Related papers: On generalized $K$-functionals in $L_p$ for $0<p<1…

We present a class of functions $\mathcal{K}$ in $C^0(\R)$ which is variant of the Knopp class of nowhere differentiable functions. We derive estimates which establish $\mathcal{K} \sub C^{0,\al}(\R)$ for $0<\al<1$ but no $K \in…

For a constant coefficient partial differential operator $P(D)$ with a single characteristic direction such as the time-dependent free Schr\"odinger operator as well as non-degenerate parabolic differential operators like the heat operator…

This paper is a contribution to semiclassical analysis for abstract Schr\"odinger type operators on locally compact spaces: Let $X$ be a metrizable seperable locally compact space, let $\mu$ be a Radon measure on $X$ with a full support.…

We establish square function estimates for integral operators on uniformly rectifiable sets by proving a local $T(b)$ theorem and applying it to show that such estimates are stable under the so-called big pieces functor. More generally, we…

In metric of spaces $L_{s}, \ 1< s\leq\infty$, we obtain exact order estimates of best approximations and approximations by Fourier sums of classes of convolutions the periodic functions that belong to unit ball of space $L_{1}$, with…

Let $G$ be a simple complex Lie group with Weyl group $W$. We give a formula for the character of $W$ on the zero weight space of any finite dimensional representation of $G$. The formula involves partition functions, generalizing Kostant's…

We prove that if $f\in L^p(\mathbb{R}^k)$ with $p<(k^2+k+2)/2$ satisfies that $\widehat{f}$ is supported on a small perturbation of the moment curve in $\mathbb{R}^k$, then $f$ is identically zero. This improves the more general result of…

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial…

We consider a problem of mixed Cauchy type for certain holomorphic partial differential operators whose principal part $Q_{2p}(D)$ essentially is the (complex) Laplace operator to a power, $\Delta^p$. We pose inital data on a singular conic…

We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of $2\pi$-periodic functions $\varphi$, such that $\|\varphi\|_2\le1$, with fixed generated kernels $\Psi_{\bar{\beta}}$,…

In this paper we give a simpler proof of the $L^p$-Schwartz space isomorphism $(0< p\leq 2)$ under the Fourier transform for the class of functions of left $\delta$-type on a Riemannian symmetric space of rank one. Our treatment rests on…

Recently, Charpentier showed that there exist holomorphic functions $f$ in the unit disk such that, for any proper compact subset $K$ of the unit circle, any continuous function $\phi$ on $K$ and any compact subset $L$ of the unit disk,…

We prove a characterization for the Peetre type $K$-functional on $\mathbb{M}$, a compact two-point homogeneous space, in terms the rate of approximation of a family of multipliers operator defined to this purpose. This extends the well…

We establish operator-valued versions of the earlier foundational factorization results for noncommutative polynomials due to Helton (Ann.~Math., 2002) and one of the authors (Linear Alg.~Appl., 2001). Specifically, we show that every…

Let f be a modular form of weight k and Nebentypus $\psi$. By generalizing a construction of Dabrowski and Delbourgo, we construct a p-adic L-function interpolating the special values of the L-function $L(s,\mathrm{Sym}^2(f)\otimes \xi)$,…

We continue our study of group algebras acting on $L^p$-spaces, particularly of algebras of $p$-pseudofunctions of locally compact groups. We focus on the functoriality properties of these objects. We show that $p$-pseudofunctions are…

We characterize those non-negative, measurable functions $\psi$ on $[0,1]$ and positive, continuous functions $\omega_1$ and $\omega_2$ on $\mathbb R^+$ for which the generalized Hardy-Ces\`aro operator $$(U_{\psi}f)(x)=\int_0^1…

Dress and Siebeneicher gave a significant generalization of the construction of Witt vectors, by producing for any profinite group $G$, a ring-valued functor $\mathbf{W}_G$. This paper gives a concrete interpretation of the rings…

We consider the Schr\"odinger operator on the real line with an even quartic potential. Our main result is a product formula of the type $\psi_k(x)\psi_k(y) = \int_{\mathbb{R}} \psi_k(z)\mathcal{K}(x,y,z)dz$ for its eigenfunctions $\psi_k$.…

We show that the boundedness of the Hardy-Littlewood maximal operator on a K\"othe function space ${\mathbb{X}}$ and on its K\"othe dual ${\mathbb{X}}'$ is equivalent to the well-posedness of the $\mathbb{X}$-Dirichlet and…