L bounds of maximal operators along variable planar curves in the Lipschitz regularity

Hilbert transforms along variable planar curves: Lipschitz regularity

“…Indeed, one can also apply Lemma 5. It was proven in [26] that, under the same assumptions as in Theorem 1.1, it holds that (5.30) }M γ f } p À }f } p for every p ą 1. Moreover, one can upgrade such scalar-valued bound into a vector-valued bound for free.…”

“…for every l P N. Here k 1 " k 1 pv x , k, l 0 ´lq and l 0 " l 0 pv x , kq. In order to prove the above vector-valued estimates, we will appeal to the L p -boundedness of the maximal operators that were established in [26]. Indeed, one can also apply Lemma 5.…”

“…In the next lemma, we prove a local smoothing estimate, which, combined with Lemma 7.2 and the Sobolev embedding inequality, further implies Lemma 7.1. We refer to [26] for a more detailed discussion. Lemma 7.3.…”

“…In the assumption piiiq of the above theorem, it is more than enough to take N " 10 3 . Some examples of curves satisfying all of the assumptions in Theorem 1.1 can be found in [26,Example 1.5]. In particular, the non-flat homogeneous curves rts α satisfy our assumptions in Theorem 1.1, where the notation rts α stands for either |t| α or sgnptq|t| α , α P p0, 1q Y p1, 8q.…”

“…Remark 1.2. In [26], for 1 ă p ď 8, Liu, Song and Yu obtained the L p -boundedness of the corresponding maximal function M γ along the variable curve pt, upx 1 , x 2 qγptqq, under the same assumptions imposed on u and γ as in Theorem 1.1. Therefore, Theorem 1.1 should be regarded as a natural continuation of [26].…”