Ψ-pseudodifferential operators and estimates for maximal oscillatory integrals

Abstract

Abstract. We define a class of pseudodifferential operators with symbols a(x, ξ) without any regularity assumptions in the x variable and explore their L p boundedness properties. The results are applied to obtain estimates for certain maximal operators associated with oscillatory singular integrals.

Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes

“…It is a pointwise bound of operators in OPL ∞ S m ρ by a maximal function and corresponds to the L p -boundedness of OPL ∞ S m ρ in the region C ∪ D of Fig. 1 obtained in [18]. Such an estimate immediately leads to weighted boundedness results of the form (1.2).…”

“…However, in the smooth case we can go on to consider end-point cases by using an interpolation technique. The results of this paper extend the applications derived in [18] and as a separate application we prove new boundedness results for the commutators of pseudodifferential operators with functions of bounded mean oscillation (written BMO, see Definition 4.1).…”

“…However, if δ ρ, then the boundedness results are known to be different [15,19], as we will see played out in this paper. The symbols of (b) in Definition 2.2 were introduced in [18], and Definition 2.2(b) is the natural generalization of this to amplitudes. They are, for example, much rougher than those considered by S. Nishigaki [22] and K. Yabuta [25] in their investigations of weighted norm inequalities for pseudodifferential operators.…”

“…We do this first in the context of symbols introduced by C.E. Kenig and W. Staubach in [18] which are only measurable in the spatial variable. Our methods also extend to amplitudes and we obtain some sharp results in this direction.…”

New Weighted Norm Inequalities for Pseudodifferential Operators and Their Commutators

“…For further information about these two classes, we refer the reader to, for example, [3,10]. e class ∞ was introduced by [11], and it is the natural generalization of the class . is class is much rougher than that considered in [6,7].…”