Pointwise Multipliers of Dyadic Besov Spaces via Haar-Fourier Coefficients

Source abstract

For a function $\varphi$ on [0,1] let $T_\varphi$ be the operator which maps the Haar--Fourier coefficients of f to the Haar-Fourier coefficients of the product $f\varphi$. We study the class $\mathcal{M}(b^{s}_{p,q})$ of those $\varphi$ for which $T_\varphi$ is bounded on the dyadic Besov sequence space $b^{s}_{p,q}$; by Triebel's theorem this is the class of pointwise multipliers of the Besov space $B^{s}_{p,q}[0,1]$ in the range 1/p-1<s<1/p, and it contains the multipliers of $\ell^{p}$ as the special case $s=1/p-1/2$. We prove that $\mathcal{M}(b^{s}_{p,q})\subset L^{\infty}$ always, with strict inclusion unless s=0; that $\mathcal{M}(b^{s}_{p,q})=\mathcal{M}(b^{-s}_{p',q'})$ with equality of norms; and we give an exact formula for the multiplier norm on $b^{s}_{1,1}$, 0<s<1, in terms of a Carleson-type condition on the dyadic oscillations of $\varphi$. Two independent sufficient conditions follow: $\varphi\in B^{s}_{1/s,1}$ (sharp in the third index) and $\varphi$ of bounded variation, the latter with explicit constants. Using Stein's interpolation theorem for analytic families we show that $\mathcal{M}(b^{s}_{1,1})\subset\mathcal{M}(b^{\sigma}_{p,q})$ whenever -s/p'<$\sigma$<s/p; consequently every function of bounded variation is a Haar multiplier of every $b^{\sigma}_{p,q}$ in Triebel's range -1/p'<$\sigma$<1/p, and $\mathrm{BV}\cup B^{1/2}_{2,1}\subset\mathcal{M}(\ell^{1})\subset\mathcal{M}(\ell^{p})\subsetneq L^{\infty}$ for $p\neq 2$.