Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms |
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Authors: | Neal Bez |
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Institution: | School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, King's Buildings, Edinburgh, EH3 9JZ, United Kingdom |
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Abstract: | For all d?2 and p∈(1,max(2,(d+1)/2)], we prove sharp Lp to Lp(Lq) estimates (modulo an endpoint) for a directional maximal operator associated to curves generated by the dilation matrices exp((logt)P), where P has real entries and eigenvalues with positive real part. For the corresponding Hilbert transform we prove an analogous result for all d?2 and p∈(1,2]. As corollaries, we prove Lp bounds for variable kernel singular integral operators and Nikodym-type maximal operators taking averages over certain families of curved sets in Rd. |
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Keywords: | Mixed-norm estimates Nonisotropic Maximal operator Hilbert transform |
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