Sobolev type inequalities for p>0
The paper deals with spaces of Sobolev type where s > 0, 0 < p ≤ ∞, and their relations to corresponding spaces of Besov type where s > 0, 0 < p ≤ ∞, 0 < q ≤ ∞, in terms of embedding and real interpolation.
In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means...
We define a Sobolev space by means of a generalized Poincaré inequality and relate it to a corresponding space based on upper gradients.
For nonquasianalytical Carleman classes conditions on the sequences and are investigated which guarantee the existence of a function in such that u(n)(a) = bn, bnKn+1Mn, n = 0,1,..., aJ. Conditions of coincidence of the sequences and are analysed. Some still unknown classes of such sequences are pointed out and a construction of the required function is suggested. The connection of this classical problem with the problem of the existence of a function with given trace at the boundary...
Let be a space of homogeneous type, i.e. X is a set, ϱ is a quasi-metric on X with the property that there are constants θ ∈ (0,1] and C₀ > 0 such that for all x,x’,y ∈ X, , and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X, . Let ε ∈ (0,θ], |s| < ε and maxd/(d+ε),d/(d+s+ε) < q ≤ ∞. The author introduces new inhomogeneous Triebel-Lizorkin spaces and establishes their frame characterizations by first establishing a Plancherel-Pólya-type inequality...
New norms for some distributions on spaces of homogeneous type which include some fractals are introduced. Using inhomogeneous discrete Calderón reproducing formulae and the Plancherel-Pólya inequalities on spaces of homogeneous type, the authors prove that these norms give a new characterization for the Besov and Triebel-Lizorkin spaces with p, q > 1 and can be used to introduce new inhomogeneous Besov and Triebel-Lizorkin spaces with p, q ≤ 1 on spaces of homogeneous type. Moreover, atomic...
In this paper, we prove new embedding theorems for generalized anisotropic Sobolev spaces, and , where is the weighted Lorentz space and is a rearrangement invariant space in . The main methods used in the paper are based on some estimates of nonincreasing rearrangements and the applications of weights.