Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^p$, Sobolev, and some new Hardy spaces naturally associated to $L$. First, we show that the...

We study weighted function spaces of Lebesgue, Besov and Triebel-Lizorkin type where the weight function belongs to some Muckenhoupt ${}_p$ class. The singularities of functions in these spaces are characterised by means of envelope functions.

2000 Mathematics Subject Classification: 42B20, 42B25, 42B35We consider the generalized shift operator, generated by the Laplace- Bessel differential operator [...] The Bn -maximal functions and the Bn - Riesz potentials, generated by the Laplace-Bessel differential operator ∆Bn are investigated. We study the Bn - Riesz potentials in the Bn - Morrey spaces and Bn - BMO spaces. An inequality of Sobolev - Morrey type is established for the Bn - Riesz potentials.* This paper has been partially supported...

Let ${ℒ}_1=-\Delta +V$ be a Schrödinger operator and let ${ℒ}_2={(-\Delta )}^2+V^2$ be a Schrödinger type operator on ${ℝ}^n$$(n\ge 5)$, where $V\ne 0$ is a nonnegative potential belonging to certain reverse Hölder class...

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.

We provide new assertions on factorization of tent spaces.

This paper deals with function spaces of varying smoothness $B_p^{,s₀}\left(ℝⁿ\right)$, where the function :x ↦ s(x) determines the smoothness pointwise. Those spaces were defined in [2] and treated also in [3]. Here we prove results about interpolation, trace properties and present a characterization of these spaces based on differences.

We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space c₀.

The paper is devoted to spaces of generalized smoothness on so-called h-sets. First we find quarkonial representations of isotropic spaces of generalized smoothness on ℝⁿ and on an h-set. Then we investigate representations of such spaces via differences, which are very helpful when we want to find an explicit representation of the domain of a Dirichlet form on h-sets. We prove that both representations are equivalent, and also find the domain of some time-changed Dirichlet form on an h-set.

We prove a restricted weak type inequality for the spherical means operator with respect to measures with finite α-energy, α ≤ 1. This complements recent results due to D. Oberlin.

We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, $L^p$ and BMO of classical ℒ-square functions.

We prove Strichartz's conjecture regarding a characterization of Hardy-Sobolev spaces.

We present a direct proof of a known result that the Hardy operator Hf(x) = 1/x ∫0x f(t) dt in the space L2 = L2(0, ∞) can be written as H = I - U, where U is a shift operator (Uen = en+1, n ∈ Z) for some orthonormal basis {en}. The basis {en} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order y' - 1/x y = g and point out some generalizations to the case with weighted Lw2(a, b) spaces.