For a convergent sequence {xi} generated by xi+l = ~(x~, xt_ l , . . . , Xi_d+l) , define the multiplicative efficiency measure E to be (log~p)/M, where p is the order of convergence and M is the number of multiplications or divisions needed to compute rp. Then, if 9 is any multivariate rational function, E < 1. Since E = 1 for the sequence {xi} generated by x~+l = x~ ~ + xi -~ with the limit -...

In this paper, we obtain the maximal estimate for Weyl sums on torus $\mathbb{T}^d$ with $d\geq 2$, which is sharp up to endpoint. We also consider two variants of problem include along rational lines and generic torus. Applications, some new upper bound Hausdorff dimension sets associated large value sums, reflect compound phenomenon between square root cancellation constructive interference. ...

We derive sparse bounds for the bilinear spherical maximal function in any dimension d ⩾ 1 $d\geqslant 1$ . When 2 2$ , this immediately recovers sharp L p × q → r $L^p\times L^q\rightarrow L^r$ bound of operator and implies quantitative weighted norm inequalities with respect to Muckenhoupt weights, which seems be first their kind operator. The key innovation is a group newly developed continu...

Let G be a non-compact connected semisimple Lie group of real rank one with finite center, K a maximal compact subgroup of G and X = G/K an associated symmetric space of real rank one. We will prove that L(G) ∗ L(G) ⊆ L(G), which is a sharp endpoint estimate for the Kunze-Stein phenomenon. We will also show that the noncentered maximal operator M2f(z) = sup z∈B 1 |B| ∫ B f(z)dz is bounded from ...

We prove a dimension-free estimate for the \(L^2({\mathbb {R}}^d)\) norm of maximal truncated Riesz transform in terms transform. Consequently, vector transforms has on {R}}^d).\) also show that function all \(L^p({\mathbb spaces, \(1<p<\infty .\)