0 30 70 65 v 3 2 A pr 2 00 4 Generalised definitions of certain functions and their uses

X iv : m at h - ph / 0 30 70 65 v 2 6 A ug 2 00 3 Generalised definitions of certain functions and their uses

Generalised definitions of exponential, trigonometric sine and cosine and hyperbolic sine and cosine functions are given. In the lowest order, these functions correspond to ordinary exponential, trigonometric sine etc. Some of the properties of the generalised functions are discussed. Importance of these functions and their possible applications are also considered. PACS No.: 02.30.Hq, 02.30.Gp...

ar X iv : 0 90 4 . 42 75 v 1 [ m at h . FA ] 2 7 A pr 2 00 9 INVERSION POSITIVITY AND THE SHARP HARDY – LITTLEWOOD – SOBOLEV INEQUALITY

We give a new proof of certain cases of the sharp HLS inequality. Instead of symmetric decreasing rearrangement it uses the reflection positivity of inversions in spheres. In doing this we extend a characterization of the minimizing functions due to Li and Zhu.

ar X iv : 0 70 8 . 37 30 v 1 [ m at h . PR ] 2 8 A ug 2 00 7 DENSITIES FOR ROUGH DIFFERENTIAL EQUATIONS UNDER HÖRMANDER ’ S CONDITION

We consider stochastic differential equations dY = V (Y ) dX driven by a multidimensional Gaussian process X in the rough path sense. Using Malliavin Calculus we show that Yt admits a density for t ∈ (0, T ] provided (i) the vector fields V = (V1, ..., Vd) satisfy Hörmander’s condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fract...

ar X iv : 0 70 4 . 15 37 v 1 [ m at h . C A ] 1 2 A pr 2 00 7 ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION

In this note, we study singular integrals with rough kernels, which belong to a class of singular Radon transforms. We prove certain estimates for the singular integrals that are useful in an extrapolation argument. As an application, we prove L p boundedness of the singular integrals under a certain sharp size condition on their kernels.

ar X iv : 0 70 4 . 33 68 v 2 [ gr - q c ] 2 6 A pr 2 00 7 Search for gravitational waves from binary inspirals in S 3 and S 4 LIGO data