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4
Vladikavkaz Mathematical Journal 2021, Volume 23, Issue 4, P. 96-108
YAK 517.928+517.968
DOI 10.46698/ e4624-8934-5248-n
LOCAL GRAND LEBESGUE SPACES*
S. G. Samko1,2 and S. M. Umarkhadzhiev2,3
1 University of Algarve, Faro 8005-139, Portugal; 2 Kh. Ibragimov Complex Institute of Russian Academy of Sciences, 21 a Staropromyslovskoe Hwy, Grozny 364051, Russia; 3 Academy of Sciences of Chechen Republic, 13 Esambaev Av., Grozny 364024, Russia E-mail: ssamko@ualg.pt, umsalaudin@gmail.com
Abstract. We introduce "local grand" Lebesgue spaces L%0fa(Q), 0 < p < to, Q C Rn, where the process of "grandization" relates to a single point x0 £ Q, contrast to the case of usual known grand spaces (Q), where "grandization" relates to all the points of Q. We define the space LX'0',a (Q) by means of the weight a(|x — x0|)£p with small exponent, a(0) = 0. Under some rather wide assumptions on the choice of the local "grandizer" a(t), we prove some properties of these spaces including their equivalence under different choices of the grandizers a(t) and show that the maximal, singular and Hardy operators preserve such a "single-point grandization" of Lebesgue spaces Lp(Q), 1 < p < to, provided that the lower Matuszewska-Orlicz index of the function a is positive. A Sobolev-type theorem is also proved in local grand spaces under the same condition on the grandizer.
Key words: grand space, Lebesgue space, Muckenhoupt weight, maximal operator, singular operator, Hardy operator, Stein-Weiss interpolation theorem, Matuszewska-Orlicz indices. Mathematical Subject Classification (2010): 46E30, 42B35.
For citation: Samko, S. G. and Umarkhadzhiev, S. M. Local Grand Lebesgue Spaces, Vladikavkaz Math. J., 2021, vol. 23, no. 4, pp. 96-108. DOI: 10.46698/e4624-8934-5248-n.
1. Introduction
We introduce the so called "local grand" Lebesgue spaces L^a (Q), where the process of "grandization" relates to a single point xq € Q, contrast to the case of usual known grand spaces Lp),e(Q), where "grandization" relates to all the points of Q. The grand spaces (Q), defined by the norm
i
r \
e
LP),9(Q) = sup £ / 1/(x)|p £ dx\ , 0<£<p-1 \ J /
V Q 7
were introduced in [1, 2] in the case of a set Q with finite measure. They were widely investigated during the last decades. We refer e.g. to [3-8]. An approach to grandize Lebesgue
#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 grant № 20-51-46003. The research of S. Umarkhadzhiev was supported by TUBITAK and Russian Foundation for Basic Research under the grant № 20-51-46003.
© 2021 Samko, S. G. and Umarkhadzhiev, S. M.
spaces on sets of infinite measure was suggested and developed in [9-13]. We refer also to [14] and references therein.
Let Q be an open set in Rra, bounded or unbounded and xq € Q, be fixed. We introduce the spaces LX0',a(Q) via the (quasi)-norm
11/llrp).9 (o) := SUP W /l/(x)|pa(|x - xo|)p£ dx ) , 0 <p< to,
Lxo,a(o) o<e<£ \ J /
o
where a(t), 0 ^ t < diam Q, is a non-negative continuous bounded function vanishing only at t = 0.
Under some rather wide assumptions on the choice of the local "grandizer" a(t), we prove some properties of the spaces LX0',a (Q) and we show that the maximal, singular and Hardy operators preserve such a "single-point grandization" of Lebesgue spaces Lp(Q), 1 < p < to.
As a motivation for the introduction of such local grand spaces, we mention the following. When we study in Lebesgue spaces such operators as Hardy and Hilbert operators, or more generally integral operators with homogeneous kernel with fixed singularity, of principal importance is the study of mapping properties near the single point x = 0, because beyond this point such operators essentially improve properties of functions.
In Section 2 we give precise definitions and prove some properties of the spaces LX0',a(Q), including their equivalence under different choices of the grandizers a(t). In Section 3 we prove the main statements on the boundedness of operators in the spaces LX0',o.(Q).
2. Definitions and Properties of Local Grand Lebesgue Spaces
2.1. Definitions. Let Q C Rra be an open set, xo € Q, |a?oI < °o and d = diamQ, 0 < d ^ to. By G(0, d) we denote the set of functions continuous and bounded on [0, d), satisfying the conditions:
a(0) = 0 and inf a(t) > 0 for every 5 € (0, d). (1)
S^t<d
Definition 2.1. Let a € G(0,d). We define the local grand Lebesgue space LX0',a(Q), where 0 < p ^ to, d > 0, by the (quasi)-norm
\\f\\LP),o(n) := sup ee( [\f(x)M\x-x0\rdx], (2)
Lxo>a(0) o<£<e \ J I
o
when p < to and
ll/llr~) (o) := sup ee sup |/(x)|a(|x - Xo|)£, (3)
L*o>a(0) o<e<i xeo
where I € (0, to) is any fixed number. By (1), the norm (2) is equivalent to
llf W/aW := o<U<M j |f (X)|Pa(|X - Xo|)p£ dX J + f ^(O^xo,5)) (4) ^ 0<lB(xo,S)
for every 5 € (0, d). Everywhere in Section 2 we take 0 < p ^ to.
The function a € G(0, d) will be referred to as grandizer.
The norm will be sometimes written as ||/\\Lp),e ^ to underline dependence on the range for t.
Lemma 2.1. The space LSQ'^Q) does not depend on the choice of l, up to equivalence of norms:
Lp)'° e (Q) < \/£ (Q) < C\/Wl?» 1 (Q)> 0 < < < ^ (5)
where
r1 - TT1QY J 1 -
ll^i
C = max < 1, ——— sup £0||a||f,oo
I i1 WaW
< In the case p < to we have
L^, (Q) =maM W/W^»'9, (Q)'E
pq ,a;i2 ' I pq ,a;tiv >
where we denoted
/ \ -
V
i,"
ll<£<l2 Let A := ||a|L^. We have
sup ¿A<( [\f(x)\r(a{lX-Xol) ll<£<l2 W V A
v Q
E := sup i |/(x)|pa(|x - Xo|)p£ dx
ll<e<l2 V
1
pe \ p
dx
< sup eöA£( / |/(x)|p fi ll<e<l2 V J V
n
< l-0a-11 sup t0A£
l1 A sup t A ||/||Lp),9
ll<£<l2 LPQ,a;ll(Q)
q.e.d.
Arguments for p = to are similar. > The embedding
Lp(Q) ^ L?0fa(Q), 0 <p < to, 0,
holds, whenever a € L^(0, d).
A natural choice of grandizers a in the case of bounded sets Q, may be:
( d ■ eV 1 1
ao(t) = t[\n —J , a\(t) = t, 02(f) = 0,3(t) = ^^2 » (6)
where v € R, though this list may be continued.
If Q is unbounded, the above functions may be modified e.g. as follows:
e\v 1 1
/ e\v 1 1
a0(t) = t[\n-J , ai(i) = i, a2(i) = a3(i) = t t £ (7)
^ t
for 0 < t ^ 1 and identically equal to 1 for 1 ^ t < to.
Definition 2.2. We define the vanishing local grand Lebesgue space 0 < p < to, as the subspace of functions f € LX0)',a(^) such that
lim epe I |f (x)|pa(|x - xo|)p£ dx = 0. (8)
£^0 J
n
Clearly, the space LX0)',a(^) contains non-integrable functions when 0 < p < 1. The same holds for p = 1, since a(0) = 0. This may happen also for p > 1, if a(t) rapidly vanishs at t = 0, e.g. a(t) = e~~t, A > 0. It is easy to check that the condition
d
sup e-dp' Í tra-1a(t)-£p' < to, 1 < p < to,
0<£<£0 J 0
guarantees the embedding LX0>',a(ü) c L1(Q).
Similar local "grandization" may be made not only with respect to a single point xo € Q, but a finite number of points x^,..., x^ € Q via the grandizer a(x) = IlfcLi afc(k — ak € G(0, d), k = 1,..., N. Such a space coincides with the algebraic sum of the "single-point" local grand spaces lX^ k = 1,..., N.
2.2. Basic properties.
Lemma 2.2. Let a, b € G(0, d). If there exists a number a > 0 such that
a(t) < Cb(t)a, t € (0, d),
then LXO'J(^) ^ LX0fa(n).
< The proof is straightforward, with Lemma 2.1 taken into account. > From Lemma 2.2 it follows that
Lg&WL* = , d< TO (9)
for all A > 0, ^ > 0.
By Lemma 2.2 we have
LX01 (Q) ^ LX01 (Q) ^ LX01 (Q) = LXO'fac(Q), (10)
where the grandizers ao, ai, a2 and a3 are from (6) or (7) and coincidence of spaces holds up to equivalence of norms. The embeddings (10) are strict, see Lemma 2.4.
The coincidence of spaces in (10) and (9) may be observed in a more general situation, as given in Theorem 2.1, where we use the notion of Matuszewska-Orlicz indices m(a) and M(a) of a non-negative function a ([15], see also [16]), where properties of these indices are given in a from convenient for us. The lower index m(a) is defined by
a(hx)
In ( lim sup -^y
/ \ v h^0 y '
mía) := sup ---
0<x<1 lnx
Note also that
m(ta) = a, m
ln
d■eN ±1
= 0, m (taa(t)) = a + m(a), m[a(t)^] = ßm(a),
t
where a € R and ß € R+.
A non-negative function a(t) on (0, d), 0 < d ^ to is called quasi-monotone, if there exist a, P € M, such that ^ is almost increasing (a.i.) and ^ is almost decreasing (a.d.). A quasi-monotone function has finite indices and m(a) = sup {a : ^ is a.i.} and M (a) =
inf {P :
a(t) ■
IF
is a
.d.}.
Theorem 2.1. Let a and b be quasi-monotone on (0,5) for some 5 € (0,d). If m(a) > 0 and m(b) > 0, then
Lg&W = O)
up to equivalence of norms.
< It suffices to refer to (4), use the fact that for an arbitrarily small e > 0 there exist constants c(e) and C(e) such that
c(e)tM(a)+£ < a(t) < C(e)tm(a)-£, t € (0,5),
where M(a) is the upper Matuszewska-Orlicz index of a, M(a) ^ m(a) (see [16, Section 6] and apply Lemma 2.2). >
Keeping in mind that the function
—^-rr belongs to the usual grand Lebesgue space
\x—xo\ p
(Q), 6 ^ 1, below we consider similar inclusion of functions u = u(|x — x0|) into the space Lxo,«(Q).
For the cone condition used in the lemma below we refer e.g. to [17]. Lemma 2.3. Let xq € Q, and assume that Q satisfies the cone condition at the point xq, when x0 lies on the boundary of Q. Let u(t) be a non-negative function on (0, d) such that ff t"-1u(t)p dt < to for every 5 € (0, d). Then the condition
sup £p0 / tra-1u(t)pa(t)p£ dt < to
0<£<£o J
(11)
for some e0 > 0 is necessary and sufficient for the inclusion
u
(|x - xo|) G LXO^Q)
< The proof is straightforward. > When Q is bounded, we put
ui(t) = -4-, u2(t) =
tp
(tn ln
-, U3 (t) =
tn (ln^) fining
(12)
correspondingly to the grandizers a1 (t), a2(t) and a3(t).
When Q is unbounded, we define the functions uj(t) for 0 < t < 1 by (12) with d = 1 and continue them for t > 1 so that J™ t"-1^(t)p dt < to (e.g. u(t) = 0, t > 1, i = 1,2,3).
Lemma 2.4. Let aj, i = 1,2,3, be the grandizers defined in (6) and uk, k = 1,2,3, be the functions (12). Then
ukeL^ak(Q), if and uk£L^ak(il), if Q<9<-k = 1,2,3, (13)
and
Ufc
G LX01 (Q), if i > k, ^ > 0.
(14)
d
1
1
p
p
t
< Let d < to. For ui and a1 we have
d d
so that the statement for ui and a1 becoms evident by Lemma 2.2. For u and a2 we have
d d _p£ epe J ira"1ui(i)pa2(i)pe dt = epe J t~l ^ln dt = to,
0 0
so that u1 / L?o,a2by Lemma 2.2 and then u1 / L?o,a3 Similarly other cases are verified. >
3. Interpolation of Sublinear Operators in Local Grand Lebesgue Spaces
Everywhere in Section 3 we take 1 ^ p < to.
3.1. On interpolation. The proof of Theorem 3.2 in this section is based on the following theorem known as Stein-Weiss interpolation theorem with change of measure (see [18]; [19, p. 17]). We formulate it in weight terms. We use the notation
Lp(Q,w) := j f : J |f (x)|pw(x) dx < to
for weighted Lebesgue spaces.
Theorem 3.1. Let pk, qk € [1, to) and vk, wk be weights on Q, k = 1,2, and T — a sublinear operator defined on LP1 (Q, wi) U LP2(Q, w2). If T : LP1 (Q, wi) ^ Lqi(Q, vi) with the norm Ki and T : LP2(Q,w2) ^ Lq2(Q,v2) with the norm K2, then
T : LPt (Q,wt) ^ Lqt(Q,vt) with the norm K < Kl-iK2, where
11 -t t 11 -t t
~ =-+ -, - =-+ -, 15
Pt Pi P2 qt qi 92
(l_t)W t£t (1-i) —
= P1w/2, vt = v\ qiv2n, 0 < i < 1. (16)
Theorem 3.2. Let Q C Rn, 1 ^ p< to, 0 > 0 and a and b be grandizers. Assume that a sublinear operator T is bounded from the space LP(Q) to the space Lq(Q) and there exists an e0 > 0 such that it is bounded from the space LP(Q,a(| ■ —x0|)p£°) to the space Lq(Q,b(| ■ —x0|)q£°). Then the operator T is bounded from L?o,a(Q) to LXj^Q) and from VL^Q) to VL^Q).
< By Theorem 3.1 we obtain
llT/Wl9(Q '6(|--xo|)£9) ^ C11/hlp(q'a(|•-x°|)£p), 0 < e < ^
where C does not depend on / and e. Hence the statements of the theorem follow for both the spaces LSO/^Q) and VLS°°''a(Q), with Lemma 2.1 taken into account. >
3.2. Boundeddness of some classical operators of harmonic analysis in local grand Lebesgue spaces. In this section we take Q = Rn and stady the action, in the frameworks of the spaces l£o?o,(Rn), of the following operators: 1) the maximal operator
J 1/(9)1'i% <17)
B(x' r)
2) singular Calderon-Zygmund operators
T/(x) = y K(x,y)/(y) dy,
with standard kernel (see [20, p. 144]), 3) the Riesz potential operator
Ia/(x) := y |x — y|a-n/(y) dy, 0 < a < n,
Rn
4) the Hardy operators
Haf{x) = \x\a~n J f{y) dy, af{x) = \x\a J t^dy. (18)
|y|<|x| |y|>|x|
We show that these operators act in the Lebesgue spaces, preserving their grandization at a single point x0 € Rn, under a wide choice of the grandizers a(t).
Maximal and singular operators. By Ap we denote the Muckenhoupt class of weights. Theorem 3.3. Let 1 <p< to, 0> 0 and a € G(R+). If there exists an e0 > 0 such that
a£° € Ap, (19)
then the maximal operator M and singular Calderon-Zygmund operators T with standard kernel, bounded in L2(Rn), are bounded in the space LX°°''«(Rra).
< It suffices to apply Theorem 3.2 and use the known fact that both M and T are bounded in Lebesgue spaces with Ap-weights (see e.g. [20, pp. 137, 144]). >
Corollary 3.1. Let 1 < p < to, d > 0, and let a(t) be quasi-monotone with m(a) > 0. Then the maximal operator M and singular Caldron-Zygmund operators T with standard kernel, bounded in L2(Rn), are bounded in the space LX°','a(Rn).
< By Theorem 2.1 we have
where
Ll°'>n) - W / llLS)°'90°(Rn),
t, 0 < t < 1, ^ 1, t > 1.
It remains to note that a0(|x|)£° € Ap under the choice e0 € (0,n(p — 1)). This is well known, if a0 was a0(t)£° = t£°, t € R+. For the truncated power function it is easily obtained from the fact that for radial weights the Muckenhoupt condition is equivalent to (see [21])
p-1
sup / r-1ao(i)£0 dtl
r>0 J
oo
tn~ ao(t) p-1 dt < oo. >
Potential operators. In the proof of Theorem 3.4 we use the known (see [21, 22]) Mucken-houpt-Wheeden class defined by the condition
sup ( t [ w(x)q dx\ ( t [ w(x) p dx ] < oo
QCRn y|Q| Q J \|Q| Q J
(20)
which goes back to [23].
Theorem 3.4. Let 0 < a < n, 1 < p < \ = ~ n- H there exists an £q > 0 such that
a£° £ then the operator Ia is bounded from LpJ<fa(Rra) to LqJf>a(Rra).
p'
< We apply Theorem 3.2. The Lp ^ Lq-boundedness holds by the well known Sobolev theorem. The weighted Lp(Rn,a(| ■ -xo|)p£0) ^ Lq(Rn,a(| ■ -xo|)q£0) holds, if a£0 €
(see [22, 23]). It remains to note that, as is known, w € Apa wq € A1+q_ (see e.g. [21]). >
p'
Corollary 3.2. Let 0 < a < n, 1 < p < j = j ~ % and let a € G(R+) be quasimonotone with m(a) > 0. Then the operator is bounded from LX0',a(Rra) to LX0',a(Rra).
< The arguments are similar to those in the proof of Corollary 3.1. > Hardy operators. In this case we take x0 = 0.
Weighted boundedness of Hardy operators in Lebesgue spaces was thoroughly studied in the one-dimensional case (see [21, 24, 25]). The multidimensional versions (18) of Hardy operators were in particular studied in the case of power weights in [26], where the sharp constants were also found.
Though the weighted Lp ^ Lq-boundedness of Hardy operators is well studied for all p, q € (1, to), we consider, for simplicity, only the case p ^ q.
By Bp>q and Bp>q we denote the classes of pairs (u, v) of weights on R+, satisfying the conditions
B
sup
xeR+
sup
xeR+
u(t) dt
u(t) dt
v(t)1-p dt < to,
;(t)1-p dt < to,
respectively. Denote
(t) =
tY0, 0 <t< 1, , t ^ 1
and v\(t) = < t ' 0 < t < 1 |tA~, t ^ 1.
It is easy to check that
(u7, vA) € Bp,q ^ < -1, Ao < p - 1,
Yo + 1 1
+ p'
Ao , + 11
— and--1—-
p q p'
—, (21) p
r
r
q
and
(u7, vA) € Bp ' q ^ Yro > —1, Ao > p — 1,
Yo + 1 1
q p'
^ and (22)
p q p' p
The known results for the one-dimensional Hardy operators
x ro
H /(x) = y /(t) dt and H/(x) = J /(t) dt, x € R+,
ox
in the case 1 < p ^ q < to state that (see [24, p. 6-7]; [25, p. 12-13 ])
\ ? / 00 \ p y |H/(x)|qu(x) dxj < Ci y |/(x)|pv(x) dx j ^ (u,v) € B,
(23)
y|H/(x)|qu(x) < C y|/(x)|pv(x) * (u,v) € Bp,q. (24)
oo
Note that norm estimates of multi-dimensional integral operators with kernel k(|x|, |y|) and radial weights reduce in a sense to similar one-dimensional estimates of spherical mens, see [26] in the case of Hardy operators and [27] in the case of operators with homogeneous kernel. In the lemma below we show this in the case of Hardy operators and arbitrary radial weights.
Lemma 3.1. Let 1 < p ^ q < to and a € R. The multi-dimensional inequality
I j |H/(x)|qU(|x|) dxl < CI j |/(x)|pV(|x|) dx
V Rn / V Rn V
with radial weights holds, if there holds the one-dimensional inequality
(25)
\ q C \Hg(t)\qu(t)dt\ <-—
J |Sn-i|
|g(t)|pv(t) dt
(26)
where
Similarly
implies were
u(
(t) = tn-i+(a-n)q u (t), v(t) = t(n-i)(i-p) V (t)
\jrg(t)\9u(t)dt]" < —— / |Sn-i |
|g(t)|pv(t) dt
|Ha/(x)|qU(|x|) dx < C / |/(x)|pV(|x|) dx
(27)
u
(t) = tn-i+aq U (t), v(t) = t-p(n-i)-iV (t).
ro
ro
ro
oo
p
1
p
ro
ro
p
1
p
q
p
< Passing to polar coordinates, we rewrite (25) as
tn-1$(t) dt
q ( CO \ p
U(r) dr } < I tn-1$p(t)V(t) dt ^ , (29)
where
$(t)= / f (ta) da, $p(t)= | |f (ta)|p da.
Sn —1 Sn—1
By Jensen inequality, |$(t)|p < |Sn-1|p-1$p(t). Therefore, (29) will be moreover satisfied, if
œ
r«- 1+(a-n
. 0
r
C i * c
J tn-1$(t) dt
0
i i ç | 9 „ i i p
n-1V (t)l^(t)|p
/ira-V(t)|$(i)rdi^ , (30)
0
which is nothing else bat (26) with g(t) = t" 1$(t). The case of the operator Ha is similarly treated. > Corollary 3.3. The conditions
(tn-1+(a-n)qU(t),t(n-1)(1-p) V(t)) € (31)
and
(tn-1+aqU(t),t-p(n-1)-1 V(t)) € BP)q (32)
are sufficient for the validity of the inequalities (25) and (28), respectively.
Theorem 3.5. Let 0 < a < n, 1 < p < f, \ = \ - % and a, b € G(E+). The Hardy
operators Ha and Ha are bounded from L^ (Rn) to L^f (Rn), 9 > 0, if there exists an £o > 0 such that
(>-1+(a-n)qb(t)£0q,t(n-1)(1-p)a(t)£0p) € Bp,q (33)
and
(tn-1+aqb(t)£0q,t-p(n-1)-1a(t)£0p) € Bp,q, (34)
respectively.
< We apply Theorem 3.2. The Lp ^ Lq boundedness of Ha and Ha is known (see [26, Section 4]). By Corollary 3.3, the weighted Lp(R",a£0p) ^ Lq(Rn, b£0p)-boundedness for the operators Ha and Ha is guaranteed by the conditions (31) with U(t) = b(t)£0q and V(t) = a(t)£0p, which proves the theorem. >
Theorem 3.6. Let 0 < a < n, 1 < p < -, 1 = 1 - -. The operators Ha and Jfa are
a q p n
bounded from LO^f (Rn) to L0^(Rn), 9 > 0, for all grandizers a, b € G(R+), quasi-monotone in a neighbourhood of the origin, having positive indices m(a) > 0 and m(b) > 0.
< By Theorem 2.1, it suffices to prove the theorem in the case
a(t) = b(t) = (t, 0 <t< 1 [1, t> 1.
Under this choice we have to verify the conditions (33) and (34) for sufficiently small e0. This verification is easily done by means of the relations (21) and (22). >
œ
r
q
4. On a Weight Generalisation
In a similar way we can consider local grandization of weighted Lebesgue spaces, defined by the norm
Hf HiS^n ,w) = £\ y |f(x)|Pw(x)a(|x - xo|)£p • £ ^ n '
It is easy to see that statements of Lemmas 2.1, 2.2, 2.3 and Theorem 2.1 hold also in the weighted case in the corresponding reformulation. In the case of radial weights w = w(|x-x0|), an extension of Lemma 2.4 may be also obtained.
As regards the boundedness of operators in the weighted local grand space Li;)',« Theorem 3.1 allows to extend all the results of Section 3 to this case. We leave this to the reader.
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Received May 17, 2021
Stefan G. Samko University of Algarve, Faro 8005-139, Portugal, Professor Jubilado;
Kh. Ibragimov Complex Institute of Russian Academy of Sciences, 21 а Staropromyslovskoe Hwy, Grozny 364051, Russia, Principal Researcher E-mail: ssamko@ualg. pt http://orcid.org/0000-0002-8022-2863
Salaudin M. Umarkhadzhiev
Kh. Ibragimov Complex Institute of Russian Academy of Sciences, 21 а Staropromyslovskoe Hwy, Grozny 364051, Russia, Principal Researcher;
Academy of Sciences of Chechen Republic, 13 Esambaev Av., Grozny 364024, Russia, Head of the Departament of Applied Semiotics E-mail: umsalaudin@gmail.com http://orcid.org/0000-0002-8283-1515
Владикавказский математический журнал 2021, Том 23, Выпуск 4, С. 96-108
ЛОКАЛЬНЫЕ ГРАНД ПРОСТРАНСТВА ЛЕБЕГА
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: ssamko@ualg.pt, umsalaudin@gmail.com
Аннотация. Мы вводим «локальные гранд» пространства Лебега Ll0',a(O), О Q Rn, где процесс «грандизации» относится к единственной точке x0 £ О, в отличие от случая обычных известных гранд пространств Lp)'e(О), где «грандизация» относится ко всем точкам О. Мы определяем пространство (О) с помощью веса a(|x — x0|)£p с малым показателем степени, а(0) = 0. При некоторых довольно
широких предположениях о выборе локального «грандизатора» a(t) мы доказываем некоторые свойства этих пространств, включая их эквивалентность при различном выборе грандизаторов a(t), и показываем, что максимальный, сингулярный операторы и операторы Харди сохраняют такую «одноточечную грандизацию» пространств Лебега Lp(Q), 1 < p < то, при условии, что нижний индекс Матушевской — Орлича функции a положительный. Доказана также теорема типа Соболева в локальных гранд пространствах при том же условии на грандизатор.
Ключевые слова: гранд-пространство, пространство Лебега, вес Макенхаупта, максимальный оператор, сингулярный оператор, оператор Харди, интерполяционная теорема Стейна — Вейса, индексы Матушевской — Орлича.
Mathematical Subject Classification (2010): 46E30, 42B35.
Образец цитирования: Samko, S. G. and Umarkhadzhiev, S. M. Local Grand Lebesgue Spaces // Владикавк. мат. журн.—2021.—Т. 23, № 4.—C. 96-108 (in English). DOI: 10.46698/e4624-8934-5248-n.
