The Bilinear Hilbert-Carleson operator along curves. The purely non-zero curvature case
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915373435060224 |
|---|---|
| author | Bényi, Árpád Hu, Bingyang Lie, Victor |
| author_facet | Bényi, Árpád Hu, Bingyang Lie, Victor |
| contents | In this paper, we provide the maximal boundedness range (up to end-points) for the Bilinear Hilbert-Carleson operator along curves in the (purely) non-zero curvature setting. More precisely, we show that the operator $$ BHC_{[\vec{a},\vecα]}(f_1,f_2)(x) := \sup_{λ\in\mathbb{R}} \left|\,p.v.\, \int_{\mathbb{R}} f_1(x - a_1 t^{α_1}) \,f_2(x - a_2 t^{α_2}) \,e^{i\,λ\,a_3 \,t^{α_3}} \,\frac{dt}{t}\right|$$ obeys the bounds $$\|BHC_{[\vec{a},\vecα]} (f_1,f_2)\|_{L^r} \lesssim_{\vec{a} \,\vecα,r,p_1,p_2} \|f_1\|_{L^{p_1}}\,\|f_2\|_{L^{p_2}}$$ whenever $\vec{a}=(a_1,a_2,a_3),\,\vecα=(α_1,α_2,α_3)\in (\mathbb{R}\setminus\{0\})^3$ with $\vecα$ having pairwise distinct coordinates and for any Hölder range $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{r}$ with $1<p_1,p_2<\infty$ and $\frac{1}{2}<r<\infty$. This result is achieved via the Rank II LGC method introduced in arXiv:2308.10706. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04467 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Bilinear Hilbert-Carleson operator along curves. The purely non-zero curvature case Bényi, Árpád Hu, Bingyang Lie, Victor Classical Analysis and ODEs Dynamical Systems Number Theory 42A16, 42A24, 42B15, 42B20 In this paper, we provide the maximal boundedness range (up to end-points) for the Bilinear Hilbert-Carleson operator along curves in the (purely) non-zero curvature setting. More precisely, we show that the operator $$ BHC_{[\vec{a},\vecα]}(f_1,f_2)(x) := \sup_{λ\in\mathbb{R}} \left|\,p.v.\, \int_{\mathbb{R}} f_1(x - a_1 t^{α_1}) \,f_2(x - a_2 t^{α_2}) \,e^{i\,λ\,a_3 \,t^{α_3}} \,\frac{dt}{t}\right|$$ obeys the bounds $$\|BHC_{[\vec{a},\vecα]} (f_1,f_2)\|_{L^r} \lesssim_{\vec{a} \,\vecα,r,p_1,p_2} \|f_1\|_{L^{p_1}}\,\|f_2\|_{L^{p_2}}$$ whenever $\vec{a}=(a_1,a_2,a_3),\,\vecα=(α_1,α_2,α_3)\in (\mathbb{R}\setminus\{0\})^3$ with $\vecα$ having pairwise distinct coordinates and for any Hölder range $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{r}$ with $1<p_1,p_2<\infty$ and $\frac{1}{2}<r<\infty$. This result is achieved via the Rank II LGC method introduced in arXiv:2308.10706. |
| title | The Bilinear Hilbert-Carleson operator along curves. The purely non-zero curvature case |
| topic | Classical Analysis and ODEs Dynamical Systems Number Theory 42A16, 42A24, 42B15, 42B20 |
| url | https://arxiv.org/abs/2507.04467 |