Some of the arguments and techniques developed by the authors in a previous paper are applied to the study of the boundedness of certain operators associated with the Ornstein–Uhlenbeck semigroup. In particular, a simple proof is given of the weak type 1 with respect to the Gaussian measure of the Riesz transforms of order 1 and the Littlewood–Paley g‐function which then is extended to show the same property for the Riesz transforms of order 2. For Riesz transforms of higher order boundedness is shown in appropriate spaces close to L1.
The weak type 1 for the Mehler maximal function is studied via a precise estimate for the ‘maximal kernel’. This, in turn, allows the geometry involved in this setting to be described.
Abstract. In this work we establish L p boundedness for maximal functions and Hilbert transforms along variable curves in the plane, via L 2 estimates for certain singular integral operators with oscillatory terms. §1. IntroductionIn this paper, we study the L p (R 2 ) boundedness for the maximal function M and the Hilbert transform H along variable curves. In our discussions, these are defined a priori on functions inandwhere S(x, y) is a suitable real-valued function vanishing on the diagonal. We shall also consider the singular integral operators T λ (acting on functions on the real line) which are of the formLocal versions of the operators T λ have been studied by Phong and Stein [PS], and by Pan [P] who proved the L p boundedness of T λ with bounds independent of λ when the mixed derivative S xy does not vanish to infinite order at any diagonal point (x 0 , x 0 ). In [S] Seeger showed that for a certain class of phases S without this finite type property, the associated operators T λ are also uniformly bounded on L 2 . Here we extend the Seeger-type result, for a different, closely related (but not always directly comparable) class of phases, to all other p, 1 < p < ∞.
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