Define weight $w(\theta) = (1+c \cos^2 \theta)^{s},$ $s>0$ $c$ some constant.
Define $\|f\|_{L^p_w(\mathbb T)}^p=\int_{\mathbb T}|f(\theta)|^p w^{p}(\theta) d\theta $
Can we expect $\|f\ast g\|_{L^p_{w}(\mathbb T)} \leq \|g\|_{L^1(\mathbb T)} \|f\|_{L^p_{w}(\mathbb T)}$?