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

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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 Rni×Mi. 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 operatorSf(x):=(∫0∞|t∇(I+t2Δ)−Mf(x)|2dtt)12 is bounded on Lp(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∈(nmin′,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 operatorsf(x):=(∫0∞|t2Δ(I+t2Δ)−Mf(x)|2dtt)12 is bounded on Lp(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 Hp do not coincide.