We establish the embedding of the critical Sobolev-Lorentz-Zygmund space into the generalized Morrey space with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in . O’Neil’s inequality and its reverse play an essential role in the proofs of the main theorems.
We prove optimal embeddings of homogeneous Sobolev spaces built over function spaces in ℝⁿ with K-monotone and rearrangement invariant norm into other rearrangement invariant function spaces. The investigation is based on pointwise and integral estimates of the rearrangement or the oscillation of the rearrangement of f in terms of the rearrangement of the derivatives of f.
We study Sobolev-type embeddings involving rearrangement-invariant norms. In particular, we focus on the question when such embeddings are optimal. We concentrate on the case when the functions involved are defined on Rn. This subject has been studied before, but only on bounded domains. We first establish the equivalence of the Sobolev embedding to a new type of inequality involving two integral operators. Next, we show this inequality to be equivalent to the boundedness of a certain Hardy operator...
This paper continues our study of Sobolev-type imbedding inequalities involving rearrangement-invariant Banach function norms. In it we characterize when the norms considered are optimal. Explicit expressions are given for the optimal partners corresponding to a given domain or range norm.
We establish the sharpness of embedding theorems for Bessel-potential spaces modelled upon Lorentz-Karamata spaces and we prove the non-compactness of such embeddings. Target spaces in our embeddings are generalized Hölder spaces. As consequences of our results, we get continuous envelopes of Bessel-potential spaces modelled upon Lorentz-Karamata spaces.
We prove some quantitatively sharp estimates concerning the Δ₂ and ∇₂ conditions for functions which generalize known ones. The sharp forms arise in the connection between Orlicz space theory and the theory of elliptic partial differential equations.
We consider a convolution operator Tf = p.v. Ω ⁎ f with , where K(x) is an (n,β) kernel near the origin and an (α,β), α ≥ n, kernel away from the origin; h(x) is a real-valued function on . We give a criterion for such an operator to be bounded from the space into itself.
Our understanding of the interplay between Poincaré inequalities, Sobolev inequalities and the geometry of the underlying space has changed considerably in recent years. These changes have simultaneously provided new insights into the classical theory and allowed much of that theory to be extended to a wide variety of different settings. This paper reviews some of these new results and techniques and concludes with an example on the preservation of Sobolev spaces by the maximal function.[Proceedings...
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.