Sharp ▫$L^p$▫ estimates of powers of the complex Riesz transform
Carbonaro, Andrea ; Dragičević, Oliver ; Kovač, Vjekoslav
Let ▫$R_{1,2}$▫ be scalar Riesz transforms on ▫${\mathbb {R}}^2$▫. We prove that the ▫$L^p$▫ norms of ▫$k$▫-th powers of the operator ▫$R_2+iR_1$▫ behave exactly as ▫$\vert k\vert ...^{1-2/p^*}(p^*-1)$▫, uniformly in ▫$k\in {\mathbb {Z}}\backslash \{0\}$▫ and ▫$1 < p < \infty $▫, where ▫$p^*$▫ is the bigger number between ▫$p$▫ and its conjugate exponent. This gives a complete asymptotic answer to a question suggested by Iwaniec and Martin in 1996. The main novelty are the lower estimates, of which we give three different proofs. We also conjecture the exact value of ▫$\Vert (R_2+iR_1)^k\Vert _p$▫. Furthermore, we establish the sharp behaviour of weak ▫$(1, 1)$▫ constants of ▫$(R_2+iR_1)^k$▫ and an ▫$L^\infty $▫ to ▫$BMO$▫ estimate that is sharp up to a logarithmic factor.
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