We show that some singular maximal functions and singular Radon transforms satisfy a weak type L log logL inequality. Examples include the maximal function and Hilbert transform associated to averages along a parabola. The weak type inequality yields pointwise convergence results for functions which are locally in L log logL. 1.Introduction Let Σ be a compact smooth hypersurface of R, and let μ...

We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in H s (R 3) for s > 4 5. The main new estimate in the argument is a Morawetz-type inequality for the solution φ. This estimate bounds φ(x, t) L 4 x,t (R 3 ×R) , whereas the well-known Morawetz-type estimate of Lin-Strauss controls ∞ 0 R 3 (φ(x,t)) 4 |x| dxdt.

This paper deals with Szegö type limits for multiplication operators on L(R) with respect to Haar orthonormal basis. Similar studies have been carried out by Morrison for multiplication operators Tf using Walsh System and Legendre polynomials [14]. Unlike the Walsh and Fourier basis functions, the Haar basis functions are local in nature. It is observed that Szegö type limit exist for a class o...

We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in H s (R 3) for s > 4 5. The main new estimate in the argument is a Morawetz-type inequality for the solution φ. This estimate bounds φ(x, t) L 4 x,t (R 3 ×R) , whereas the well-known Morawetz-type estimate of Lin-Strauss controls ∞ 0 R 3 (φ(x,t)) 4 |x| dxdt.

Small hairpin RNAs (shRNAs) having duplex lengths of 25-29 bp are normally processed by Dicer into short interfering RNAs (siRNAs) before incorporation into the RNA-induced silencing complex (RISC). However, shRNAs of ≤ 19 bp [short shRNAs (sshRNAs)] are too short for Dicer to excise their loops, raising questions about their mechanism of action. sshRNAs are designated as L-type or R-type accor...