For every closed subset C in the dual space ${Ĥ}_n$ of the Heisenberg group $H_n$ we describe via the Fourier transform the elements of the hull-minimal ideal j(C) of the Schwartz algebra $S\left(H_n\right)$ and we show that in general for two closed subsets $C_1,C_2$ of ${Ĥ}_n$ the product of $j\left(C_1\right)$ and $j\left(C_2\right)$ is different from $j\left(C_1\cap C_2\right)$.

For every countable ordinal α, we construct an $l_1$-predual $X_{\alpha }$ which is isometric to a subspace of $C\left({\omega }^{{\omega }^{{\omega }^{\alpha }+2}}\right)$ and isomorphic to a quotient of $C\left({\omega }^{\omega }\right)$. However, $X_{\alpha }$ is not isomorphic to a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$.

We prove that there exists an absolute constant c such that for any positive integer n and any system Φ of $2^n$ characters of a compact abelian group, $2^{-n/2}{t}_{\Phi }\left(T\right)\le c{n}^{-1/2}{t}_n\left(T\right)$, where T is an arbitrary operator between Banach spaces, $t_{\Phi }\left(T\right)$ is the type norm of T with respect to Φ and $t_n\left(T\right)$ is the usual Rademacher type-2 norm computed with n vectors. For the system of the first $2^n$ Walsh functions this is even true with c=1. This result combined with known properties of such type norms provides easy access to quantitative...

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...

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.

It is shown that under certain conditions on $c_{jk}$, the rectangular partial sums $s_{mn}(x,y)$ converge uniformly on $T^2$. These conditions include conditions of bounded variation of order (1,0), (0,1), and (1,1) with the weights |j|, |k|, |jk|, respectively. The convergence rate is also established. Corresponding to the mentioned conditions, an analogous condition for single trigonometric series is ${\sum }_{\left|k\right|=n}^{\infty }\left|\Delta c_k\right|=o(1/n)$ (as n → ∞). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition:...

Every separable, infinite-dimensional Banach space X has a biorthogonal sequence $z_n,z{*}_n$, with $spanz{*}_n$ norming on X and $\parallel z_n\parallel +\parallel z{*}_n\parallel$ bounded, so that, for every x in X and x* in X*, there exists a permutation π(n) of n so that $x\in \overline{conv}finitesubseriesof{\sum }_{n=1}^{\infty }z{*}_n\left(x\right)z_{n}andx{*}_n\left(x\right)={\sum }_{n=1}^{\infty }z{*}_{\pi \left(n\right)}\left(x\right)x*\left(z_{\pi \left(n\right)}\right)$.

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.

A holomorphic family $f_z$, |z|<1, of injections of a compact set E into the Riemann sphere can be extended to a holomorphic family of homeomorphisms $F_z$, |z|<1, of the Riemann sphere. (An earlier result of the author.) It is shown below that there exist extensions $F_z$ which, in addition, commute with some holomorphic families of holomorphic endomorphisms of $ℂ̅̅̅̅̅̅f_z\left(E\right)$, |z|<1 (under suitable assumptions). The classes of covering maps and maps with the path lifting property are discussed. ...

Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if $Q{̃}_p{\in }^{inj}(X,X{̃}_p)$, where $X{̃}_p$ is the completion of the normed space $X_p=X/p^{-1}\left(0\right)$ and $Q{̃}_p$ is the canonical map. p is said to be a Groth()- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map $Q{̃}_{pq}:X{̃}_q\to X{̃}_p$ belongs to $(X{̃}_q,X{̃}_p)$. It is well known that when is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only...

We give a characterization of the pairs of weights (v,w), with w in the class $A_{\infty }$ of Muckenhoupt, for which the fractional maximal function is a bounded operator from $L^p(X,vd\mu )$ to $L^q(X,wd\mu )$ when 1 < p ≤ q < ∞ and X is a space of homogeneous type.

For U open in a locally convex space E it is shown in [31] that there is a complete locally convex space G(U) such that $G{\left(U\right)}_i^{\text{'}}=(\mathscr{H}\left(U\right),{\tau }_{\delta })$. Here, we assume U is balanced open in a Fréchet space and give necessary and sufficient conditions for G(U) to be Montel and reflexive. These results give an insight into the relationship between the ${\tau }_0$ and ${\tau }_{\omega }$ topologies on ℋ (U).

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....

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 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 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...

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.