Certain Schur-Hadamard multipliers in the spaces $C_{p}$

Abstract:

Let $f$ be a continuously differentiable function on $[ - 1,1]$ satisfying $\left | {f’(t)} \right | \leqslant C{\left | t \right |^\alpha }$ for some $0 < C$, $\alpha < \infty$ and all $- 1 \leqslant t \leqslant 1$, and let $\lambda = ({\lambda _i}) \in {l_r}$ satisfy $- 1 \leqslant {\lambda _i} \leqslant 1$ for all $i$. Then \[ {a_{f,\lambda }} = \left ( {\frac {{f({\lambda _i}) - f({\lambda _j})}} {{{\lambda _i} - {\lambda _j}}}} \right )\] is a Schur-Hadamard multiplier from ${C_{{p_1}}}$ into ${C_{{p_2}}}$ for all ${p_1}$, ${p_2}$ satisfying $1 \leqslant {p_2} \leqslant 2 \leqslant {p_1} \leqslant \infty$ and $p_2^{ - 1} \leqslant p_1^{ - 1} + \alpha /r$.