Our aim is to establish Sobolev type inequalities for fractional maximal functions \({M_{\mathbb{H},\nu }}f\) and Riesz potentials \({I_{\mathbb{H},\alpha}}f\) in weighted Morrey spaces of variable exponent on the half space \(\mathbb{H}\). We also obtain Sobolev type inequalities for a C¹ function on \(\mathbb{H}\). As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents Φ(x, t) = t^p(x) + (b(x)t)^q(x), where p(·) and q(·) satisfy log-Hölder conditions, p(x) > q(x) for \(x \in \mathbb{H}\), and b(·) is nonnegative and Hölder continuous of order θ ∈ (0,1].

我们的目标是建立分数极大函数\({M_{\mathbb{H},\nu }}f\)和 Riesz 势\({I_{\mathbb{H},\alpha}}f\ 的Sobolev 型不等式）在半空间\(\mathbb{H}\)上的可变指数加权莫里空间中。我们还获得了\(\mathbb{H}\)上C ¹函数的 Sobolev 型不等式。作为一个应用，我们获得了具有可变指数 Φ( x, t ) = t ^{p ( x )} + ( b ( x ) t ) ^{q ( x )}的双相泛函的 Sobolev 型不等式，其中p (·) 和q (·) ) 满足 log-Hölder 条件，p ( x ) > q ( x ) 对于\(x \in \mathbb{H}\)，并且b (·) 是非负且阶数θ ∈ (0,1] 的 Hölder 连续。