We prove a maximal Fourier restriction theorem for the sphere S d−1 in R d for any dimension d ≥ 3 in a restricted range of exponents given by the Stein-Tomas theorem. The proof consists of a simple observation. When d = 3 the range corresponds exactly to the full Stein-Tomas one, but is otherwise a proper subset when d > 3. We also present an application regarding the Lebesgue points of functions in F (L p ) when p is sufficiently close to 1.
We apply Christ's method of refinements to the ℓ p -improving problem for discrete averages A N along polynomial curves in Z d . Combined with certain elementary estimates for the number of solutions to certain special systems of diophantine equations, we obtain some restricted weak-type p → p ′ estimates for the averages A N in the subcritical regime. The dependence on N of the constants here obtained is sharp, except maybe for an ǫ-loss.
Let Ω be a collection of disjoint dyadic squares ω, let πω denote the non-smooth bilinear projection onto ω πωpf, gqpxq :"¨1ωpξ, ηq p f pξqp gpηqe 2πipξ`ηqx dξdη and let r ą 2. We show that the bilinear Rubio de Francia operatoŕ ÿ ωPΩ |πωpf, gq| r¯1 {r is L pˆLq Ñ L s bounded with constant independent of Ω whenever 1{p`1{q " 1{s, r 1 ă p, q ă r, and r 1 {2 ă s ă r{2.1991 Mathematics Subject Classification. 42A45.
We apply Christ’s method of refinements to the $$\ell ^p$$
ℓ
p
-improving problem for discrete averages $${\mathcal {A}}_N$$
A
N
along polynomial curves in $${\mathbb {Z}}^d$$
Z
d
. Combined with certain elementary estimates for the number of solutions to certain special systems of diophantine equations, we obtain some restricted weak-type $$p \rightarrow p'$$
p
→
p
′
estimates for the averages $${\mathcal {A}}_N$$
A
N
in the subcritical regime. The dependence on N of the constants here obtained is sharp, except maybe for an $$\epsilon $$
ϵ
-loss.
We consider a class of multiparameter singular Radon integral operators on the Heisenberg group H 1 where the underlying variety is the graph of a polynomial. A remarkable difference with the euclidean case, where Heisenberg convolution is replaced by euclidean convolution, is that the operators on the Heisenberg group are always L 2 bounded. This is not the case in the euclidean setting where L 2 boundedness depends on the polynomial defining the underlying surface. Here we uncover some new, interesting phenomena. For example, although the Heisenberg group operators are always L 2 bounded, the bounds are not uniform in the coefficients of polynomials with fixed degree. When we ask for which polynoimals uniform L 2 bounds hold, we arrive at the same class where uniform bounds hold in the euclidean case.1991 Mathematics Subject Classification. 42B15, 42B20, 43A30, 43A80.
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