Powers of unimodular homogeneous multipliers

Abstract: We study asymptotically sharp estimates for the $L^p\to L^p$ norms of multipliers associated with unimodular homogeneous symbols of degree 0, i.e. multipliers associated with symbols $\xi\mapsto \exp(i\lambda\Phi(\xi/|\xi|))$, where $\lambda$ is a real number and $\Phi \in C^{\infty}(S^{n-1})$. We show that that the powers of a generic multiplier in that class exhibit asymptotically maximal order of growth. As a consequence, we disprove Maz'ya's conjecture regarding the asymptotically sharp estimates of such multipliers in all dimensions and solve the problem posed by Dragicevic, Petermichl, and Volberg concerning the sharp lower estimate of a certain multiplier falling within the mentioned class.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

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