Extension of Lorentz-Shimogaki and Boyd’sresults to the weighted Lorentz spaces
E. Agora
Instituto Argentino de Matemática “Alberto P. Calderón ”IAM-CONICET Buenos Aires, Argentina
Joint work with: J. Antezana, M. J. Carro and J. Soria
bg=whiteClassical operators of Harmonic Analysis
DefinitionThe Hilbert transform H is defined
Hf (x) = limε→0+
∫|x−y|>ε
f (y)x− y
dy ,
whenever this limit exists almost everywhere.
DefinitionThe Hardy-Littlewood maximal function is defined by:
Mf (x) = supx∈I
1|I|
∫I|f (y)|dy,
where I is an interval of the real line and the supremum is considered overall intervals containing x ∈ R.
bg=whiteWeighted Lorentz spaces
Definition (Lorentz, 1950)Let p > 0, u ∈ L1
loc(R), w ∈ L1loc(R+). The weighted Lorentz spaces are:
Λpu(w) = {f ∈M(R) : ||f ||p
Λpu(w)
=∫ ∞
0(f ∗u (t))pw(t)dt <∞},
Λp,∞u (w) = {f ∈M(R) : ||f ||Λp,∞
u (w) = supt>0
W1/p(t)f ∗u (t) <∞},
where f ∗u (t) = inf{s > 0 : u({x ∈ R : |f (x)| > s}) ≤ t} and W(t) =∫ t
0 w.
Examples:
(i) If w = 1, we recover Λpu(1) = Lp(u) and Λp,∞
u (1) = Lp,∞(u);(ii) If u = 1, we obtain Λp(w) and Λp,∞(w).
bg=whiteCharacterization of H : Λpu(w)→ Λp
u(w)
Recall that given an operator T , by means of
T : Λpu(w)→ Λp
u(w),
we express||Tf ||Λp
u(w) ≤ C||f ||Λpu(w),
for some positive constant C.
ProblemLet p > 1. Find necessary and sufficient conditions on u ∈ L1
loc(R) andw ∈ L1
loc(R+) so that the boundedness of H on Λpu(w) holds, i.e.:
H : Λpu(w)→ Λp
u(w).
bg=whiteThe case of weighted Lebesgue spaces
Hunt-Muckenhoupt-Wheeden 1973, Coifman-Fefferman 1974 proved:
TheoremLet 1 < p <∞. The boundedness H : Lp(u)→ Lp(u) holds if and only if
u ∈ Ap : supI⊂R
(1|I|
∫Iu(x)dx
)(1|I|
∫Iu−1/(p−1)(x)dx
)p−1
<∞.
Muckenhoupt 1972, proved that Ap characterizes also the boundedness of M:
TheoremFor p > 1, the following statements are equivalent:
(i) u ∈ Ap;
(ii) M : Lp(u)→ Lp(u);
(iii) M : Lp(u)→ Lp,∞(u).
bg=whiteThe case of classical Lorentz spacesBoundedness of M
It is known that (Mf )∗(t) ≈ Pf ∗(t), where
Pf (t) =1t
∫ t
0f (r)dr =
∫ 1
0f (ts)ds =
∫ 1
0Es f (t)ds,
where Es f (t) = f (ts). For s ∈ (0, 1], define
‖Es‖pΛp(w)→Λp(w) = sup
‖f∗‖Lp(w)≤1‖Es f ∗‖p
Lp(w) = supt>0
W(st)W(t)
=: W(s).
By Minkowski inequality we obtain
||M||Λp(w)→Λp(w) = sup‖f‖Λp(w)≤1
‖Mf‖Λp(w) ≈ sup‖f∗‖Lp(w)≤1
‖Pf ∗‖Lp(w)
= sup‖f∗‖Lp(w)≤1
∥∥∥∥∥∫ 1
0Es f ∗ds
∥∥∥∥∥Lp(w)
≤∫ 1
0W1/p(s)ds.
Proposition
If∫ 1
0W1/p(s)ds <∞, then M : Λp(w)→ Λp(w).
bg=whiteThe case of classical Lorentz spacesLorentz-Shimogaki theorem for Λp(w)
Define the function W by
W(µ) = sups>0
W(µs)W(s)
,
and the upper Boyd index by
αΛp(w) = limµ→∞
log W1/p(µ)logµ
.
Theorem (Lorentz 1955 - Shimogaki 1965)The boundedness M : Λp(w)→ Λp(w) holds if and only if αΛp(w) < 1.
bg=whiteThe case of classical Lorentz spacesBoundedness of H
Bennett and Rudnick proved in 1980
(Hf )∗(t) . (P + Q)f ∗(t) . (Hg)∗(t), t > 0,
where f ∗ is equimeasurable with f and
Qf (t) =∫ ∞
tf (s)
dss
=∫ ∞
1f (ts)
dss
=∫ ∞
1Es f (t)
dss.
By Minkowski inequality
||H||Λp(w)→Λp(w) = sup||f ||Λp(w)≤1
||Hf ||Λp(w) . sup||f∗||Lp(w)≤1
||(P + Q)f ∗||Lp(w)
= sup||f∗||Lp(w)≤1
∣∣∣∣∣∣∣∣∣∣∫ 1
0Es f ∗ds +
∫ ∞1
Es f ∗dss
∣∣∣∣∣∣∣∣∣∣Lp(w)
.∫ 1
0W1/p(s)ds +
∫ ∞1
W1/p(s)dss.
Proposition
If∫ 1
0W1/p(s)ds +
∫ ∞1
W1/p(s)dss<∞, then H : Λp(w)→ Λp(w).
bg=whiteThe case of classical Lorentz spacesBoyd theorem for Λp(w)
Recall that
W(µ) = sups>0
W(µs)W(s)
,
and define the lower Boyd index
βΛp(w) = limλ→0
log W1/p(λ)logλ
.
Theorem (Boyd 1950)It holds that
H : Λp(w)→ Λp(w) if and only if 0 < βΛp(w) and αΛp(w) < 1.
Corollary
H : Λp(w)→ Λp(w)⇔ 1 + logst
.W(s)W(t)
.( s
t
)p−ε, t ≤ s.
bg=whiteGeneralized Lorentz Shimogaki theorem
The generalized upper Boyd index is:
αΛpu(w) = lim
µ→∞
log W1/pu (µ)
logµ,
where the function Wu : [1,∞)→ [1,∞) is defined by:
Wu(λ) := sup{
W(u(∪j∈JIj))W(u(∪j∈JSj))
: Sj ⊆ Ij and |Ij| = λ|Sj| for every j ∈ J}.
Theorem (Lerner, Pérez 2007- Carro, Raposo, Soria 2007)If p > 0, then M : Λp
u(w)→ Λpu(w) ⇔ αΛp
u(w) < 1.
Theorem (A., Antezana, Carro, Soria 2014)If p > 1, then M : Λp
u(w)→ Λp,∞u (w) ⇔ αΛp
u(w) < 1.
Remark: If w = 1, then αΛpu(w) < 1 is equivalent to the Ap condition.
bg=whiteGeneralized Boyd theoremThrough the Weighted Lorentz spaces
The generalized lower Boyd index is:
βΛpu(w) = lim
λ→0
log Wu1/p(λ)
logλ,
where the function Wu : (0, 1]→ (0, 1] is defined as follows:
Wu(λ) := sup{
W(u(∪j∈JSj))W(u(∪j∈JIj))
: Sj ⊆ Ij and |Sj| = λ|Ij|, for every j ∈ J}.
Theorem (A., Antezana, Carro, Soria 2014)If p > 1, then H : Λp
u(w)→ Λpu(w) ⇔ 0 < βΛp
u(w) and αΛpu(w) < 1.
Remark:(i) If w = 1, then 0 < βΛp
u(w) is just the A∞ condition.(ii) If w = 1, then the condition 0 < βΛp
u(w) and αΛpu(w) < 1 recover the
conditions over the classical Boyd indices.
bg=whiteSome references
[1] E. Agora, J. Antezana, M. J. Carro and J. Soria, Lorentz-Shimogaki andBoyd Theorems for weighted Lorentz spaces, London Math. Soc. 89 (2014)321-336.
[2] E. Agora, M. J. Carro and J. Soria, Complete characterization of theweak-type boundedness of the Hilbert transform on weighted Lorentzspaces, J. Fourier Anal. Appl. 19 (2013) 712-730.
[3] M. J. Carro, J. A. Raposo and J. Soria, Recent developments in the theoryof Lorentz Spaces and Weighted Inequalities, Mem. Amer. Math. Soc. 187(2007) no. 877, xii+128.
[4] A. Lerner and C. Pérez, A new characterization of the Muckenhoupt Ap
weights through an extension of the Lorentz-Shimogaki theorem, IndianaUniv. Math. J. 56 (2007) no. 6, 2697-2722.
bg=whiteThe end
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