We define Sobolev space J V ' , ~ for 1 < p < (x, on an arbitrary metric space with finite diameter and equipped with finite, positive Bore1 measure. In the Euclidean case it coincides with standard Sobolev space. Several classical imbedding theorems are special cases of general results which hold in the metric case. We apply our results to weighted Sobolev space with Muckenhoupt weight. Mathem...

A hybrid inversion scheme for through-the-wall imaging is proposed in this paper. The approach based on a linearized model of the inverse-scattering problem, employing Green’s function developed layered background. reconstruction obtained by means Landweber-like iterative method performing regularization framework variable-exponent Lebesgue spaces. Thanks to non-conventional geometrical propert...

where ⊂RN (N ≥ ) is a smooth bounded domain such that ξi ∈ , i = , , . . . ,k, k ≥ , are different points, ≤ μi < μ̄ := (N–  ), L := – · – ∑k i=μi · |x–ξi| , η,λ,σ ≥ , a,a,a ∈ R,  < α, β < ∗ – , α + β = ∗. We work in the product space H ×H , where the space H :=H ( ) is the completion of C∞  ( ) with respect to the norm ( ∫ |∇ · | dx)   . In resent years many publications...

We develop an axiomatic approach to the theory of Sobolev spaces on metric measure spaces and we show that this axiomatic construction covers the main known examples (Hajtasz Sobolev spaces, weighted Sobolev spaces, Upper-gradients, etc). We then introduce the notion of variational p-capacity and discuss its relation with the geometric properties of the metric space. The notions of p-parabolic ...