On off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients

We investigate off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients on $$\mathrm {L}^2_{\sigma } ({\mathbb {R}}^d)$$ L σ 2 ( R d ) . Such estimates are well-known for elliptic equations in the form of pointwise heat kernel bounds and for elliptic systems in the form of integrated off-diagonal estimates. On our way to unveil this off-diagonal behavior we prove resolvent estimates in Morrey spaces $$\mathrm {L}^{2 , \nu } ({\mathbb {R}}^d)$$ L 2 , ν ( R d ) with $$0 \le \nu < 2$$ 0 ≤ ν < 2 .

Estimates for Fractional Integral Operators and Linear Commutators on Certain Weighted Amalgam Spaces

In this paper, we first introduce some new classes of weighted amalgam spaces. Then, we give the weighted strong-type and weak-type estimates for fractional integral operators Iγ on these new function spaces. Furthermore, the weighted strong-type estimate and endpoint estimate of linear commutators b,Iγ generated by b and Iγ are established as well. In addition, we are going to study related problems about two-weight, weak-type inequalities for Iγ and b,Iγ on the weighted amalgam spaces and give some results. Based on these results and pointwise domination, we can prove norm inequalities involving fractional maximal operator Mγ and generalized fractional integrals ℒ−γ/2 in the context of weighted amalgam spaces, where 0<γ<n and L is the infinitesimal generator of an analytic semigroup on L2Rn with Gaussian kernel bounds.

Operators with Gaussian kernel bounds on mixed Morrey spaces

Let L be an analytic semigroup on L2(Rn) with Gaussian kernel bound, and let L-?/2 be the fractional operator associated to L for 0 < ? < n. In this paper, we prove some boundedness properties for the commutator [b,L-?/2] on Mixed Morrey spaces Lq,? (0,T,Lp,?(Rn)), when b belongs to BMO(Rn) or to suitable homogeneous Lipschitz spaces.