Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups

Let be a Schrödinger operator on the Heisenberg group , where is the sublaplacian on and the nonnegative potential V belongs to the reverse Hölder class with . Here is the homogeneous dimension of . Assume that is the heat semigroup generated by. The Lusin area integral and the Littlewood–Paley–Stein function associated with the Schrödinger operator are defined, respectively, bywhereandWhere is a parameter. In this article, the author shows that there is a relationship between and the operator and for any , the following inequality holds true:Based on this inequality and known results for the Lusin area integral , the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function on . In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator on . By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator acting on the Morrey spaces for an appropriate choice of . It can be shown that the same conclusions hold for the operator on generalized Morrey spaces as well.

Lp(·)–Lq(·) boundedness of some integral operators obtained by extrapolation techniques

Given a matrix A such that {A^{M}=I} and {0\leq\alpha<n}, for an exponent p satisfying {p(Ax)=p(x)} for a.e. {x\in\mathbb{R}^{n}}, using extrapolation techniques, we obtain {L^{p(\,\cdot\,)}\rightarrow L^{q(\,\cdot\,)}} boundedness, {\frac{1}{q(\,\cdot\,)}=\frac{1}{p(\,\cdot\,)}-\frac{\alpha}{n}}, and weak type estimates for integral operators of the formT_{\alpha}f(x)=\int\frac{f(y)}{|x-A_{1}y|^{\alpha_{1}}\cdots|x-A_{m}y|^{\alpha% _{m}}}\,dy,where {A_{1},\dots,A_{m}} are different powers of A such that {A_{i}-A_{j}} is invertible for {i\neq j}, {\alpha_{1}+\cdots+\alpha_{m}=n-\alpha}. We give some generalizations of these results.

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.