engleski

Asymptotic behavior of L^p estimates for a class of multipliers with homogeneous unimodular symbols

We study Fourier multiplier operators associated with symbols ξ↦exp(iλϕ(ξ/|ξ|)), where λ is a real number and ϕ is a real-valued C^∞ function on the standard unit sphere S^(n−1)⊂R^n. For p between 1 and ∞ we investigate asymptotic behavior of norms of these operators on L^p(R^n) as |λ|→∞. We show that these norms are always O((p∗−1)|λ|^(n|1/p−1/2|)), where p∗ is the larger number between p and its conjugate exponent. More substantially, we show that this bound is sharp in all even-dimensional Euclidean spaces R^n. In particular, this gives a negative answer to a question posed by Maz'ya. Concrete operators that fall into the studied class are the multipliers forming the two-dimensional Riesz group, given by the symbols rexp(iφ)↦exp(iλcosφ). We show that their L^p norms are comparable to (p∗−1)|λ|^(2|1/p−1/2|) for large |λ|, solving affirmatively a problem suggested in the work of Dragičević, Petermichl, and Volberg.

Fourier multiplier ; singular integral ; spherical harmonic

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