Abstract
This paper is dedicated to study weighted inequalities for pseudodifferential operators with amplitudes and their commutators by using the new class of weights and the new BMO function space BMO∞which are larger than the Muckenhoupt class of weights and classical BMO space BMO, respectively. The obtained results therefore improve substantially some well-known results.
Highlights
Introduction and the Main ResultsFor ffff ff ffff∞ 0 (Rnnnn) a pseudodifferential operator given formally by TTTTaaaaffff (xxxx) = 1 (2ππππ)nnnn Rnnnn aaaa xxxxxx xxxxxx xxxx eeeeiiiixxxxiixxxxxxxxxxiiffff xxxx ddddxxxx ddddxxxxxx (1)where the amplitude aaaa satis es certain growth conditions. e boundedness of pseudodifferential operators has been studied extensively by many mathematicians; see, for example, [1,2,3,4,5,6,7] and the references therein
If aaaa bb bbbb∞SSSSmρρmρmρm with ρρρρ bb δδδδδδδδδδ and mmmm mmγγγγγγγγγγγγγγ, the authors in [5] proved that the pseudodifferential operator TTTTaaaa and BMO functions δδbbbbρρ ρρρρaaaaδδ are bounded on its commutators LLLLpppp(wwwwpp for pp pp pppp with mm m and wwww bb bbbbpppp; see [5, eorems 3.3 and 4.5]
Let us consider the family of balls {BBBBBBBBBB, jjjjBB) LL BBBB jj Rnnnn}
Summary
For ffff ff ffff∞ 0 (Rnnnn) a pseudodifferential operator given formally by TTTTaaaaffff (xxxx). By using eorem 3, we reobtain wwww bb bbbb∞ pppp the boundedness of and obtain the TTTTaaaa on LLLLpppp(wwwwpp for pp new result on the pp pppp mm m and boundedness of its commutator with BMO∞ functions. If aaaa bb bbbb∞SSSSmρρmρmρm with ρρρρ bb δδδδδδδδδδ and mmmm mmγγγγγγγγγγγγγγ, the authors in [5] proved that the pseudodifferential operator TTTTaaaa and BMO functions δδbbbbρρ ρρρρaaaaδδ are bounded on its commutators LLLLpppp(wwwwpp for pp pp pppp with mm m and wwww bb bbbbpppp; see [5, eorems 3.3 and 4.5]. En we have the following: the pppp mm commutator m and wwww bb bbbδbδb∞ pbppbpbρρ .ρρρρaaaa δδ is It was proved in [5, eorem 3.7] that if aaaa bb bbbb∞AAAAmρρmρmρm with θθ ≤ on ρρρρ LLLLpppp.
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