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Let      k k       be a positive integer and      1 &gt; p &gt; ∞  1 &gt; p &gt; \infty      . It is shown that if      T T       is a multiplier operator on      L p {L^p}       of the line with weight      | x | k p −  1 |x{|^{kp-1}}      , then      T f Tf       equals a constant times      f f       almost everywhere. This does not extend to the periodic case since      m ( j ) = 1 / j , j ≠  0 m(j) = 1/j, j \ne 0      , is a multiplier sequence for      L p {L^p}       of the circle with weight      | x | k p −  1 |x{|^{kp-1}}      . A necessary and sufficient condition is derived for a sequence      m ( j ) m(j)       to be a multiplier on      L 2 {L^2}       of the circle with weight      | x | 2 k −  1 |x{|^{2k - 1}}      .

𝐿^{𝑝} multipliers with weight 𝑥^{𝑘𝑝-1}