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Vladikavkaz Mathematical Journal 2020, Volume 22, Issue 4, P. 104-118
YAK 517.968
DOI 10.46698/c3825-5071-7579-i
GRAND MORREY TYPE SPACES#
S. G. Samko1,2 and S. M. Umarkhadzhiev2,3
1 University of Algarve, Faro 8005-139, Portugal; 2 Kh. Ibragimov Complex Institute of the Russian Academy of Sciences, 21 a Staropromyslovskoe Hwy, Grozny 364051, Russia; 3 Academy of Sciences of Chechen Republic, 13 Esambaev Av., Grosny 364024, Russia E-mail: ssamko@ualg. pt, umsalaudin@gmail. com
Dedicated to the 75-th anniversary of Professor S. S. Kutateladze
Abstract. The so called grand spaces nowadays are one of the main objects in the theory of function spaces. Grand Lebesgue spaces were introduced by T. Iwaniec and C. Sbordone in the case of sets Q with finite measure |Q| < to, and by the authors in the case |Q| = to. The latter is based on introduction of the notion of grandizer. The idea of "grandization" was also applied in the context of Morrey spaces. In this paper we develop the idea of grandization to more general Morrey spaces Lp,q,w(Rn), known as Morrey type spaces. We introduce grand Morrey type spaces, which include mixed and partial grand versions of such spaces. The mixed grand space is defined by the norm
sup S) sup
£,s xeE
(
V
w(r)q sb(r)i
I
\|x-y|<r
\
\
1
q-E
\№\P £a(y)pdy
)
dr r
/
with the use of two grandizers a and b. In the case of grand spaces, partial with respect to the exponent q, we study the boundedness of some integral operators. The class of these operators contains, in particular, multidimensional versions of Hardy type and Hilbert operators.
Key words: Morrey type space, grand space, grand Morrey type space, grandizer, partial grandization, mixed grandization, homogeneous kernel, Hardy type operator, Hilbert operator. Mathematical Subject Classification (2010): 46E30, 42B35.
For citation: Samko, S. G. and Umarkhadzhiev, S. M. Grand Morrey Type Spaces, Vladikavkaz Math. J., 2020, vol. 22, no. 4, pp. 104-118. DOI: 10.46698/c3825-5071-7579-i.
1. Introduction
Last decades there were widely investigated the so called grand Lebesgue spaces, introduced in [1]; see for instance, [2-5] and references therein, where such spaces and operators on them were studied in the case of finite measure underlying set. An approach to grand Lebesgue spaces on sets of infinite measure was suggested and developed in [6-10].
#The research of S. Samko was supported by Russian Foundation for Basic Research under the grant 19-01-00223 and TUBITAK and Russian Foundation for Basic research under the grant20-51-46003. The research of S. Umarkhadzhiev was supported by TUBITAK and Russian Foundation for Basic Research under the grant 20-51-46003.
© 2020 Samko, S. G. and Umarkhadzhiev, S. M.
That idea of "grandization" was also applied in the context of Morrey spaces defined by the norm
\/ ILp^(n) = sup
v ' xen
1
<p(r)
1/(y)\p dy
B(x,r)
(1)
LTC(0, diamQ)
we refer for instance to [11-14], see also references therein.
Our goal is to extend the notion of grandization to the spaces Lp>q>w(Rn), with the norm of the type (1), where L^-norm is replaced by Lq-norm, 1 ^ q < oo, see precise definitions in Section 2. Such spaces are usually referred to as Morrey type spaces. These spaces were introduced and studied in [15] and [16]. In the case w(r) = r-X, A > 0, these space first appeared, though as an episode, in [17, p. 44]. For further studies of operators on Lp'q'w(Rn)-spaces we refer, for instance, to [18, 19] and references therein, see also the surveying papers [20, 21] and [22].
We study various approaches to the grandization of Morrey type spaces, with respect to the exponents p and q. This includes partial grandization and mixed grandization. To this end, we deal with "grandizers" a(y) and b(r) in the corresponding variables, see Definitions 3.1, 3.2, 3.3.
We find conditions on the grandizers a and b, which ensure embedding of Morrey type spaces into the introduced grand Morrey type spaces.
In the case of partial grandization with respect the exponent q, we study, in grand Morrey type spaces, the boundedness of a certain class of integral operators K with a kernel homogeneous of degree — n. This class includes, in particular, multidimensional versions of Hardy type and Hilbert operators. Within the frameworks of generalized Morrey spaces, corresponding to the case q = oo, a more general class of operators with homogeneous kernel was studied in [23].
We first study such operators in Morrey type spaces (not grand ones) and obtain sufficient conditions and also some necessary conditions for their boundedness. In fact, we obtain a result stronger than just boundedness: we estimate the norms ||K/1|Lp,q,w(Rn) via similar one-dimensional norms of spherical means of /. Then we apply the obtained results on the boundedness to grand Morrey type spaces.
In application to the Hardy operators with power weights, the obtained conditions have a form of criterion when w(r) = r-A, A > 0.
The paper is organized as follows. Section 2 contains necessary definitions. In Section 3.1 we discuss varions ways of grandization of Morrey type spaces Lp>q>w(Rn). In Section 3.2 we provide conditions on grandizers ensuring embedding of Morrey type spaces into grand Morrey type spaces. In Section 4.1 we study the operators K in the Morrey type spaces Lp>q>w(Rn), and in Section 4.2 in grand Morrey type spaces Lpq)'w(Rn), both with application to the Hardy and Hilbert type operators.
2. Preliminaries
Following the known definitions, we introduce the spaces Lp'q'w(Rn), defined by the norm
|lp>9>w (Rn) • sup
( ) xeE
oo (
w(r)q
\ f \
\/(y)\P dy
\|x-y|<r
dr r
(2)
/ /
q
where E is an arbitrary set in Rn, E C Rn, 1 ^ p< to, 1 ^ q< to, w €
{OO v
w(r)q
w : w is a weight and / -dr < to for some t > 0 >.
In the cases E = {0} and E = Rn we have the local and global Morrey type spaces, respectively. We do not indicate dependence of the space on the choice of the set E, since it is unessential for our consideration. Only in Section 4 we choose E = {0}. In the special case w(r) = r-A, A > 0, we also use the notation
Lp'q'A(Rn):= Lp'q'w(Rn) |w=r-A
without danger of confusion of notation.
For a function w(r) defined on R+, we will use the notation
*t \ w(tr) , , „ w(tr)
w (t) := sup ——— and w*(t) := mf —-—, t > 0. reR+ w(r) x€R+ w(r)
Observe that = —¡r^y. Obviously w*(r) = w*(r) = w(r), when w(r) = r~x, A € M.
However, in the case of piece-wise power function
{r ^ r < 1
Y' 1
r ', r > 1,
where A, 7 € R, we have a gap between w* and w*:
r- max(A,Y} r < 1 | r- mln{A,Y} r < 1
1
see e. g. [24, p. 715 ].
w*Y(r) = < . x ' ' and (wa,7)*(r) = < ' (3)
A>7W I r- min|A,Y}, r > 1, I r- max|A,Y}, r> 1,
3. Grand Morrey Type Spaces
3.1. Grandization of Morrey type spaces. Everywhere in the sequel, a = a(y) and b = b(r) are weights on Rn and R+, respectively. Definition 3.1. Let
1 <p< to, 1 < q < to, w € Qq(R+) (4)
and
€ L°(RP;q), ^(e, 5) > 0 for (e, 5) € Rp,q and lim <^(e, 5) = 0,
where RP;q := {(e, 5) € R+ : 0 < e < p — 1, 0 < 5 < q — 1}.
We define the mixed grand Morrey type space La)q,)'w(Rn) as the space of functions with the finite norm
/ ^ \ 1
( CO ( \ P-s \
f Lp),q),w (Rn) := SUP ^(e,5)SUp
a'h (R ) (£,i)eRp,q xee
jw(r)q-sb(r)* 0
q-S
dr
r
J \f(y)\P Ea(y)pdy
\|x-y|<r J J
We also say that La)q,)'w(Rn) is the mixed gandization of the space Lp'q'w(Rn). Note that mixed coordinate-wise grandization of mixed Lebesgue spaces was studied in [25].
Definition 3.2. We define partial grandizations L0),q'w(Rn), 1 < p < to, 1 ^ q < to, and L?'q)'w(Rn), 1 ^ p < to, 1 < q < to, of the space Lp>q>w(Rn) as the spaces of functions
with the finite norm
f Lp),,,w(Rn) := sup ^(e)sup
a (R ) 0<£<p-\ x&E
CO (
w(r)q
V
\^ \
f(y)\ a(y)pdy
\|x-y|<r
dr r
where ^ € LC(0,p — 1), <^(e) > 0 and lim£^0 ^(e) = 0, and
f \\Lv,q)w(Rn) := sup y(S) sup
0<5<qr-1 xeE
(
o
w(r)q~5b(r
/ /
\ f \
i
qr-ä
If (y)|p dy
\|x-y|<r
dr r
^ y
where ^ € L^(0, q — 1), ^>(5) > 0 and lim^o <p(S) = 0, respectively.
Definitions 3.1 and 3.2 may be generalized in the following direction. Let U C Rp,q be an arbitrary measurable set of points in Rp,q, such that (0,0) is a limiting point for U.
Definition 3.3. Let 1 < p < to, 1 < q < to, ^ € L~(U), p(e,5) > 0 for (e,5) € U and limu3(e,5)^(0)0) <p(e,5) = 0. We define the U-grandization ULpo^^'w(Rn) of the Morrey type space as the space of functions with finite norm
f\\u Lp),,),w (Rn) := suP T ^M)suP
(e,S)GU
xeE
/
V0
(
w(r)q~5b(ry
. ti \
\ p-£
i
qr-ä
/(y) a(y)pdy
\Jx-y|<r
dr r
/ /
Under the choice U = Rp,q we have the mixed grand Morrey type space introduced in Definition 3.1. Partial grandization from Definition 3.2 formally correspond to the case where U = {(e, 5) : 0 < e < p — 1,5 = 0} and U = {(e, 5) : e = 0,0 <5 <q — 1}. In the sequel we use the notation
/ / \ — \ ^
^ / \ P —
N(f ; e, ¿) := sup
xeE
w(r)q~sb(r)«
V
\|x-y|<r
/(y) a(y)pdy
dr r
for brevity, assuming that p, q, w, a and b are fixed. In case of partial grandization we have N(f; e, 0) and N(f; 0,5) for b = 1 and a = 1, respectively.
Lemma 3.1. I. Let (eo,5o) € U£oAl := {(e,5) € U : 0 <e<eo, 0 <5< So}, a € L1(Rn) and b e Ll(R+,f), then
sup p(e,5) N(f;e,5) < C sup p(e,5) N(f;e,S), M)eu M)eu£oAo
(Rn),
ILHR+.t1)^
where C = C(eo,5o, ||a||Li
II. In the case of partial grandization with respect to the variable r, similarly
sup p(5) N(f; 0,5) < C sup p(5) N(f ;0,S),
o<5<q-1 o<5<50
where C = C(S0, ||&||Li(R+>«k)).
< We have to estimate
where
sup <^(e, 5) N(/; e, 5) =ma^ SI, £12, £2},
M)€U\USQ,g0
Si := sup p(e,5) N(/; e, 5), S2 := sup p(e,5) N(/; e, 5),
(e,5)€Ui (e,5)€U2
S12 := sup p(e,5)N(/;e,5), U1 := {(e,5) € U : e < eo, 5 > 5o},
(£,5)eUi2
U2 := {(e,5) € U : e > eo, 5 < 5o}, U12 := {(e,5) € U : e > eo, 5 > 5o}.
S
and ^f) respectively and obtain
For S12, we apply the Holder inequalities in the variables y and r with the expends ^^
____1___
p-s p-£q 11 l 11 q-5o
S12 < sup |l6llLiiR dL)
(s,s)eu12 y +' r'
\ q-So
/ \ p-£o
r
i 1
izio \ q-«o
w(r)q-Sob(r)f
\f(y)\P £°a(y) p° dy
dr r
o
sup pM)|M|£(yfeo |h|Ll(R+;i£)
\|x-y|<r / y
___i_ _1___i_
p-£ p-£Q ||/1||<?-'5 <?-5o
^ (s,s)eu 12-— - g die,
suP ^(e,5) M)eu£Q
(e,5)eU£0 ,«0
Estimation of S1 and S2 is easier via similar use of the Holder inequality in one variable only. We omit details. >
3.2. Embedding of Morrey type spaces into grand Morrey type spaces. Lemma 3.2. If
/SL » (¡r
b(r)A(x, r)p i-1 — < 00, (5)
r
o
where A(x,r) := Jjx_y|<r a(y)dy, then
Lp,q,w(Mn) ^ ULa!bq)'W(Rn) and ||/Hu£a).,).«(RB) < Coy/yLP,q,w(RB). (6)
< It suffices to apply the Holder inequalities with the exponents and in the inner and outer integrals in the definition of the norm in Definition 3.3. >
Theorem 3.1. The conditions a € L1(Rn) and b € Ll[are sufficient for the embedding (6) for any choice of the set U. The condition b € L1(R+,^f) is sufficient for embedding of Lp'q'w(Rn) into the partial grand space LP'q)'w(Rn).
< It is easy to see that the change of a(y) by Aa(y), A = const > 0, keeps the grand space (up to equivalents of norms). Consequently, we may assume that |M|Li(Rn) = 1. Then A(r) ^ 1 and the statement follows from (5).
The embedding into the partial grand space follows from Lemma 3.1. >
x
Remark 3.1. Note that the condition (5) does not assume that aeL^R"-), beL1(R+, ^f). In particular, in the case of "partial" grandization in the direction 5 = ^e, the embedding (6) holds, if b(r)A(r)q/p € Ll{R+, f).
By ACloc(R+) we denote the class of functions ^ : R+ — R, absolutely continuous on every interval [0, N], N > 0.
In Theorem 3.2 we impose the condition
i
£ \ q-s
[ ^- dt) < oo (7)
0 ^ )
on the grandizer b.
Lemma 3.3. Let ^ . Grandizer b of the form
where
6W = (Tnr^(r)'
lnr -, r < 1, it,* (r) = { r
ln* r, r ^ 1,
satisfies the condition (7) if 0 < ^ < v < to and t, a € R. < Let first t = a = 0. We have
r
f 7 f-i c
~dr = I —--dr = } (8)
(1 + r)
f Ö'
q
so that (7) is satisfied.
In the case of the presence of the logarithmic factor lT;0-(r), it suffices to observe that ^t,<t(V) is dominated by with arbitrarily small exponents rji > rfc > 0 and the estimate
by | in (8) does not depend on ¡i and v, provided 0 < ¡j, < v. >
Theorem 3.2. Let 1 ^ p < to and 1 < q < to. Let the grandizer b satisfy the condition (7). The embedding Lp'q'w(Rn) c LP'q)'w(Rn) is strict if w(r) satisfies one of the following assumptions:
i) w is decreasing, limr^o w(r) = oo and € ACioc(R+);
ii) there exist numbers v1 and v2, 0 < v2 < v1 ^ to, such that w(r)rVl is almost increasing and w(r)rV2 is almost decreasing.
The corresponding counterexample is f (x) = f0(|x|), where
fo(r) = ( rV ^
1
w(r)p
for the case i ) and
fo(r) = —1
fPw{r) in the case ii ).
r
< The case i). For f = f0(|x|) we have
r
/ 1/oflvDl' *v = c]±
|y|<r
w(p)p
dp, c = is
n— 1 I
By the absolute continuity of -j^ we obtain that
|fo(|y|)|P dy
\ p
Cv
\|y|<r
/
w(r)'
Therefore,
m(f-,o,5) = Cr
°° i \ 1-5 1 1 f b(r)i
■ dr
from which it follows that / € Lp'q)'w(Rn), but / € Lp'q'w(Rn). The case ii). In this case we have
|f (y)|p dy = C
dp
|y|<r
Pw(P)P"
From the assumptions in ii) it is easy to obtain that
Cl < [ dp < C2
w(r)p J pw(p)p w(r)p' o
From the equivalence (9) we obtain that
00 i \
9t(/;0,5) ~ I f^^dr
r
o
which completes the proof. >
4. Operators with Homogenous Kernel
In this section we choose E = {0} in the definition of the space Lp'q'w(Rn). We consider integral operators
Kf (x)= / K(|x|, |y|)f (y) dy,
(9)
where the kernel is homogeneous of degree —n, i. e.
K(t|x|,t|y|) = t-nK(|x|, |y|).
1
1
r
r
r
1
4.1. Operator K in Morrey type spaces. In the sequel we use the notation
ntf (x) := f (tx), x € Rn, t > 0.
It is not hard to check that
-JT^- \\f\\LP'i'w(Rn) ^ \\^-tf\\LP'i'w(Rn) ^ -JT^- \\f\\LP'i'w(Rn)- (10)
tP tP
If w(r) = r-A, then w*(t) = w*(t) = t-A and ||nty£p,,,w(Rn} = tA-n/p. In the theorem below we also use the notation
oo
x*(n):= l^"1! j sf~l[X{l,s)\w* Q ) ds 0
and
x*(n):= IS™-1! I \jf{l,s)\w* ( - ) ds,
o
— — 1 I 1/1
¡p' (1. ,sh w*
s 0
where |Sn 1| should be replaced by 1 in the one-dimensional case of R+.
The following one-dimensional theorem is an immediate consequence of (10). Theorem 4.1. Let 1 ^ p< to, 1 ^ q< to and w € (R+). The condition k*(1) < to is sufficient for the boundedness of the operator
oo
Kf (x) = / K(x,y)f (y) dy, x € R+,
0
where K(tx,ty) = t-1K(x,y), t > 0, in the space Lp>q>w(R+) and
||Kf yLP.i.w(R+) ^ K*(1)||f yLP.i.w(R+). (11)
< We have
oo
Kf (x) = y K(1,y)f (xy) dy. 0
Then by the Minkowsky inequality we obtain
oo
IIKf || / K(1,y)|nyf II dy,
0
whence (11) follows by (10). >
For the multi-dimensional case, in the next theorem we provide a statement stronger than just the boundedness in the space Lp'q'w(Rn). More precisely, we estimate the norm ||Kf ||Lp,q,w(Rn) via one-dimensional norms of spherical means of f.
Theorem 4.2. Let 1 ^ p< to, 1 ^ q< to and w € Qq(R+). If x*(n) < to, n ^ 1, then
f oo / r \ P \
f^y- I J tn~l\$(t)\p dt J dr 00
where
= f f(ta)da.
Sn-1
< Note that K/ is a radial function for any /. It is easily seen that for any radial function g(x) = G(|x|) one has
i
Gp\\Lp>I>w (R+)'
— js"7" 1|p ||Cp||Lp,q,ro(-R (13)
where Gp(p) = p(n-1)/pG(p).
Furthermore, passing to polar coordinates we have
X
n-l
Kf(x) = IS1"1! J t~?~J(r(\x\,t)$p(t)dt,
o
where
n-l i p
= Ki^ri I f (to) de-
ls n-1|
Then by (13) we have
Sn-1
where
/n-l n-l
JTi (p,i)$p(i)di, = P p t p' Jf(p,t).
o
The kernel K1(p,t) is homogeneous of degree -1, i. e. K1(sp,st) = s_1K1(p,t), s > 0. Therefore, we can apply Theorem 4.1 and obtain (12) after easy calculation. >
Remark 4.1. The estimate (12) is stronger than the boundedness in Lp'q'w(Rra). Indeed,
/00 / r \ I \ 1
w(r)
q
r in_1|$(i)|pdi J dr
oo
by Jensen inequality
11
< Is1-11 p II f II (14)
^ Is I I/ IILp,q,w(Rn) v-1-^
J ¡{per) da j J U(pv)\pda.
|Sn-1| J JKf J ^ |Sn-1|
Sn-1 / Sn-1
Clearly, the left-hand side in (14) may be finite when the right-hand side is infinite (e. g. when /(x) = /1(p)/2(r), x = pa, H = 1, with /2 € L1(Sn_1), but /2 € Lp(Sn-1)).
In the necessity part of Theorem 4.3 we shall use the following minimizing sequence
„ . . Me(|X|^ . . | r£, r < 1,
¡s{x) = , where ¡i£(r) = { e > 0.
|a;|pw(|a;|) I r , r > 1,
Lemma 4.1. Let w satisfy the condition that rôw(r) is almost decreasing for some 5 > 0. Then
r
[ V>ep(t) dt < C/X£p(r)
iw(i)*
w(r)p
for all e € [0, e0], where 0 < e0 <5 and c = c(e0) does not depend on e. < The proof is straightforward. >
Theorem 4.3. Let 1 ^ p< to, 1 ^ q< to and w € Qq(R+). If x*(n) < to, then
IlKf
"(Rn)
If K (|x|, |y|) ^ 0 and w satisfies the assumption of Lemma 4.1, then the condition x*(n) < to is necessary for such a boundedness; in particular, when w(r) = r-A, A > 0, the operator K is bounded if and only if
oo
[ t7+X~\jf(l,t) dt < TO.
< Sufficiency of the condition x*(n) < to follows from Theorem 4.2 by Remark 4.1. To prove the necessity, we choose f (x) = f£(x). By using Lemma 4.1, it is easy to check that f£(x) € Lp'q'w(Rn) for all e € (0,eo]. We have
Kfe(x)= f Jf(l,\y\)fe(\X\y)dy = -±w [ J \x\p J
K(1, |y|)
1
\X\PJ„ \v\PW(\x\
K(1, |y|)
■^(|x| ■ |y|) dy
\%\pw(\x\)Jn \y\p
It is easy to check that ^£(rp) ^ ^£(r)^£(p), so that
K/£(x) ^ x*(n,e)/£(x)
where
K (1,
w*(|y|)^£(|x| ■ |y|) dy.
= J ■
= Is
n— 1 I
w*(|ylK(|y|) dy
Hence
1 oo
j p7+£~l,j(i(1, p)w* ^^ dp + J ppT~£~1Jf(l,p)w^ ) dp I . 01
||K|| ^
It remains to apply Fatou theorem when passin to the limit as e — 0. > In the corollary below we consider the Hardy operators
Haf(x) = \x\a-n J Mdy and ^/(x) = \xf J
M<M
M>M
m
\y\n+ß
dy
p
and the Hilbert type operator
as examples of the operator K.
Corollary 4.1. The operators Ha and H13 are bounded in the space Lp'q'A(Rn), 1 ^ p < oo, 1 ^ q < to, A > 0, if and only if a < y + A and /3 > A — respectively
and \\H<*\\ = and H^ll =
pp
The operator H is bounded in the space Lp>q>A(Rra), 1 ^ q < to, A > 0, if and only if 1 ^ p < j and
||H|| = &(2.+ AWl_Ay
n \p' n/ \p n) < In the case of the operator Ha we have
^ x fta, t < 1,
K (1,t) = ^
0, t > 1,
so that K*(n) = K*(n) = • Arguments for .Jf13 are similar.
p
For the operator H we have
OO n
x\n)=it*(n) = l^"1! JtP'
j+A—1
dt,
1 + tn o
where it remains to pass the Beta function via the change —> t. > 4.2. Operator K in grand Morrey type spaces. Lemma 4.2. Let f0(x) = and K(\x\, \y\) ^ 0. Then
K/o(x) ^ K*(n)/o(x).
< We have
oo
dy I-?™"1! f i-i dp
^ _n__^ w(t)
^ —^-!— I pp' ^(1, p) inf . dp = x*(n)fo(x). >
l^li^d^D 7 Jt>ow{pt)
oo
^n-ll r „
Theorem 4.4. Let 1 ^ p< to, 1 <q< to and w € Qq (R+). If
'1
sup f tv' 1 |jf(l,i)| w* (-
0<&<&0 J \t
0
'qV
S
b* .
t
q(q-S)
dt < to (15)
for some 5o € (0, q — 1), then the operator K is bounded in the grand space Lp"q)'w(Rn).
< By Lemma 3.1 we take
r P'riw ffn* — sup 11/WhP'q-S'wS (Rn) ,
(R ) 0<5<5q v ;
where w(r) — w(r)[6(r)]5/q(q-5). By using Theorem 4.2 and Remark 4.1, we get
I|K/y (¿)||/(Rn),
(R ^ 0<5<5q v ;
where
™ 1 i , / 1
C(5) = (S1-11 I i"1 I JT(1 ,t) I - ) dt. >
0
Corollary 4.2. Let assumptions of Theorem 4.4 be satisfied and, b(r) — rM(1 + r) v, 0 < p < v .If there exists e0 > 0 such that
i
^so-l I ^ I / 1 \ ^ , f I ^Y-, ^ I „..* ' 1
k£0 := I I Jf (1, i) dt+ / iF™"1 I JT(1, i) I w* ( - ) dt < oo, (16)
i
then the operator K is bounded in the grand space (Rn) and
||K/(Rn) ^ k£q ||f (Rn)-
< For b(r) = rM(1 + r) v, by (3) we have
b
„ , I t-^, t < 1,
tj t> 1.
Then it is easy to see that (16) implies (15). >
Theorem 4.5. The Hardy operators and H^ are bounded in the grand space L^'X(Rn), 1 ^ p < oo, 1 < q < oo, A > 0, with the grandizer b(r) = , 0 < ¡i < v,
if a < ^ + A and /3 > A — respectively. If <p(5) ^ c^1/9, tiien tiiese conditions are also necessary for such a boundedness. < For the operator we have
1
k£0= [ t7+X-a-£0~l dt,
which is finite under the choice eo € (0, y + A — a). This ensures the boundedness of Ha when a < £ + A.
Similarly, the sufficiency of the condition /3 > A — ^ for the boundedness of Jf13 is checked.
To prove the necessary, we choose / = fo(x) =: so that
J | fo (y) |p dy = cirAp
|y|<r
and then
P,q),A(Rn) = SUP
= sup ^>(5)
0<5<q-1
B
0<5<q-1
fj.5 vb fj,5s Q ' Q q ,
Hi . .-nl dt t"(l+t) " j
i
q-S
1
q-S
^ c sup
p(5)
< oo.
0<(5<g—1 ¿q
Thus /0 € LP'q)'A(Rn). On the other hand, direct colculation shows that
Hafo(x) = cfo(x), c = IS
n— 11
ff+X-a~ldt,
which implies that /0^ tra/p'+A a must be finite.
The case of the operator H^ is analogously treated. >
b
References
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Received July 13, 2020
Stefan G. Samko University of Algarve, Faro 8005-139, Portugal, Professor Jubilado;
Kh. Ibragimov Complex Institute of the Russian Academy of Sciences,
21 a Staropromyslovskoe Hwy, Grozny 364051, Russia,
Principal Researcher
E-mail: ssamko@ualg. pt
https://orcid.org/0000-0002-8022-2863
Salaüdin M. Umarkhadzhiev
Kh. Ibragimov Complex Institute of the Russian Academy of Sciences, 21 a Staropromyslovskoe Hwy, Grozny 364051, Russia, Principal Researcher;
Academy of Sciences of Chechen Republic, 13 Esambaev Av., Grosny 364024, Russia, Head of the Department of Applied Semiotics E-mail: [email protected] https://orcid.org/0000-0002-8283-1515
Владикавказский математический журнал 2020, Том 22, Выпуск 4, С. 104-118
ГРАНД-ПРОСТРАНСТВА ТИПА МОРРИ
Самко С. Г.1'2, Умархаджиев С. М.2'3
1 Университет Алгарво, Португалия, 8005-139, Фаро; 2 Комплексный научно-исследовательский институт им. Х. Ибрагимова РАН, Россия, 364051, Грозный, Старопромысловское шоссе, 21а; 3 Академия наук Чеченской Республики, Россия, 364024, Грозный, пр. им. М. Эсамбаева, 13 E-mail: [email protected], [email protected]
Аннотация. Так называемые гранд-пространства в настоящее время являются одним из основных объектов в теории функциональных пространств. Гранд-пространства Лебега были введены в работах T. Iwaniec и C. Sbordone в случае множеств О конечной меры |О| < то, и авторами в случае |О| = то.
Последнее основано на введении понятия грандизатора. Идея «грандизации» была также применена в контексте пространств Морри. В этой статье мы развиваем идею грандизации до более общих пространств Морри Ьр'4'™ (К"), известных как пространства типа Морри. Мы вводим гранд-пространства типа Морри, что включает смешанные и частные гранд версии таких пространств. Смешанное гранд-пространство определяется нормой
sup ifi(e, 5) sup
s,S xEE
(
Vе
w(r)q 5
г
b(r) 4
(
\|x-y|<r
\ — \
1
q-e
If(y)\P Sa{y)"dy
/
dr r
/
с использованием двух грандизаторов а и b. В случае гранд-пространств, частных относительно показателя q, мы изучаем ограниченность некоторых интегральных операторов. Класс этих операторов содержит, в частности, многомерные версии операторов типа Харди и операторов Гильберта.
Ключевые слова: пространство типа Морри, гранд-пространство, гранд-пространство типа Морри, грандизатор, частная грандизация, смешанная грандизация, однородное ядро, оператор типа Харди, оператор Гильберта.
Mathematical Subject Classification (2010): 46E30, 42B35.
Образец цитирования: Samko, S. G. and Umarkhadzhiev, S. M. Grand Morrey Type Spaces // Вла-дикавк. мат. журн.—2020.—Т. 22, № 4.—C. 104-118 (in English). DOI: 10.46698/c3825-5071-7579-i.
