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Abstract
We consider a class of nondoubling manifolds M that are the connected sum of a finite number of Ndimensional manifolds of the form R^{ni}×M_{i}. Following on from the work of Hassell and the second author [20], a particular decomposition of the resolvent operators (Δ+k^{2})^{−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 weaktype (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 weaktype (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.
Original language  English 

Pages (fromto)  41102 
Number of pages  62 
Journal  Journal of Differential Equations 
Volume  358 
DOIs  
Publication status  Published  15 Jun 2023 
Keywords
 Nondoubling spaces
 Resolvent estimates
 Square functions
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Harmonic analysis of rough oscillations
Sikora, A., Portal, P., Hassell, A., Guillarmou, C. & van Neerven, J.
30/05/16 → …
Project: Research

Nonlinear harmonic analysis and dispersive partial differential equations
Sikora, A., Guo, Z., Hauer, D. & Tacy, M.
8/04/20 → 31/12/23
Project: Research