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ISSN 1064–5624, Doklady Mathematics, 2006, Vol. 73, No. 1, pp. 59–62. © Pleiades Publishing, Inc., 2006.
Original Russian Text © E.I. Berezhnoi, L. Maligranda, 2006, published in Doklady Akademii Nauk, 2006, Vol. 406, No. 4, pp. 439–442.
The role played by exact estimates of classical oper-ators in harmonic analysis and adjacent areas is well known. In recent years, because of new problems of analysis, estimates of operators on some cones in spaces rather than on the entire spaces have become very popular (see, e.g., [1–4]). On the other hand, in the theory of integral operators with positive kernels, the extrapolation theorem of Schur (see, e.g., [5]) is well known; it says that an integral operator Kx(t) = (t, s)x(s)ds with k(t, s) ≥ 0 is bounded in Lp if and only if there exists a positive function u(t) such that it is finite almost everywhere and the operator is bounded in the pairs K: → and K: → , where v = u1/p – 1. In relation to various problems of analysis, the interest in extrapolation theorems has increased [6–8]. For this reason, it is natural to pass from the Lebesgue space Lp to cones of Lebesgue spaces in these theorems. In this paper, we suggest a reduction of estimating operators on cones to estimating them on new spaces, which are constructed from the cones and the initial spaces, for the most important cones in the Lebesgue spaces. Such a reduction makes it possible to apply the whole apparatus developed for obtaining exact mates on weight Lebesgue spaces to obtain exact esti-mates of operators on cones. Using the reduction, we prove a new extrapolation theorem for a certain class of operators defined on cones in Lebesgue spaces.
Let S(µ) = S(R+, Σ, µ) [where R+ = (0, +∞)] be the space of measurable functions x: R+→R. Recall that a Banach space X = (X, ||·|X||) consisting of measurable functions is said to be ideal [9] if, for any y∈ X and any measurable x such that |x(t)|≤|y(t)| almost everywhere on R+, x∈X and ||x|X||≤||y|X||. As usual, the symbol Lp
k
∫
Lu ∞ Lu ∞ Lv 1 Lv 1(where 1 ≤ p ≤ ∞) denotes the classical Lebesgue space.
Let w: R+→R+ be a positive function (weight). For an ideal space X, we use Xw to denote the new ideal space with norm ||x|Xw|| = ||wx|X||.
Definition 1. Let X be an ideal space in S(µ), and let
K be a cone in S(µ). As usual, K∩X denotes the inter-section of the cone K with the cone X+.
Let K(↓) denote the cone in S(µ) consisting of the functions x: R+ → R+ that do not increase, i.e., satisfy the condition x(t + h) ≤ x(t) for h ≥ 0, and let K(↑) be the cone of nondecreasing functions in S(µ); by K(↓, ↑) we denote the cone in S(µ) consisting of the concave func-tions x: R+→ R+ satisfying the additional conditions
(t) = 0 and x(t) = 0.
Theorem 1. Suppose that p∈ (1, ∞) and w is a
weight function such that
(1)
and
(2)
Let Q be the operator defined by
Finally, let v be a new function for which
(3)
[κ(D) denotes the characteristic function of the set D]. x t→0 lim t–1 t→0 lim wp( )s ds 1 ∞
∫
= ∞ wp( )s ds 0 t∫
<∞ for any t>0 Qx t( ) x( ) ττ d . t ∞∫
= κ(0 t, )w Lp κ(t,∞)1 v ---- Lp' ⋅ ≡1MATHEMATICS
Representability of Cones in Weighted Lebesgue Spaces
and Extrapolation Operators on Cones
E. I. Berezhnoi* and L. Maligranda**
Presented by Academician S.M. Nikol’skii March 30, 2005 Received September 23, 2005
DOI: 10.1134/S1064562406010169
* Yaroslavl State University, Sovetskaya ul. 14, Yaroslavl, 150000 Russia
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DOKLADY MATHEMATICS Vol. 73 No. 1 2006
BEREZHNOI, MALIGRANDA Then, the following assertions are valid: the
opera-tor Q acts and is bounded in the pair
(4) there exists a constant c > 0 such that, for any y∈ K(↓) ∩ with ||y| || = 1, there exists a function x ∈ with
||x| || = 1 such that
(5) for any t∈ (0, ∞).
Note that condition (4) holds by virtue of the classi-cal estimates of the operator Q in the spaces [the weight v in (3) was chosen so as to ensure this]; see, e.g., [4, 10]. For y ∈ K(↓) ∩ , the function x in (5) is defined constructively.
Remark 1. Theorem 1 has a complete analogue for
the cones K(↑), K(ϕ, ↓) = {x: R+→ R+: ϕ(t)x(t)↓}, and K(ϕ, ↑) = {x: R+→ R+: ϕ(t)x(t)↑} with the only differ-ence that, instead of Q, the operators
should be considered.
Let us exemplify the applications of Theorem 1.
Theorem 2. Suppose that 1 ≤ p0 < p1 < ∞ and u and
w are weight functions satisfying conditions (1) and (2). Then,
i.e., these cones do not coincide for any weight func-tions satisfying the assumpfunc-tions of the theorem.
We say that an operator T: S(µ) → S(µ) is sublinear if |T(x + y)(t)|≤ T|x|(t) + T|y|(t) and |T(λx)(t)|≤λ|Tx(t)| for λ≥ 0.
Theorem 1 immediately implies the following asser-tion.
Theorem 3. Suppose that p∈ (1, ∞) and w is a
weight function satisfying conditions (1) and (2). Let Y be an ideal Banach space in S(µ).
A sublinear operator T acts and is bounded as an operator from K(↓) ∩ to Y if and only if the super-position operator TQ acts and is bounded as an opera-tor from to Y.
Using the technique for estimating operators L: → Y (see, e.g., [2, 4, 11, 12]), we can obtain various
esti-Q: Lv p ( )+ K( )↓ Lw p ; ∩ → Lw p Lp w Lv p Lv p Qx ( )( )t ≥cy t( ) Lu p Lw p Px t( ) x s( )ds, Qϕx t( ) 0 t
∫
1 ϕ( )t --- x s( )ds, t ∞∫
= = Pϕx t( ) ϕ1 t ( ) --- x s( )ds 0 t∫
= K( ) ∩ ↓ Lu p0 K( ) ∩ ↓ Lu p1 ; ≠ Lw p Lv p Lw pmates for operators on the cone of monotone functions in Lebesgue spaces by applying Theorem 3.
Consider the cone K(↓, ↑). For nonnegative func-tions on R+, we define the operators
It is easy to show by simple integration by parts that, if a function x ∈ K(↓, ↑) has absolutely continuous first derivative, then
where z(s) is a nonnegative function. We can set z(s) ≡ –x"(s).
Theorem 4. Suppose that p∈ (1, ∞) and w is a
weight function such that, for any t ∈ R+,
(6)
Consider the cone K(↓, ↑) ∩ .
Let w0 and w1 be the new weight functions defined by the equalities
(7)
and
(8)
for all t > 0, and let
Then, the following assertions are valid: the sum of operators Q1 + P1 acts and is bounded in the pair
(9)
for any y∈ K(↓, ↑) ∩ with ||y| || = 1, there exists a function x∈ with ||x| || = 1 such that
(10) Q1x t( ) t x s( )ds and P1x t( ) t ∞
∫
sx s( )ds. 0 t∫
= = x t( ) z( ) ττ d s ∞∫
⎝ ⎠ ⎜ ⎟ ⎛ ⎞ s d 0 t∫
= = t z s( )ds t ∞∫
sz s( )ds 0 t∫
+ = Q1z t( )+P1z t( ), min 1 s t --, ⎩ ⎭ ⎨ ⎬ ⎧ ⎫ ⎝ ⎠ ⎜ ⎟ ⎛ ⎞p wp( )s ds 0 ∞∫
<∞. Lw p κ(t,∞) 1 w0( )s --- Lp' ⋅ κ(0 t, )sw s( ) Lp ≡1, κ(0 t, ) s w1( )s --- Lp' ⋅ κ(t,∞)w s( ) Lp ≡1 v( )t = max w{ 0( )t ,w1( )t }. Q1+P1 ( ): Lv p ( )+ K(↓ ↑, ) ∩ Lw p ; → Lw p Lw p Lv p Lv p Q1+P1 ( )x ( )( )t ≥cy t( )DOKLADY MATHEMATICS Vol. 73 No. 1 2006
REPRESENTABILITY OF CONES IN WEIGHTED LEBESGUE SPACES 61
for all t ∈ (0, ∞), where c > 0 is a constant, if and only if
(11)
Condition (6) ensures that the extreme functions
min 1, belong to the cone K(↓, ↑) of the space . Condition (7) is necessary and sufficient for the bound-edness of Q1 as an operator from in , and condi-tion (8) is necessary and sufficient for the boundedness of P1 as an operator from to . Therefore, if v is chosen as in the statement of the theorem, then condi-tion (9) does hold.
Condition (11) in Theorem 4 simply ensures the possibility that condition (10) holds for the family of
extreme functions min 1, of the cone K(↓, ↑).
For a function y ∈ K(↓, ↑) ∩ , the function x in (10) is defined constructively.
Theorem 4 has an analogue for the cones K(ϕ, ψ) = {x: R+→ R+: x(t) · ϕ is nondecreasing and ψ(t) · x(t) is nonincreasing}.
Condition (11) does not always hold. There are var-ious sufficient conditions under which (11) holds. In particular, on the power scale, i.e., for w(t) = tα, condi-tions (11) are satisfied for α∈ – 1, .
The following theorem exemplifies the applications of Theorem 4.
Theorem 5. Suppose that p∈ (1, ∞) and w is a
weight function satisfying (6). Let w0, w1, and v be functions such that condition (11) holds for this w, and let Y be an ideal Banach space in S(µ).
A sublinear operator T acts and is bounded as an operator from K(↓, ↑) ∩ to Y if and only if the superposition operator T(Q1 + P1) acts and is bounded as an operator from to Y.
Now, we proceed to extrapolation theorems for operators on cones. We need some additional construc-tions.
Let X0 and X1 be two ideal spaces in S(µ). inf t κ(0 t, ) s v( )s --- Lp' t κ(t,∞) s v( )s --- Lp' + ⎝ ⎠ ⎛ ⎞ ⎩ ⎨ ⎧ × κ(0 t, )s t --w s( ) Lp + κ(t,∞)w s( ) Lp ⎝ ⎠ ⎛ ⎞ ⎭ ⎬ ⎫ ∞ . < ⎩ ⎨ ⎧ s t --⎭ ⎬ ⎫ Lw p Lw0 p Lw p Lw1 p Lw p ⎩ ⎨ ⎧ s t --⎭ ⎬ ⎫ Lw p 1 p ---– ⎝ ⎛ 1 p ---– ⎠⎞ Lw p Lv p
Take 0 < θ < 1. Consider the new ideal space (the Calderon–Lozanovskii construction) con-sisting of those x ∈ S(µ) for which the norm
(12)
is finite. The space was introduced by Cal-deron [13] in studying the complex interpolation method.
If K is a cone in S(µ), then we can define a new cone (K ∩ X0)θ(K ∩ X1)1 – θ, by analogy with , namely, by taking only decompositions into elements of the cone in (12). The following theorem is an inter-polation result; it is well known for the cone of nonne-gative functions (see, e.g., [14, 15]).
Theorem 6. Suppose that T is a positive operator and K0 and K1 are two cones in S(µ)+. Let X0, X1, Y0, and Y1 be ideal Banach spaces in S(µ). Suppose that an operator T acts and is bounded as an operator T: Xi ∩ K0→ Yi∩ K1 for i = 0, 1. Let θ ∈ (0, 1).
Then, the operator T acts and is bounded as an operator T: (K0 ∩ X0)θ(K0 ∩ X1)1 – θ→ (K1 ∩ Y0)θ(K1 ∩ Y1)1 – θ.
Remark 2. As is usual in interpolation theory, for an
arbitrary cone K, the equality (K ∩ )θ(K ∩ )1 – θ = K ∩ (( )θ( )1 – θ) does not always hold, even for the cone K(↓).
Theorem 7. Suppose that p∈ (1, ∞), w is a weight
function satisfying conditions (1) and (2), and v is a function constructed for w according to (3). Let θ = . Suppose that T is a linear positive operator T acting and bounded in the pair
Then, there exist functions v0, v1, u0, and u1 such that (13) the operator TQ acts and is bounded in the pairs
Combining Theorems 6 and 7, we obtain the follow-ing extrapolation theorem for operators on the cone K(↓).
Theorem 8. Suppose that p∈ (1, ∞); w is a weight
function satisfying condition (6); w0, w1, and v are functions constructed for w; and condition (11) holds. Let θ = . Suppose that T is a linear positive operator acting and bounded in the pair
X0θX11–θ x X 0 θX 1 1–θ inf λ 0: x t( ) λ x0( )t θ x1( )t 1–θ ⋅ ≤ > { = for any t Ω; x0 X 0 1, x1 X1 1 ≤ ≤ ∈ } X0θX11–θ X0 θ X1 1–θ Lv0 1 Lv1 ∞ Lv0 1 Lv1 ∞ 1 p ---T : K( )↓ Lw p ∩ Lw p . → v0θ( )t ⋅v11–θ( )t v( )t , u0 θ t ( ) u1 1–θ t ( ) ⋅ u t( ); ≡ ≡ TQ: Lv0 1 Lu0 1 , TQ: Lv1 ∞ Lu1 ∞ . → → 1 p ---T : K(↓ ↑, ) Lw p ∩ Lu p . →
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DOKLADY MATHEMATICS Vol. 73 No. 1 2006
BEREZHNOI, MALIGRANDA Then, there exist functions v0, v1, u0, and u1 such that
relations (13) hold and the operator T(Q1 + P1) acts and is bounded in the pairs
Combining Theorems 6 and 7, we obtain an extrap-olation theorem for operators on the cone K(↓, ↑).
ACKNOWLEDGMENTS
This work was supported by the grant of the Swed-ish Royal Academy of Sciences for collaboration with Russia (project no. 35160). E.I. Berezhnoi acknowl-edges the support of the Russian Foundation for Basic Research (project no. 05-01-00206).
REFERENCES
1. E. T. Sawyer, Stud. Math. 96, 145–158 (1990).
2. E. I. Berezhnoi, Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 204, 3–36 (1993).
3. H. Heinig and L. Maligranda, Stud. Math. 116, 133–165 (1995).
4. A. Kufner and L.-E. Persson, Weighted Inequalities of Hardy Type (World Sci., Singapore, 2003).
5. V. B. Korotkov, Integral Operators (Nauka, Novosi-birsk, 1983) [in Russian].
6. J. Garcia-Cuerva and J. Rubio De Francia, Weighted Norm Inequalities and Related Topics (North-Holland, Amsterdam, 1985).
7. E. I. Berezhnoi, Dokl. Akad. Nauk 344, 727–730 (1995). 8. E. I. Berezhnoi and L. Maligranda, Dokl. Akad. Nauk
393, 583–586 (2003) [Dokl. Math. 68, 392–395 (2003)].
9. S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpo-lation of Linear Operators (Nauka, Moscow, 1978) [in Russian].
10. V. G. Maz’ja, Sobolev Spaces (Springer-Verlag, Berlin, 1985).
11. E. I. Berezhnoi, Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 201, 26–42 (1991).
12. E. I. Berezhnoi, Proc. Am. Math. Soc. 127 (1), 79–87 (1999).
13. A. P. Calderon, Matematika 9 (3), 56–129 (1965). 14. E. I. Berezhnoi, in Qualitative and Approximate
Meth-ods for the Study of Operator Equations (Yaroslav. Gos. Univ., Yaroslavl, 1981), pp. 3–12 [in Russian].
15. L. Maligranda, Orlicz Spaces and Interpolation (Campi-nas, 1989). T Q( 1+P1): Lv0 1 Lu0 1 , T Q( 1+P1): Lv1 ∞ Lu1 v . → →
