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Twisted bilinear spherical maximal functions

Ankit Bhojak, Surjeet Singh Choudhary, Saurabh Shrivastava

Abstract

We obtain $L^p-$estimates for the full and lacunary maximal functions associated to the twisted bilinear spherical averages given by \[\mathfrak{A}_t(f_1,f_2)(x,y)=\int_{\mathbb S^{2d-1}}f_1(x+tz_1,y)f_2(x,y+tz_2)\;dσ(z_1,z_2),\;t>0,\] for all dimensions $d\geq1$. We show that the estimates for such operators in dimensions $d\geq2$ essentially relies on the method of slicing. The bounds for the lacunary maximal function in dimension one is more delicate and requires a trilinear smoothing inequality which is based on an appropriate sublevel set estimate in this context.

Twisted bilinear spherical maximal functions

Abstract

We obtain estimates for the full and lacunary maximal functions associated to the twisted bilinear spherical averages given by for all dimensions . We show that the estimates for such operators in dimensions essentially relies on the method of slicing. The bounds for the lacunary maximal function in dimension one is more delicate and requires a trilinear smoothing inequality which is based on an appropriate sublevel set estimate in this context.

Paper Structure

This paper contains 17 sections, 16 theorems, 153 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Main results
  3. Full maximal function: Proof of \ref{['full']}
  4. Lacunary maximal function : Proof of \ref{['lacunary']}
  5. Fiberwise bilinear Calderón-Zygmund theory: Proof of \ref{['estimateGrowth']}
  6. Trilinear Smoothing Estimate: Proof of \ref{['Trilinearsmoothing']}
  7. Localization of the operator to isolate the degeneracies of $\cos t$ and $\sin t$
  8. Localization of the operator in the space variables $(x, y)$
  9. Reduction of the $(2,2,1)$-estimate to an $(\infty, \infty, 1)$-estimate
  10. Spatial decomposition of the functions
  11. Decomposition of the operator based on the degeneracy at $t=0$
  12. Frequency Pruning and a structural decomposition of the functions.
  13. Conclusion of the proof of \ref{['Trilinearsmoothing']}
  14. Proof of \ref{['lemma:oscillatory']}
  15. Proof of \ref{['lemma:reductiontosublevel']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $1\leq p_1,p_2\leq \infty$ and $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. Then we have for all $f_1\in L^{p_1}({\mathbb {R}}^{2d}),\;f_2\in L^{p_2}({\mathbb {R}}^{2d}),$ and Moreover, we have the following restricted weak-type inequality for all $f_1\in L^{p_1}({\mathbb {R}}^{2d}),\;f_2\in L^{p_2}({\mathbb {R}}^{2d}),$ and

Figures (3)

  • Figure 1: Sketch of proof of \ref{['lacunary']}
  • Figure 2: Sketch of proof of \ref{['Trilinearsmoothing']}
  • Figure 3: Necessary conditions for $\mathfrak{A}_1$

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Proposition 4.1
  • Lemma 4.2
  • ...and 10 more