The relation between Riesz potential and heat kernel on the Heisenberg group is studied. Moreover , the Hardy-Littlewood-Sobolev inequality is established.

New definitions are suggested for frequencies which may be instantaneous or not. The Heisenberg-Gabor inequality and the Shannon sampling theorem are briefly discussed.

The Meda inequality for rearrangements of the convolution operator on the Heisenberg group Hn is proved. By using the Meda inequality, an O’Neil-type inequality for the convolution is obtained. As applications of these results, some sufficient and necessary conditions for the boundedness of the fractional maximal operator MΩ,α and fractional integral operator IΩ,α with rough kernels in the spac...

We discuss the Heisenberg uncertainty inequality for groups of the form K Rn , K is a separable unimodular locally compact group of type I. This inequality is also proved for Gabor transform for several classes of groups of the form K Rn . Mathematics subject classification (2010): Primary 43A32; Secondary 43A30, 22D10, 22D30, 22E25.

We show that the sharp constant in the Hardy-Littlewood-Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range.

In this work, we study the Dirichlet problem for a class of semi-linear subelliptic equations on the Heisenberg group with a singular potential. The singularity is controlled by Hardy’s inequality, and the nonlinearity is controlled by Sobolev’s inequality. We prove the existence of a nontrivial solution for a homogenous Dirichlet problem.

In this thesis we consider the first Heisenberg group and study spectral properties of the Dirichlet sub-Laplacian, also known as Heisenberg Laplacian, which is a sum-ofsquares differential operator of left-invariant vector fields on the first Heisenberg group. In particular, we consider the bound for the trace of the eigenvalues which reflects the correct geometrical constant and order of grow...

1. INTRODUCTION. When studying a metric space, it is valuable to have a mental picture that displays distance accurately. When the space is Z, Q, or R, we usually form such a picture by imagining points on the " number line ". When the space is X = Z 2 , Q 2 , R 2 , or C we use a planar picture in which nonempty discs (sets of the form {x ∈ X : d(x, b) ≤ γ } or {x ∈ X : d(x, b) < β}, with metri...