Let (M, g) be a complete Riemannian manifold which satisfies a Sobolev inequality of dimension n, and on which the volume growth is comparable to the one of R for big balls; if there is no non-zero L harmonic 1-form, and the Ricci tensor is in L n 2 −ε ∩ L∞ for an ε > 0, then we prove a Gaussian estimate on the heat kernel of the Hodge Laplacian acting on 1-forms. This allows us to prove that, ...

We prove L estimates for the maximal Riesz transform in terms of the Riesz transform itself, for 1 < p ≤ ∞. We show that the corresponding weak L1 estimate fails for the maximal Riesz transform, but surprisingly does hold for the maximal Beurling transform.

Suucient conditions for the convergence in distribution of an inn-nite convolution product 1 2 : : : of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions that satisfy T = where T is a reenement operator constructed from a measure and a dilation automorphism A. The existence of A implies G is nilpoten...

Let λ be a positive number, and let (xj : j ∈ Z) ⊂ R be a fixed Riesz-basis sequence, namely, (xj) is strictly increasing, and the set of functions {R ∋ t 7→ ej : j ∈ Z} is a Riesz basis (i.e., unconditional basis) for L2[−π,π]. Given a function f ∈ L2(R) whose Fourier transform is zero almost everywhere outside the interval [−π, π], there is a unique square-summable sequence (aj : j ∈ Z), depe...

In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (...

Sufficient conditions for the convergence in distribution of an infinite convolution product μ1 ∗μ2 ∗ . . . of measures on a connected Lie group G with respect to left invariant Haar measure are derived. These conditions are used to construct distributions φ that satisfy Tφ = φ where T is a refinement operator constructed from a measure μ and a dilation automorphism A. The existence of A implie...

The only quadrature operator of order two on L2(R) which covaries with orthogonal transforms, in particular rotations is (up to the sign) the Riesz transform. This property was used for the construction of monogenic wavelets and curvelets. Recently, shearlets were applied for various signal processing tasks. Unfortunately, the Riesz transform does not correspond with the shear operation. In thi...

The Riesz transform is a natural multidimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of...