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.

Two kinds of orthogonal decompositions of the Sobolev space W̊₂¹ and hence also of $W{₂}^{-1}$ for bounded domains are given. They originate from a decomposition of W̊₂¹ into the orthogonal sum of the subspace of the ${\Delta }^k$-solenoidal functions, k ≥ 1, and its explicitly given orthogonal complement. This decomposition is developed in the real as well as in the complex case. For the solenoidal subspace (k = 0) the decomposition appears in a little different form. In the second kind decomposition the ${\Delta }^k$-solenoidal...

Let $𝔾$ be a $k$-step Carnot group. The first aim of this paper is to show an interplay between volume and $𝔾$-perimeter, using one-dimensional horizontal slicing. What we prove is a kind of Fubini theorem for $𝔾$-regular submanifolds of codimension one. We then give some applications of this result: slicing of $B{V}_{𝔾}$ functions, integral geometric formulae for volume and $𝔾$-perimeter and, making use of a suitable notion of convexity, called$𝔾$-convexity, we state a Cauchy type formula for $𝔾$-convex sets. Finally,...

If p ∈ Rn, then we have the radial projection map from Rn {p} onto a sphere. Sometimes one can construct similar mappings on metric spaces even when the space is nontrivially different from Euclidean space, so that the existence of such a mapping becomes a sign of approximately Euclidean geometry. The existence of such spherical mappings can be used to derive estimates for the values of a function in terms of its gradient, which can then be used to derive Sobolev inequalities, etc. In this paper...