Sharp endpoint $L^p-$estimates for Bilinear spherical maximal functions

Abstract

In this article, we address endpoint issues for the bilinear spherical maximal functions. We obtain borderline restricted weak type estimates for the well studied bilinear spherical maximal function $$\mathfrak{M}(f,g)(x):=\sup_{t>0}\left|\int_{\mathbb S^{2d-1}}f(x-ty_1)g(x-ty_2)\;dσ(y_1,y_2)\right|,$$ in dimensions $d=1,2$ and as an application, we deduce sharp endpoint estimates for the multilinear spherical maximal function. We also prove $L^p-$estimates for the local spherical maximal function in all dimensions $d\geq 2$, thus improving the boundedness left open in the work of Jeong and Lee (https://doi.org/10.1016/j.jfa.2020.108629). We further study necessary conditions for the bilinear maximal function, \[\mathcal M (f,g)(x)=\sup_{t>0}\left|\int_{\mathbb S^{1}}f(x-ty)g(x+ty)\;dσ(y)\right|\] to be bounded from $L^{p_1}(\mathbb R^2)\times L^{p_2}(\mathbb R^2)$ to $L^p(\mathbb R^2)$ and prove sharp results for a linearized version of $\mathcal M$.

Figures (6)

• Figure 1: The figure denotes the region of $L^{p_1,1}\times L^{p_2,1}\to L^{p,\infty}$ boundedness of $\mathcal{M}$ on the line segments $U_1U_2\cup U_2U_3$ and $V_1V_2$ in dimensions one and two respectively.
• Figure 2: The figure denotes the $L^{p_1}({\mathbb {R}}^d)\times L^{p_1}({\mathbb {R}}^d)\to L^p({\mathbb {R}}^d)$ boundedness of $\mathfrak{M}_{loc}$ when $\left(\frac{1}{p_1},\frac{1}{p}\right)$ is contained in the convex hull generated by the points $A=\left(\frac{1}{2},0\right),\;E=\left(\frac{2d-3}{2(d-1)},\frac{d-2}{d(d-1)}\right),\;F=\left(\frac{(2d-1)^2}{2(2d^2-d+1)},\frac{2d-3}{2d^2-d+1}\right),\;B'=\left(\frac{2d^2-d+1}{2(d^2+1)},\frac{d-1}{d^2+1}\right),\;C=\left(\frac{2d-1}{2d},\frac{1}{d}\right),$ and $D=\left(\frac{2d-1}{2d},\frac{2d-1}{d}\right)$. The light grey region $OAECD$ was previously obtained in MaximalEstimatesForTheBilinearSphericalAveragesAndTheBilinearBochnerRieszOperators and the blue region $FB'C$ remains open.
• Figure 3: The figure denotes the region of $L^p({\mathbb {R}}^d)\to L^q({\mathbb {R}}^d)$ boundedness of $\mathfrak{A}^r$ when $\left(\frac{1}{p},\frac{1}{q}\right)$ is contained in the closed convex hull generated by the points $O,A,P,Q,R$ excluding the points $P,Q,R$.
• Figure 4: The figure denotes the region of $L^p({\mathbb {R}}^d)\to L^q({\mathbb {R}}^d)$ boundedness of $\mathfrak{A}^r$ when $\left(\frac{1}{p},\frac{1}{q}\right)$ is contained in the closed convex hull generated by the points $O,A,P,Q,R$ excluding the points $P,Q,R$.
• Figure 5: The table prescribes the values of the exponents used for the interpolation Lemma \ref{['interpolation']} to obtain restricted weak type bounds for $\mathfrak{A}^r$.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2: DosidisRamos
• Corollary 1.3
• Theorem 1.4: MaximalEstimatesForTheBilinearSphericalAveragesAndTheBilinearBochnerRieszOperators
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8: Schlag
• Theorem 1.9
• Theorem 1.10: GIKL