Lp estimates for multilinear convolution operators defined with spherical measure

Abstract

Let σ = (σ1, σ2, . . . , σn) ∈ S n−1 and dσ denote the normalized Lebesgue measure on S n−1 , n 2. For functions f1, f2, . . . , fn defined on R, consider the multilinear operator given by

Sparse bounds for the bilinear spherical maximal function

“…The region $$\mathcal {R}_1$$ was first investigated by Oberlin in [36] in the case $$p=q$$. The results were further improved in [3] and then very recently in [42], where some necessary and sufficient conditions for the $$L^p$$ improving estimates for $$\mathcal {A}_1$$ were obtained. However, these conditions do not yet match and one still only has a partial description of $$\mathcal {R}_1$$.…”

“…However, these conditions do not yet match and one still only has a partial description of $$\mathcal {R}_1$$. As these conditions are fairly complicated to state, we omit them here and refer the reader to [42, theorems 2.1 and 2.3].…”

“…As detailed in the “Bad–Good” case in Section 5 below, such assumptions are not necessary once the boundedness region for $$(1/p, 1/q, 1/r)$$ satisfies the following property: for all $$1<p<\infty$$, there exists some $$q^\ast$$ such that $$(1/p, 1/q^\ast , 1/p)$$ is in the interior of the region; symmetrically, for all $$1<q<\infty$$, there exists some $$p^\ast$$ such that $$(1/p^\ast , 1/q, 1/q)$$ is in the interior of the region. Even though the full description of the boundedness region $$\mathcal {R}_1$$ here is not yet available, it is proved in [42, theorem 2.3] that the region contains the convex hull of the points $$(0,0,0), (0,1,1), (1,0,1), (\frac{3}{5}, \frac{3}{5}, \frac{2}{5})$$, hence it has this property.…”

“…It is well known that $$\mathcal {A}_1$$ maps boundedly from $$L^2\times L^2$$ into $$L^1$$ (see e.g. [13, 36, 42]). Fix a parameter $$k\in \mathbb {Z}$$, which shall be chosen later, and decompose $$f_1$$ to get $$$$\begin{align*} \Vert \mathcal {A}_1(f_1 -\tau _h f_1, f_2)\Vert _1\lesssim\, &\Vert \mathcal {A}_1(P_kf_1 - \tau _h P_kf_1, f_2)\Vert _1\\ &+\sum _{i> k}\Vert \mathcal {A}_1(Q_if_1, f_2)\Vert _1+\sum _{i> k}\Vert \mathcal {A}_1(\tau _h Q_if_1, f_2)\Vert _1\\ &=I+II+III.$$…”

“…For instance, the boundedness result for $$\mathcal {M}$$ obtained in [28] only applies in $$d\geqslant 2$$, and such Lebesgue space bounds in $$d=1$$ were only very recently obtained, independently, in [13, 20] via somewhat similar approaches involving parametrizing pieces of the circle. For the single scale operators, the study of the $$L^p$$ improving estimates for $$\mathcal {A}_t$$ in $$d=1$$ was initiated by Oberlin [36] several decades ago but one still only has limited knowledge on the optimal range for such estimates (see [3, 42]). When it comes to the localized maximal function $$\tilde{\mathcal {M}}$$ defined in () below, as far as the authors are aware, there is no known result on its boundedness outside the Hölder range ($$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$$) in the literature.…”