A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type

Given a bounded measurable function σ on R n , we let T be the operator obtained by multiplication Fourier transform . Let 0 < s 1 ≤ 2 ⋯ and ψ Schwartz real line whose ˆ is supported in [ − / ] ∪ which satisfies ∑ j ∈ Z ( ξ ) = for all ≠ In this work provide sharp form of Marcinkiewicz multiplier theorem L p finding an almost optimal space with property that, if … ↦ ∏ i I ∂ belongs to it uniformly then when | ∞ case where + was proved [12] that Lorentz sought. Here address significantly more difficult general certain indices might have We obtain version replaced appropriate associated concave related number terms among equal Our result up arbitrarily small power logarithm defining space.