3,161 research outputs found
A short proof of Stein's universal multiplier theorem
We give a short proof of Stein's universal multiplier theorem, purely by
probabilistic methods, thus avoiding any use of harmonic analysis techniques
(complex interpolation or transference methods)
Bi-stochastic kernels via asymmetric affinity functions
In this short letter we present the construction of a bi-stochastic kernel p
for an arbitrary data set X that is derived from an asymmetric affinity
function {\alpha}. The affinity function {\alpha} measures the similarity
between points in X and some reference set Y. Unlike other methods that
construct bi-stochastic kernels via some convergent iteration process or
through solving an optimization problem, the construction presented here is
quite simple. Furthermore, it can be viewed through the lens of out of sample
extensions, making it useful for massive data sets.Comment: 5 pages. v2: Expanded upon the first paragraph of subsection 2.1. v3:
Minor changes and edits. v4: Edited comments and added DO
estimates for the Hilbert transforms along a one-variable vector field
Stein conjectured that the Hilbert transform in the direction of a vector
field is bounded on, say, whenever is Lipschitz. We establish a wide
range of estimates for this operator when is a measurable,
non-vanishing, one-variable vector field in \bbr ^2. Aside from an
estimate following from a simple trick with Carleson's theorem, these estimates
were unknown previously. This paper is closely related to a recent paper of the
first author (\cite{B2}).Comment: 25 page
Estimates for compositions of maximal operators with singular integrals
We prove weak-type (1,1) estimates for compositions of maximal operators with
singular integrals. Our main object of interest is the operator
where is Bourgain's maximal multiplier operator and is the
sum of several modulated singular integrals; here our method yields a
significantly improved bound for the operator norm when . We
also consider associated variation-norm estimates
Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group: an expanded version
Marcinkiewicz multipliers are L^{p} bounded for 1<p<\infty on the Heisenberg
group H^{n}\simeqC^{n}\timesR (D. Muller, F. Ricci and E. M. Stein) despite the
lack of a two parameter group of automorphic dilations on H^{n}. This lack of
dilations underlies the inability of classical one or two parameter Hardy space
theory to handle Marcinkiewicz multipliers on H^{n} when 0<p\leq1. We address
this deficiency by developing a theory of flag Hardy spaces H_{flag}^{p} on the
Heisenberg group, 0<p\leq1, that is in a sense `intermediate' between the
classical Hardy spaces H^{p} and the product Hardy spaces H_{product}^{p} on
C^{n}\timesR. We show that flag singular integral operators, which include the
aforementioned Marcinkiewicz multipliers, are bounded on H_{flag}^{p}, as well
as from H_{flag}^{p} to L^{p}, for 0<p\leq1. We characterize the dual spaces of
H_{flag}^{1} and H_{flag}^{p}, and establish a Calder\'on-Zygmund decomposition
that yields standard interpolation theorems for the flag Hardy spaces
H_{flag}^{p}. In particular, this recovers the L^{p} results by interpolating
between those for H_{flag}^{p} and L^{2} (but regularity sharpness is lost).Comment: At 113 pages, this is an expanded version of the paper that includes
much detai
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