The paper deals with spaces $L_p^s\left(ℝⁿ\right)$ of Sobolev type where s > 0, 0 < p ≤ ∞, and their relations to corresponding spaces $B_{p,q}^s\left(ℝⁿ\right)$ 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 $\left\{{\widehat{M}}_n\right\}$ and $\left\{M_n\right\}$ are investigated which guarantee the existence of a function in $C_J\left\{{\widehat{M}}_n\right\}$ such that u(n)(a) = bn,    bnKn+1Mn,    n = 0,1,...,    aJ. Conditions of coincidence of the sequences $\left\{{\widehat{M}}_n\right\}$ and $\left\{M_n\right\}$ 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 ${(X,\varrho ,\mu )}_{d,\theta }$ 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, $|\varrho (x,y)-\varrho (x^{\text{'}},y)|\le C₀\varrho {(x,x^{\text{'}})}^{\theta }{[\varrho (x,y)+\varrho (x^{\text{'}},y)]}^{1-\theta }$, and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X, $\mu \left(y\in X:\varrho (x,y)<r\right)\sim r^d$. Let ε ∈ (0,θ], |s| < ε and maxd/(d+ε),d/(d+s+ε) < q ≤ ∞. The author introduces new inhomogeneous Triebel-Lizorkin spaces $F_{\infty q}^s\left(X\right)$ 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, $W_{{\Lambda }^{p,q}\left(w\right)}^{r_1,\cdots ,r_n}$ and $W_X^{r_1,\cdots ,r_n}$, where ${\Lambda }^{p,q}\left(w\right)$ is the weighted Lorentz space and $X$ is a rearrangement invariant space in ${ℝ}^n$. The main methods used in the paper are based on some estimates of nonincreasing rearrangements and the applications of $B_p$ weights.