Vertical and horizontal square functions on a class of non-doubling manifolds

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×M_i. Following on from the work of Hassell and the second author [20], a particular decomposition of the resolvent operators (Δ+k²)^−M, for M∈N^⁎, will be used to demonstrate that the vertical square function operator [Formula presented] is bounded on L^p(M) for 1<p<n_min=min_i⁡n_i and weak-type (1,1). In addition, it will be proved that the reverse inequality ‖f‖_p≲‖Sf‖_p holds for p∈(n_min^′,n_min) and that S is unbounded for p≥n_min provided 2M<n_min. Similarly, for M>1, this method of proof will also be used to ascertain that the horizontal square function operator [Formula presented] is bounded on L^p(M) for all 1<p<∞ and weak-type (1,1). Hence, for p≥n_min, the vertical and horizontal square function operators are not equivalent and their corresponding Hardy spaces H^p do not coincide.