The classical Whitney extension theorem states that every function in Lip$(\beta ,F)$, $F\subset {\mathbf{R}}^n$, $F$ closed, $k\<\beta \le k+1$, $k$ a non-negative integer, can be extended to a function in Lip$(\beta ,{\mathbf{R}}^n)$. Her Lip$(\beta ,F)$ stands for the class of functions which on $F$ have continuous partial derivatives up to order $k$ satisfying certain Lipschitz conditions in the supremum norm. We formulate and prove a similar theorem in the $L^p$-norm. The restrictions to ${\mathbf{R}}^d$, $d\<n$, of the Bessel potential spaces in ${\mathbf{R}}^n$ and the Besov or generalized Lipschitz...

In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator $T:L^p\left({ℝ}^{+}\right)\to L^p\left({ℝ}^{+}\right)$ defined by $\left(Tf\right)\left(x\right)≔v\left(x\right){ʃ}_0^{\infty }u\left(t\right)f\left(t\right)dt$ when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3].

We determine the set of all triples 1 ≤ p,q,r ≤ ∞ for which the so-called Marcinkiewicz-Zygmund inequality is satisfied: There exists a constant c≥ 0 such that for each bounded linear operator $T:L_q\left(\mu \right)\to L_p\left(\nu \right)$, each n ∈ ℕ and functions $f_1,...,f_n\in L_q\left(\mu \right)$, $\left(ʃ\right({\sum }_{k=1}^n|T{f}_k{|}^r{)}^{p/r}{d\nu )}^{1/p}\le c\parallel T\parallel \left(ʃ\right({\sum }_{k=1}^n|f_k{|}^r{{)}^{q/r}d\mu )}^{1/q}$. This type of inequality includes as special cases well-known inequalities of Paley, Marcinkiewicz, Zygmund, Grothendieck, and Kwapień. If such a Marcinkiewicz-Zygmund inequality holds for a given triple (p,q,r), then we calculate the best constant c ≥ 0 (with the...

Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If $L^{p,\infty }(\Omega ,\sum ,\mu )$ is isomorphic, as a Banach space, to $L^{p,\infty }({\Omega }^{\text{'}},{\sum }^{\text{'}},{\mu }^{\text{'}})$ for some purely atomic measure space (Ω’,∑’,μ’), then there is a measurable partition $\Omega ={\Omega }_1\cup {\Omega }_2$ such that $({\Omega }_1,\Sigma \cap {\Omega }_1,\mu {|}_{\Sigma \cap {\Omega }_1})$ is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable $\sigma \subseteq {\Omega }_2$. In particular, $L^{p,\infty }(\Omega ,\sum ,\mu )$ is isomorphic to ${\ell }^{p,\infty }$.

We consider the Volterra integral operator $T:L^p\left({ℝ}^{+}\right)\to L^p\left({ℝ}^{+}\right)$ defined by $\left(Tf\right)\left(x\right)=v\left(x\right){ʃ}_0^{x}u\left(t\right)f\left(t\right)dt$. Under suitable conditions on u and v, upper and lower estimates for the approximation numbers $a_n\left(T\right)$ of T are established when 1 < p < ∞. When p = 2 these yield $li{m}_{n\to \infty }n{a}_n\left(T\right)={\pi }^{-1}{ʃ}_0^{\infty }\left|u\left(t\right)v\left(t\right)\right|dt$. We also provide upper and lower estimates for the ${\ell }^{\alpha }$ and weak ${\ell }^{\alpha }$ norms of (an(T)) when 1 < α < ∞.

This work deals with various questions concerning Fourier multipliers on $L^p$, Schur multipliers on the Schatten class $S^p$ as well as their completely bounded versions when $L^p$ and $S^p$ are viewed as operator spaces. For this purpose we use subsets of ℤ enjoying the non-commutative Λ(p)-property which is a new analytic property much stronger than the classical Λ(p)-property. We start by studying the notion of non-commutative Λ(p)-sets in the general case of an arbitrary discrete group before turning...

Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if ${ʃ}_{\Omega }xyd\mu \le {\varphi }^{-1}\left({ʃ}_{\Omega }\varphi \circ xd\mu \right){\psi }^{-1}\left({ʃ}_{\Omega }\psi \circ xd\mu \right)$ for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then...

The authors obtain some multiplier theorems on $H^p$ spaces analogous to the classical $L^p$ multiplier theorems of de Leeuw. The main result is that a multiplier operator ${\left(Tf\right)}^{(}x)=\lambda \left(x\right)f̂\left(x\right)$ $\left(\lambda \in C\left({ℝ}^n\right)\right)$ is bounded on $H^p\left({ℝ}^n\right)$ if and only if the restriction ${\lambda \left(\epsilon m\right)}_{m\in \Lambda }$ is an $H^p\left(T^n\right)$ bounded multiplier uniformly for ε>0, where Λ is the integer lattice in ${ℝ}^n$.

We investigate the Fourier transforms of functions in the Sobolev spaces $W_1^{r_1,...,r_n}$. It is proved that for any function $f\in W_1^{r_1,...,r_n}$ the Fourier transform f̂ belongs to the Lorentz space $L^{n/r,1}$, where $r=n{({\sum }_{j=1}^{n}1/r_j)}^{-1}\le n$. Furthermore, we derive from this result that for any mixed derivative $D^{s}f(f\in C_0^{\infty },s=(s_1,...,s_n))$ the weighted norm $\parallel {\left(D^{s}f\right)}^{\wedge }{\parallel }_{L^1\left(w\right)}(w\left(\xi \right)={\left|\xi \right|}^{-n})$ can be estimated by the sum of $L^1$-norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

Let A be a pseudodifferential operator on ${ℝ}^N$ whose Weyl symbol a is a strictly positive smooth function on $W={ℝ}^N\times {ℝ}^N$ such that $|{\partial }^{\alpha }a|\le C_{\alpha }{a}^{1-\varrho }$ for some ϱ>0 and all |α|>0, ${\partial }^{\alpha }a$ is bounded for large |α|, and $li{m}_{w\to \infty }a\left(w\right)=\infty$. Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin. ...

Given a real-valued continuous function ƒ on the half-line [0,∞) we denote by P*(ƒ) the set of all probability measures μ on [0,∞) with finite ƒ-moments ${ʃ}_0^{\infty }ƒ\left(x\right){\mu }^{*n}\left(dx\right)$ (n = 1,2...). A function ƒ is said to have the identification propertyif probability measures from P*(ƒ) are uniquely determined by their ƒ-moments. A function ƒ is said to be a Bernstein function if it is infinitely differentiable on the open half-line (0,∞) and ${(-1)}^n{ƒ}^{(n+1)}\left(x\right)$ is completely monotone for some nonnegative integer n. The purpose...