We establish the necessary and sufficient conditions for those symbols $b$ on the Heisenberg group $mathbb H^{n}$ for which the commutator with the Riesz transform is of Schatten class. Our main result generalises classical results of Peller, Janson-
-Wolff and Rochberg--Semmes, which address the same question in the Euclidean setting. Moreover, the approach that we develop bypasses the use of Fourier analysis, and can be applied to characterise that the commutator is of the Schatten class in other settings beyond Euclidean.
For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=frac{1}{N}sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $ell^{p}$-improving inequality begin{e
quation*} |A_N f|_{ell^q(mathbb{Z})} lesssim_{P,p,q} N^{-d(frac{1}{p}-frac{1}{q})} |f|_{ell^p(mathbb{Z})}, qquad N inmathbb{N}, end{equation*} where $1leq p leq q leq infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. For degree three and higher, the inequalities are close to being sharp. In the quadratic case, we appeal to discrete fractional integrals as studied by Stein and Wainger. In the higher degree case, we appeal to the Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and Guth.
Consider averages along the prime integers $ mathbb P $ given by begin{equation*} mathcal{A}_N f (x) = N ^{-1} sum_{ p in mathbb P ;:; pleq N} (log p) f (x-p). end{equation*} These averages satisfy a uniform scale-free $ ell ^{p}$-improving estimate.
For all $ 1< p < 2$, there is a constant $ C_p$ so that for all integer $ N$ and functions $ f$ supported on $ [0,N]$, there holds begin{equation*} N ^{-1/p }lVert mathcal{A}_N frVert_{ell^{p}} leq C_p N ^{- 1/p} lVert frVert_{ell^p}. end{equation*} The maximal function $ mathcal{A}^{ast} f =sup_{N} lvert mathcal{A}_N f rvert$ satisfies $ (p,p)$ sparse bounds for all $ 1< p < 2$. The latter are the natural variants of the scale-free bounds. As a corollary, $ mathcal{A}^{ast} $ is bounded on $ ell ^{p} (w)$, for all weights $ w$ in the Muckenhoupt $A_p$ class. No prior weighted inequalities for $ mathcal{A}^{ast} $ were known.
For 1<p<infty and for weight w in A_p, we show that the r-variation of the Fourier sums of any function in L^p(w) is finite a.e. for r larger than a finite constant depending on w and p. The fact that the variation exponent depends on w is necessary.
This strengthens previous work of Hunt-Young and is a weighted extension of a variational Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses weighted adaptation of phase plane analysis and a weighted extension of a variational inequality of Lepingle.
The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp boundsare obtained for both the fractional integral operators and the associated fr
actional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.
It is shown that product BMO of Chang and Fefferman, defined on the product of Euclidean spaces can be characterized by the multiparameter commutators of Riesz transforms. This extends a classical one-parameter result of Coifman, Rochberg, and Weiss,
and at the same time extends the work of Lacey and Ferguson and Lacey and Terwilleger on multiparameter commutators with Hilbert transforms. The method of proof requires the real-variable methods throughout, which is new in the multi-parameter context.
When is the composition of paraproducts bounded? This is an important, and difficult question, related to to a question of Sarason on composition of Hankel matrices, and the two-weight problem for the Hilbert transform. We consider randomized variant
s of this question, finding non-classical characterizations, for dyadic paraproducts.
Let $ v$ be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform operatorname H_{v, epsilon}f(x) := text{p.v.}int_{-epsilon}^ epsilon f(x-yv
(x)) frac{dy}y where $ epsilon $ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.thinspace M.thinspace Stein, that if $ v$ is Lipschitz, there is a positive $ epsilon $ for which the transform above is bounded on $ L ^{2}$. Our principal result gives a sufficient condition in terms of the boundedness of a maximal function associated to $ v$. This sufficient condition is that this new maximal function be bounded on some $ L ^{p}$, for some $ 1<p<2$. We show that the maximal function is bounded from $ L ^{2}$ to weak $ L ^{2}$ for all Lipschitz maximal function. The relationship between our results and other known sufficient conditions is explored.
We study the Walsh model of a certain maximal truncation of Carlesons operator, related to the Return Times Theorem from Ergodic Theory.
Let h_R denote an L ^{infty} normalized Haar function adapted to a dyadic rectangle R contained in the unit cube in dimension d. We establish a non-trivial lower bound on the L^{infty} norm of the `hyperbolic sums $$ sum _{|R|=2 ^{-n}} alpha(R) h_R (
x) $$ The lower bound is non-trivial in that we improve the average case bound by n^{eta} for some positive eta, a function of dimension d. As far as the authors know, this is the first result of this type in dimension 4 and higher. This question is related to Conjectures in (1) Irregularity of Distributions, (2) Approximation Theory and (3) Probability Theory. The method of proof of this paper gives new results on these conjectures in all dimensions 4 and higher. This paper builds upon prior work of Jozef Beck, from 1989, and first two authors from 2006. These results were of the same nature, but only in dimension 3.
