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On the Norm of the Beurling–Ahlfors Operator in Several Dimensions

  • Tuomas P. Hytönen (a1)

Abstract

The generalized Beurling–Ahlfors operator $S$ on ${{L}^{p}}({{\mathbb{R}}^{n}};\,\Lambda )$ , where $\Lambda \,:=\,\Lambda ({{\mathbb{R}}^{n}})$ is the exterior algebra with its natural Hilbert space norm, satisfies the estimate

$$||S||L\left( {{L}^{p}}\left( {{\mathbb{R}}^{n}};\Lambda \right) \right)\le \left( n/2+1 \right)\left( {{p}^{*}}-1 \right),\,\,\,{{p}^{*}}:=\,\max \{p,\,{{p}^{'}}\}.$$
.

This improves on earlier results in all dimensions $n\,\ge \,3$ . The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms.

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References

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On the Norm of the Beurling–Ahlfors Operator in Several Dimensions

  • Tuomas P. Hytönen (a1)

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