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.

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...

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).

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 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.

The paper studies the relation between asymptotically developable functions in several complex variables and their extensions as functions of real variables. A new Taylor type formula with integral remainder in several variables is an essential tool. We prove that strongly asymptotically developable functions defined on polysectors have $C^{\infty }$ extensions from any subpolysector; the Gevrey case is included.

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 $j\left(C_1\cap C_2\right)$.

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...

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.

$L^p$-$L^q$ 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.

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 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.

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:...

This paper characterizes the commutant of certain multiplication operators on Hilbert spaces of analytic functions. Let $A=M_z$ be the operator of multiplication by z on the underlying Hilbert space. We give sufficient conditions for an operator essentially commuting with A and commuting with $A^n$ for some n>1 to be the operator of multiplication by an analytic symbol. This extends a result of Shields and Wallen.

Let X be a normed space. A set A ⊆ X is approximately convexif d(ta+(1-t)b,A)≤1 for all a,b ∈ A and t ∈ [0,1]. We prove that every n-dimensional normed space contains approximately convex sets A with $\mathscr{H}(A,Co\left(A\right))\ge lo{g}_{2}n-1$ and $diam\left(A\right)\le C\surd n{\left(lnn\right)}^2$, where ℋ denotes the Hausdorff distance. These estimates are reasonably sharp. For every D>0, we construct worst possible approximately convex sets in C[0,1] such that ℋ(A,Co(A))=(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.

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 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...