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We prove an inequality of the type $$\parallel |x{|}^{r_f}{\parallel }_{L^p({\mathbf{R}}^n)}\le c(n,p,q,r)\parallel |x{|}^{\tau +\mu }\Delta f{\parallel }_{L^q({\mathbf{R}}^n)}.$$ This is then used to derive the unique continuation property for the differential inequality $\left|\Delta f\right(x\left)\right|\le \left|v\right(x\left)\right|\left|f\right(x\left)\right|$ under suitable local integrability assumptions on the function $v$.