Vertical and horizontal Square Functions on a Class of Non-Doubling Manifolds

Abstract

We consider a class of non-doubling manifolds M that are the connected sum of a finite number of N -dimensional manifolds of the form R n i × Mi. Following on from the work of Hassell and the second author [19], a particular decomposition of the resolvent operators (∆ + k 2 ) −M , for M ∈ N * , will be used to demonstrate that the vertical square function operatoris bounded on L p (M) for 1 < p < nmin = mini ni and weak-type (1, 1). In addition, it will be proved that the reverse inequality f p Sf p holds for p ∈ (n ′ min , nmin) and that S is unbounded for p ≥ nmin provided 2M < nmin.Similarly, for M > 1, this method of proof will also be used to ascertain that the horizontal square function operatoris bounded on L p (M) for all 1 < p < ∞ and weak-type (1, 1). Hence, for p ≥ nmin, the vertical and horizontal square function operators are not equivalent and their corresponding Hardy spaces H p do not coincide.