Let N > 1 be a real number and ε > 0 be given. In this paper, we will prove that, for a measurable subset S of [0,N] with positive density ε , there must be patterns of the formwhere γ is convex and has some curvature constraints, t > δ (ε,γ)γ −1 (N) and δ (ε,γ) is a positive constant depending only on ε and γ , γ −1 is the inverse function of γ . Our result extends Bourgain's result [2] to the general curve γ . We use Bourgain's energy pigeonholing argument and Li's σ -uniformity argument.

We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard singular integrals, fractional integral operators, and square functions. These bounds are known to be sharp in many cases, and our main new result is the optimal bound [w,σ ] Mathematics subject classification (2010): 42B20, 42B25.

In this paper we prove several weighted estimates for iterated product commutators generated by BMO -functions and the bilinear fractional integral operators on Morrey spaces. As a corollary we obtain new weighted estimates for Adams type inequality.

Let Ω ∈ L s (S n−1) for s 1 be homogeneous functions of degree zero and b be BMO functions. In this paper, we obtain some boundedness of the Marcinkiewicz integral operator μ Ω and its commutator [b, μ Ω ] on Herz-type Hardy spaces with variable exponent.

In this paper, we consider the mapping property of both m -linear p -adic Hardy operator H p m and Hardy-Littlewood-Pólya operators T p m on p -adic central Morrey-type spaces. The obtained bounds turn to be sharp when λ 1 p 1 = ••• = λ m p m .