Hardy-type operators with general kernels and characterizations of dynamic weighted inequalities

In this paper, we prove some new characterizations of the weighted functions such that norm dynamic inequalities of mixed type involving operators of Hardy’s type with general kernels, of the form$$ \Vert \mathcal {H}_{\mathcal {K}}f \Vert _{\mathbb {L}_{u}^{q}([a,\infty )_{\mathbb {T}})}\leq A \Vert f \Vert _{\mathbb {L}_{\upsilon }^{p}([a,\infty )_{\mathbb {T}})}, $$ hold for $1 \lt p\leq q \lt \infty $ and $1 \lt q \lt p \lt \infty ,$ where $\mathcal {H}_{\mathcal {K}}f (x ) :=\int _{a}^{\sigma (x ) }\mathcal {K} (\sigma (x ) ,y ) f(y)\,\varDelta y$ (here $u$ and $\upsilon $ are the weight functions). Corresponding results are also obtained for the adjoint operator $\mathcal {H}_{\mathcal {K}}^{\ast }f ( x ) :=\int _{x}^{\infty }\mathcal {K} (x,\sigma (y) ) f(y)\,\varDelta y,$ where $\sigma (x)$ is the forward jump operator on time scales. Our results include some well known inequalities in the literature.