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Supposing that the metric space in question supports a fractional diffusion, we prove that after introducing an appropriate multiplicative factor, the Gagliardo seminorms ${\left|\right|f\left|\right|}_{W^{\sigma ,2}}$ of a function f ∈ L²(E,μ) have the property $1/Cℰ(f,f)\le lim in{f}_{\sigma \nearrow 1}{\left(1-\sigma \right)\left|\right|f\left|\right|}_{W^{\sigma ,2}}\le lim su{p}_{\sigma \nearrow 1}{\left(1-\sigma \right)\left|\right|f\left|\right|}_{W^{\sigma ,2}}\le Cℰ(f,f)$, where ℰ is the Dirichlet form relative to the fractional diffusion.

We outline the construction of Brownian motion on certain self-similar fractals and introduce the notion of walk dimension. We then show how the probabilistic approach relates to the theory of function spaces on fractals.

We obtain interpolation inequalities for derivatives: $\int M_{q,\alpha }\left(\right|\nabla f\left(x\right)\left|\right)dx\le C[\int M_{p,\beta }\left(\Phi ₁\right(x,\left|f\right|,|{\nabla }^{\left(2\right)}f\left|\right))dx+\int M_{r,\gamma }\left(\Phi ₂\right(x,\left|f\right|,|{\nabla }^{\left(2\right)}f\left|\right)\left)dx\right]$, and their counterparts expressed in Orlicz norms: ||∇f||²(q,α) ≤ C||Φ₁(x,|f|,|∇(2)f|)||(p,β) ||Φ₂(x,|f|,|∇(2)f|)||(r,γ)$,$where ${\left|\right|·\left|\right|}_{(s,\kappa )}$ is the Orlicz norm relative to the function $M_{s,\kappa }\left(t\right)=t^s{\left(ln(2+t)\right)}^{\kappa }$. The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces with higher...

Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities ${\int }_{ℝ₊}M\left(\omega \left(x\right)\right|u\left(x\right)\left|\right)exp(-\phi \left(x\right))dx\le C{\int }_{ℝ₊}M\left(\right|u^{\text{'}}\left(x\right)\left|\right)exp(-\phi \left(x\right))dx$, where u belongs to some set of locally absolutely continuous functions containing $C{₀}^{\infty }\left(ℝ₊\right)$. We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.

We derive inequalities of Gagliardo-Nirenberg type in weighted Orlicz spaces on ℝⁿ, for maximal functions of derivatives and for the derivatives themselves. This is done by an application of pointwise interpolation inequalities obtained previously by the first author and of Muckenhoupt-Bloom-Kerman-type theorems for maximal functions.

We obtain Hardy type inequalities $${\int }_0^{\infty }M\left(\omega \left(r\right)\left|u\left(r\right)\right|\right)\rho \left(r\right)dr⩽C_1{\int }_0^{\infty }M\left(\left|u\left(r\right)\right|\right)\rho \left(r\right)dr+C_2{\int }_0^{\infty }M\left(\left|u^{\text{'}}\left(r\right)\right|\right)\rho \left(r\right)dr,$$ and their Orlicz-norm counterparts $${∥\omega u∥}_{L^M({ℝ}_{+},\rho )}⩽{\tilde{C}}_1{∥u∥}_{L^M({ℝ}_{+},\rho )}+{\tilde{C}}_2{∥u^{\text{'}}∥}_{L^M({ℝ}_{+},\rho )},$$ with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.

Given a nondegenerate harmonic structure, we prove a Poincaré-type inequality for functions in the domain of the Dirichlet form on nested fractals. We then study the Hajłasz-Sobolev spaces on nested fractals. In particular, we describe how the "weak"-type gradient on nested fractals relates to the upper gradient defined in the context of general metric spaces.

We obtain new variants of weighted Gagliardo-Nirenberg interpolation inequalities in Orlicz spaces, as a consequence of weighted Hardy-type inequalities. The weights we consider need not be doubling.