We establish the sharp growth order, up to epsilon losses, of the $L^2$-norm of the maximal directional averaging operator along a finite subset $V$ of a polynomial variety of arbitrary dimension ...$m$, in terms of cardinality. This is an extension of the works by C\'ordoba, for one-dimensional manifolds, Katz for the circle in two dimensions, and Demeter for the 2-sphere. For the case of directions on the two-dimensional sphere we improve by a factor of $\sqrt{\log N}$ on the best known bound, due to Demeter, and we obtain a sharp estimate for our model operator. Our results imply new $L^2$-estimates for Kakeya type maximal functions with tubes pointing along polynomial directions. Our proof technique is novel and in particular incorporates an iterated scheme of polynomial partitioning on varieties adapted to directional operators, in the vein of Guth, Guth-Katz, and Zahl.
This article develops a novel approach to the representation of singular integral operators of Calderón–Zygmund type in terms of continuous model operators, in both the classical and the ...bi-parametric setting. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calderón–Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of
T
(1) theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the
A
2
theorem; that is, sharp dependence of the Sobolev norm of
T
on the weight characteristic is obtained in the full range of exponents. In the bi-parametric setting, where local average sparse domination is not generally available, we obtain quantitative
A
p
estimates which are best known, and sharp in the range
max
{
p
,
p
′
}
≥
3
for the fully cancellative case.
A metric approach to sparse domination Conde-Alonso, José M.; Di Plinio, Francesco; Parissis, Ioannis ...
Annali di matematica pura ed applicata,
08/2022, Letnik:
201, Številka:
4
Journal Article
Recenzirano
We present a general approach to sparse domination based on single-scale
L
p
-improving as a key assumption. The results are formulated in the setting of metric spaces of homogeneous type and avoid ...completely the use of dyadic-probabilistic techniques as well as of Christ–Hytönen–Kairema cubes. Among the applications of our general principle, we recover sparse domination of Dini-continuous Calderón–Zygmund kernels on spaces of homogeneous type, we prove a family of sparse bounds for maximal functions associated to convolutions with measures exhibiting Fourier decay, and we deduce sparse estimates for Radon transforms along polynomial submanifolds of
R
n
.
Let
Θ
⊂
S
1
be a lacunary set of directions of order
D
. We show that the maximal directional Hilbert transform
H
Θ
f
(
x
)
:
=
s
u
p
v
∈
Θ
|
p
.
v
∫
R
f
(
x
+
t
v
)
d
t
t
|
obeys the bounds ||
H
Θ
...||
L
p
→
L
p
≃
p
,
D
(log#
Θ
)
1/2
, for all 1 <
p
< ∞. For vector fields v
D
with range in a lacunary set of order
D
and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field v
D
,
H
v
D
,
1
f
(
x
)
:
=
p
.
v
.
∫
|
t
|
≤
1
f
(
x
+
t
v
D
(
x
)
)
d
t
t
,
satisfies the bounds ||
H
v
D
,1||
L
p
→
L
p
≲
p
,
D
1 for all 1 <
p
< ∞. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele.
We prove
L
p
bounds for the extensions of standard multilinear Calderón–Zygmund operators to tuples of
UMD
spaces tied by a natural product structure. The product can, for instance, mean the ...pointwise product in
UMD
function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond
UMD
on each space—in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance, we prove new versions of fractional Leibniz rules via our results concerning the boundedness of multilinear singular integrals in non-commutative
L
p
spaces. Our proof techniques combine a novel scheme of induction on the multilinearity index with dyadic-probabilistic techniques in the
UMD
space setting.
Endpoint bounds for the bilinear Hilbert transform Di Plinio, Francesco; Thiele, Christoph
Transactions of the American Mathematical Society,
June 1, 2016, 20160601, 2016-6-00, Letnik:
368, Številka:
6
Journal Article
Recenzirano
Odprti dostop
valid for all tuples of sets F_j\subset \mathbb{R} such that \vert f_j\vert \leq \boldsymbol {1}_{F_j}, with the additional restriction that f_3 of F_3 \{F_j:j=1,2,3\} We employ our endpoint results ...to describe the blow-up rate of weak-type and strong-type estimates for \mathrm {BHT} \vec \alpha \mathcal H , improving on results of M. Carro et al. The main technical novelty in our article is an enhanced version of the multi-frequency Calder>ó
We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one‐dimensional subspace by positive sparse forms involving Lp‐averages. This class includes the ...adjoint forms to the bilinear Hilbert transforms. Our result strengthens the Lp‐boundedness proved by Muscalu, Tao and Thiele, and entails as a corollary a novel rich multilinear weighted theory. A particular case of this theory is the Lq(v1)×Lq(v2)‐boundedness of the bilinear Hilbert transform when the weight vj belong to the class Aq+12∩RH2. Our proof relies on a stopping time construction based on newly developed localized outer‐Lp embedding theorems for the wave packet transform. In the Appendix, we show how our domination principle can be applied to recover the vector‐valued bounds for the bilinear Hilbert transforms recently proved by Benea and Muscalu.
Maximal subspace averages Di Plinio, Francesco; Parissis, Ioannis
Advances in mathematics,
12/2022, Letnik:
410
Journal Article
Recenzirano
Odprti dostop
We consider maximal operators associated to singular averages along finite subsets Σ of the Grassmannian Gr(d,n) of d-dimensional subspaces of Rn. The well studied d=1 case corresponds to the ...directional maximal function with respect to arbitrary finite subsets of Gr(1,n)=Sn−1. We provide a systematic study of all cases 1≤d<n and prove essentially sharp L2(Rn) bounds for the maximal subspace averaging operator in terms of the cardinality of Σ, with no assumption on the structure of Σ. In the codimension 1 case, that is n=d+1, we prove the precise critical weak (2,2)-bound.
Drawing on the analogy between maximal subspace averages and (d,n)-Nikodym maximal averages, we also formulate the appropriate maximal Nikodym conjecture for general 1<d<n by providing examples that determine the critical Lp-space for the (d,n)-Nikodym problem. Unlike the d=1 case, the maximal Kakeya and Nikodym problems are shown not to be equivalent when d>1. In this context, we prove the best possible L2(Rn)-bound for the (d,n)-Nikodym maximal function for all combinations of dimension and codimension.
Our estimates rely on Fourier analytic almost orthogonality principles, combined with polynomial partitioning, but we also use spatial analysis based on the precise calculation of intersections of d-dimensional plates in Rn.
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