurable functions f(cc, y) for which the Lebesgue integral
/ / /(*> V) q {®, y)dxdy
is finite. R
Consider a sequence of measurable functions Фп{х, у) (n = 1, 2, ...) essentially bounded in the rectangle В = [a, &; c, d]. I t will be assumed th at for each rectangle
(
1
)the estimates
В г(0) = [a-f ô, b; c, d]
B 2{ô) = [a, b\ c + <5, d]
(0 < ô < b — a), (0 < ô < d — c) (2) ess sup \Фп($), y)\ < Fv(ô) (v = 1 , 2 ; w = 1 , 2 , . . . ) hold, with positive F v(ô) depending only on <5. Write
(3) J n(f) = / J / ( ® , У)Фп(я, y)e(æ, y)dxdy for f e L Q{B).
R
In the sequel, we shall examine the convergence of singular integrals.
(3) for / ’s continuous at (a, c), and for functions of bounded variation in a neighbourhood of this point. The supremum of |/ | over В will be- signified by $ ( / ; B).
2. Results. We start with the following
T
h e o r em1. For any f e L (B), continuous at (a, c) from the rectangle В
(4) lim J n( f ) = f ( a , c )
i f and only i f n~*°°
1° lim J J Фп(оо, у)
q(x ,y)dxdy = 1,
n~>0° iî
2° lim J J Фп(со, y) q (< ü , y)dxdy = 0
n-+°° Rv(ô)
for all B v(d) (v = 1, 2) defined above, and
3° J J №n(<c,y)\Q(æ,y)da>dy^K
R
where К is a positive constant.
(n = 1 , 2 , ...),
Double singular integrals
1 5 7P r o o f . The necessity of 1° and 2° is evident. In order to show the necessity of 3° let us denote by C(R) the Banach space of all real-valued functions f continuous in R, with norm
II/1| = max \f{x, y)\.
R
Clearly, integrals (3) may be treated as some linear functionals on
€(R).
If condition 3° is not fulfilled, we can find a sequence of functions fneC(R) such that
Ш 1 < 1 and \ i m\ Jn(fn)\ = oo.
n-> oo
Then, there exists an enumerable set of functions f eC(R) for which lim \Jn(f)\ = oo
n—>00
(see [1], p. 263 and 247). This contradicts (4).
The proof of sufficiency of l°-3° runs as follows. Consider an arbitrary fixed f e L e( R), continuous at (a, c) from R. Choose, for a given s > 0,
a positive <3 = <5(e) such th at
<5) \f(x, y ) —f{a, c)\ < s when {x, y)eQ, Q = [a, a + <5; с, c+<5].
Taking a function geC(R) satisfying the condition
<
<6) f \ If(%, y ) —g(x, y)\g{x, y) dxdy ^ ,
J J max{¥/1(<3), УЛ/с))}
let us write
(7) J n(f) = f f {/(®, y ) ~ f { a , с)}Фп(ж, y) g{x, y) dxdy +
R
+/(«» 0) f f фп (x, у) g{x, y) dxdy — A n + Bn.
R
By 1°, B n c) as n -» oo. Applying (5), (6) and 3°, we have U J < £ f f \®n(æiy)\Q(æ, y ) dædy +
Q
+ ! f f {/(я, y ) ~ f ( a , с)}Фп(х, у) д{ х, y)dxdy\
R \ Q
< e(I£ + l) + | / f g{x, У)Фп(®, y ) e №, y) dxdy I + + 1 f { a, o) f f < P n( x , y ) g { x , y ) d x d y \ .
7?\ П I
R \ Q
R \ Q
In view of 2°, the last integral tends to zero as n->oo. Further,.
b d a + <5 d
\ § g{x,y)<&n{œ, y)Q{x, y)dxdy = J f + f J = U n+ V n.
R \ Q а+д с c c+S
Given an arbitrary positive A, split the rectangle [a + ô, b ; c, d], by means of the stright lines parallel to axes, into two-dimensional inter
vals rk (k = 1, 2, .. . , s) such th a t
Os c g ( æ, y ) <A (к = 1 , 2 , ..., s) .
r k
P ut mk = m ing(æ, y), M = тах|</(#, у )|. Since
rk R
S
u n = J T f f {g{x, у ) - т к}Фп{а?, у)д(а>, y)âxdy +
к - l r k
s
+ я * . (a),y)g(æ, y)dædyr к-l r k
we have
b d s
\Un\ < A f f \Фп{я>, y)\Q{æ, y ) d x d y + M ^ | y)g{æ, y)dxdy\ .
a + à с k= 1 тк
Hence, by (2) and 2°,
\ü n\ < f f
q(s û,y)dcody+M}
R
for sufficiently large n, i.e.
lim Un = 0.
n->oo
Analogously, lim Vn = 0. Consequently,
n—>OQ
\An\ < e{K~\-2) if n is large enough, and th e desired relation (4) is proved.
Denoting an arbitrary two-dimensional interval of B = [a, b ; c, d]
by r, we shall now give a related
T
h eo r em2. P or any f e h Q(B) of bounded variation in the sense of Hardy over a certain neighbourhood of (a, e), being continuous at this point from B, relation (4) is true if and only i f conditions l°-2° of Theorem 1 together with
3° sup | J J Фп{ос, у) q ( oc , y)dwdy\ < К (n — 1 , 2 , . . . ) instead of 3°, hold.
P r o o f . We first prove the necessity of 3°, only. For this purpose
let us consider the Banach space E of all functions / of bounded variation
Double singular integrals
159in the sense of H ardy over R = I 1x Z 2, continuous at (a, c) from R, with norm
ll/lI = « ( / ;
R )+ V( f ;
R )+
V { f ( ; c ) - ,I J + Ff/fa, •);*«)•
Assuming th a t f e E, we have
\Jn( f ) \ < K n \\f\\ in =1,2,...),
where
K n = f f \0n( x, y)\ Q(x, y) dxdy.
в
Hence, functionals (3) are linear in E.
If condition 3° is not fulfilled, then there exists a sequence {ги} of two-dimensional intervals rn a R for which
l i m
J J Фп( х, у)
q(x,y)dxdy =
oo.00
r rn
The functions
1/5 0 belong to E and ||/J| < 1 (n = 1, 2,
when { x , y ) e r n, otherwise, ...). Moreover,
= l \ i m \ j j 0 n( x, y)Q(x, y) dxdy I =
oo.П -+ 0 0 71-+O Q „
Tn
Thus, by [1], p. 247, there is an f e E such th at lim \Jn(f)\ =
oo.n->OQ
This contradicts (4).
To prove the sufficiency of our conditions, let us start with identity- (7). Choose, for a given e > 0, a positive ô < min(& — a, d — c) such th a t (5>
is satisfied and
V( f ; Q) + V{ f ( - , c ) ; l ' ) + V ( j { a , •); i " ) < e, where I ' = (a, a + <5>, I " = <c, c + ô), Q — I' x I ”.
As previously, B n -+f(a, c) as n -»
oo,and
An = (JJ + У) ~ / ( а, с)} Фп(х, y)
q{x, y) dx dy = P n + Wn.
Q R \ Q
In view of 3°,
|P J < 2 1 f e for n = 1 , 2 , . . .
Considering a function geC{R) for which inequality (6) holds, we have
I / / {/(«, У ) - д ( х , У)}Фп(®, y)Q(x, y)dxdy |< e,
R\Q
uniformly in n. Hence
| W J < e + f f { g { æ , y ) - f { a , c ) } 0 n{æ,y)Q{æ,y)dxdy.
R \Q
By 2° the last integral tends to zero as n->oo. Thus,
\An\ < e(2K + 2) if n is large enough, and the result follows.
3. Appendix. Now let
q(x,y)
=1 when (x, y)e R = [a, 5; c, d].
Suppose th a t the functions Фп(х, y) (n = 1, 2, ...) are Lebesgue-integrable in R and that, for each rectangles (1), the sums
Я(Ф„; B,(ô))+V{<Pni R ,(ô)) (, = 1 ,2 ) together with
<« + ô, &>) + У[Фп{ •, c + <5); &)), У( фп(а + д ,
•);<c, dy) + V ^ n(a,
•); < c + < 5 , d > ):are uniformly bounded in n. Then, condition 2° implies lim (T) f f f { x, у) Фп{х, y)dxdy = 0
П->оо д,,((3)
for any f e T ( R ) (see [2], p. 99, 105).
I t can easily be observed th at, under the above restrictions, Theorems 1, 2 remain valid for any/eT(.R ) continuous at (a, c) from R. Sequence (3) is defined
п ол уby the double T-integrals.
References