For any subanalytic $C^k$-Whitney field (k finite), we construct its subanalytic $C^k$-extension to ${ℝ}^n$. Our method also applies to other o-minimal structures; e.g., to semialgebraic Whitney fields.

We prove an abstract comparison principle which translates gaussian cotype into Rademacher cotype conditions and vice versa. More precisely, let 2 < q < ∞ and T: C(K) → F a continuous linear operator. (1) T is of gaussian cotype q if and only if ${\left({\sum }_k{(\left(\parallel T{x}_k{\parallel }_F\right)/\left(\surd log(k+1)\right))}^q\right)}^{1/q}\le c\parallel {\sum }_k{\varepsilon }_k{x}_k{\parallel }_{L_2\left(C\left(K\right)\right)}$, for all sequences ${\left(x_k\right)}_{k\in \mathbb{N}}\subset C\left(K\right)$ with ${\left(\parallel T{x}_k\parallel \right)}_{k=1}^n$ decreasing. (2) T is of Rademacher cotype q if and only if $({\sum }_k{\left(\parallel T{x}_k{\parallel }_F\surd \left({\left(log(k+1)\right)}^q\right)\right)}^{1/q}\le c\parallel {\sum }_k{g}_k{x}_k{\parallel }_{L_2\left(C\left(K\right)\right)}$, for all sequences ${\left(x_k\right)}_{k\in \mathbb{N}}\subset C\left(K\right)$ with ${\left(\parallel T{x}_k\parallel \right)}_{k=1}^n$ decreasing. Our method allows a restriction to a fixed number of vectors and complements the corresponding results of...

This paper gives upper and lower bounds for moments of sums of independent random variables $\left(X_k\right)$ which satisfy the condition ${P\left(\right|X|}_k\ge t)=exp(-N_k\left(t\right))$, where $N_k$ are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which $N\left(t\right)={\left|t\right|}^r$ for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapień who have done the same for convex functions N.

We investigate a scale of $BM{O}_{\psi }$-spaces defined with the help of certain Lorentz norms. The results are applied to extrapolation techniques concerning operators defined on adapted sequences. Our extrapolation works simultaneously with two operators, starts with $BM{O}_{\psi }$-$L_{\infty }$-estimates, and arrives at $L_p$-$L_p$-estimates, or more generally, at estimates between K-functionals from interpolation theory.

For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from ${\ell }^{q̅}\left(L_v^{p̅}\right)$ into ${\ell }^q\left(L_u^p\right)$. For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted $L^p$-spaces. Amalgams of the form ${\ell }^q\left(L_w^p\right)$, 1 < p,q < ∞ , q ≠ p, $w\in A_p$, are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.

We give a representation of the spaces $C^{\infty }\left({ℝ}^N\right)\cap H^{k,p}\left({ℝ}^N\right)$ as spaces of vector-valued sequences and use it to investigate their topological properties and isomorphic classification. In particular, it is proved that $C^{\infty }\left({ℝ}^N\right)\cap H^{k,2}\left({ℝ}^N\right)$ is isomorphic to the sequence space $s^{\mathbb{N}}\cap l^2\left(l^2\right)$, thereby showing that the isomorphy class does not depend on the dimension N if p=2.

We prove that every quotient algebra of a unital Banach function algebra A has a unique complete norm if A is a Ditkin algebra. The theorem applies, for example, to the algebra A (Γ) of Fourier transforms of the group algebra $L^1\left(G\right)$ of a locally compact abelian group (with identity adjoined if Γ is not compact). In such algebras non-semisimple quotients $A\left(\Gamma \right)/\overline{J\left(E\right)}$ arise from closed subsets E of Γ which are sets of non-synthesis. Examples are given to show that the condition of Ditkin cannot be relaxed....

We give an example of a compact set K ⊂ [0, 1] such that the space ℇ(K) of Whitney functions is isomorphic to the space s of rapidly decreasing sequences, and hence there exists a linear continuous extension operator $L:ℇ\left(K\right)\to C^{\infty }[0,1]$. At the same time, Markov’s inequality is not satisfied for certain polynomials on K.

We consider a convolution operator Tf = p.v. Ω ⁎ f with $\Omega \left(x\right)=K\left(x\right)e^{ih\left(x\right)}$, where K(x) is an (n,β) kernel near the origin and an (α,β), α ≥ n, kernel away from the origin; h(x) is a real-valued $C^{\infty }$ function on ${ℝ}^n\setminus 0$. We give a criterion for such an operator to be bounded from the space $H_0^p\left({ℝ}^n\right)$ into itself.

$L^p$-$L^q$ boundedness properties are obtained for operators defined by convolution with measures supported on certain curves on the Heisenberg group. We find the curvature condition for which the type set of these operators can be the full optimal trapezoid with vertices A=(0,0), B=(1,1), C=(2/3,1/2), D=(1/2,1/3). We also give notions of right curvature and left curvature which are not mutually equivalent.

A new characterization is given for the pairs of weight functions v, w for which the fractional maximal function is a bounded operator from $L_v^p\left(X\right)$ to $L_w^q\left(X\right)$ when 1 < p < q < ∞ and X is a homogeneous space with a group structure. The case when X is n-dimensional Euclidean space is included.

We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable $L^1$ to $L^1$ Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and $L^1$...

We study various characterizations of the Hardy spaces $H^p\left(\mathbb{Z}\right)$ via the discrete Hilbert transform and via maximal and square functions. Finally, we present the equivalence with the classical atomic characterization of $H^p\left(\mathbb{Z}\right)$ given by Coifman and Weiss in [CW]. Our proofs are based on some results concerning functions of exponential type.

Let D be a bounded strictly pseudoconvex domain of ${ℂ}^n$ with smooth boundary. We consider the weighted mixed-norm spaces $A_{\delta ,k}^{p,q}\left(D\right)$ of holomorphic functions with norm $\parallel f{\parallel }_{p,q,\delta ,k}=({\sum }_{\left|\alpha \right|\le k}{ʃ}_0^{r_0}({ʃ}_{\partial D_r}|D^{\alpha }{f|}^{p}d{\sigma }_r{{)}^{q/p}{r}^{\delta q/p-1}dr)}^{1/q}$. We prove that these spaces can be obtained by real interpolation between Bergman-Sobolev spaces $A_{\delta ,k}^p\left(D\right)$ and we give results about real and complex interpolation between them. We apply these results to prove that $A_{\delta ,k}^{p,q}\left(D\right)$ is the intersection of a Besov space $B_s^{p,q}\left(D\right)$ with the space of holomorphic functions on D. Further, we obtain several properties...

We costruct functions in $H_w^1$ ($w\in A_1$) whose Fourier integral expansions are almost everywhere non-summable with respect to the Bochner-Riesz means of the critical order.

Let E be a Banach space. Let $L{¹}_{\left(1\right)}({ℝ}^d,E)$ be the Sobolev space of E-valued functions on ${ℝ}^d$ with the norm ${ʃ}_{{ℝ}^d}\parallel f{\parallel }_{E}dx+{ʃ}_{{ℝ}^d}\parallel \nabla f{\parallel }_{E}dx=\parallel f\parallel ₁+\parallel \nabla f\parallel ₁$. It is proved that if $f\in L{¹}_{\left(1\right)}({ℝ}^d,E)$ then there exists a sequence $\left(g_m\right)\subset L_{\left(1\right)}¹({ℝ}^d,E)$ such that $f={\sum }_m{g}_m$; ${\sum }_m(\parallel g_m\parallel ₁+\parallel \nabla g_m\parallel ₁)<\infty$; and $\parallel g_m{\parallel }_{\infty }^{1/d}\parallel g_m\parallel {₁}^{(d-1)/d}\le b\parallel \nabla g_m\parallel ₁$ for m = 1, 2,..., where b is an absolute constant independent of f and E. The result is applied to prove various refinements of the Sobolev type embedding $L_{\left(1\right)}¹({ℝ}^d,E)\hookrightarrow L²({ℝ}^d,E)$. In particular, the embedding into Besov spaces $L{¹}_{\left(1\right)}({ℝ}^d,E)\hookrightarrow B_{p,1}^{\theta (p,d)}({ℝ}^d,E)$ is proved, where $\theta (p,d)=d(p^{-1}+d^{-1}-1)$ for 1 < p ≤ d/(d-1), d=1,2,... The latter embedding in the scalar case is due to Bourgain...