Abstract
We prove that the maximal operator associated with variable homogeneous planar curves (t,ut ^{α} ) _{t∈ℝ} , α≠1 positive, is bounded on L ^{p} (ℝ ^{2} ) for each p>1, under the assumption that u:ℝ ^{2} → ℝ is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with (t,ut ^{α} ) _{t∈ℝ} , α≠1 positive, is bounded on L ^{p} (ℝ ^{2} ) for each p>1, under the assumption that u:ℝ ^{2} →ℝ is a measurable function and is constant in the second variable. Our proofs rely on stationary phase methods, TT∗ arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.
Original language  English 

Pages (fromto)  177219 
Number of pages  43 
Journal  Proceedings of the London Mathematical Society 
Volume  115 
Issue number  1 
Early online date  5 Apr 2017 
DOIs  
Publication status  Published  1 Jul 2017 
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Dive into the research topics of 'Maximal operators and Hilbert transforms along variable nonflat homogeneous curves'. Together they form a unique fingerprint.Profiles

Jonathan Hickman
 School of Mathematics  Lecturer in Mathematical Sciences
Person: Academic: Research Active (Teaching)