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...

The purpose of this paper is to describe a unified approach to proving vector-valued inequalities without relying on the full strength of weighted theory. Our applications include the Fefferman-Stein and Córdoba-Fefferman inequalities, as well as the vector-valued Carleson operator. Using this approach we also produce a proof of the boundedness of the classical bi-parameter multiplier operators, which does not rely on product theory. Our arguments are inspired by the vector-valued restricted type...

In this paper we introduce some new modified cosine sums and then using these sums we study $L^1$-convergence of trigonometric cosine series.

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.

We study $L^p$ norm convergence of Bochner-Riesz means $S_R^{\delta }f$ associated with certain non-negative differential operators. When the kernel $S_R^m(x,y)$ satisfies a weak estimate for large values of m we prove $L^p$ norm convergence of $S_R^{\delta }f$ for δ > n|1/p-1/2|, 1 < p < ∞, where n is the dimension of the underlying manifold.

We are interested in Banach space geometry characterizations of quasi-Cohen sets. For example, it turns out that they are exactly the subsets E of the dual of an abelian compact group G such that the canonical injection $C\left(G\right)/C_{E^c}\left(G\right)\hookrightarrow L{²}_E\left(G\right)$ is a 2-summing operator. This easily yields an extension of a result due to S. Kwapień and A. Pełczyński. We also investigate some properties of translation invariant quotients of L¹ which are isomorphic to subspaces of L¹.