Related papers: Split functions, Fourier transforms and multiplier…

We prove boundedness results for integral operators of fractional type and their higher order commutators between weighted spaces, including $L^p$-$L^q$, $L^p$-$BMO$ and $L^p$-Lipschitz estimates. The kernels of such operators satisfy…

Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes…

Given a charge and current distribution with compact support, the associated potentials and fields are generally not integrable in the classical sense. However, it is convenient to be able to define their Fourier transform in order to…

The present paper, is devoted to investigation of operator--valued Fourier multiplier theorems from $B_{q_{1},r}^{s}$ to $B_{q_{2},r}^{s}$, optimal embedding of Besov spaces, the separability and positivity of differential operators. Here,…

The Fourier operator truncated on a finite symmetric interval is considered. The limiting behavior of its spectrum is discussed as the length of the interval tends to infinity.

We generalize the idea of a multiplier in two different ways and generalize a recent result of Geiss, Montomery-Smith and Saksman. First of all, we consider multipliers in the form of a vector acting on a scalar function. Using this…

We introduce the Segal-Bargmann transform associated to the Mittag Leffler Fock space and study how it will be connected to the Fourier transform. We will discuss also the counterpart of the creation and annihilation operator in this…

Let $f$ be a normalized primitive Hecke eigen cusp form of even integral weight $k$ for the full modular group $SL(2,\mathbb{Z})$. For integers $j \geq 2$, let $\lambda_{sym^j f}(m)$ denote the $m$th Fourier coefficient of the $j$th…

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp boundsare obtained for both the fractional integral operators and the…

For $s > 0$, $s \neq 1$, bilinear Fourier multipliers of the form $e^{i (|\xi|^s + |\eta|^s+ |\xi + \eta|^s)} \sigma (\xi, \eta)$ are considered, where $\sigma(\xi, \eta)$ belongs to the H\"ormander class $S^{m}_{1, 0}(\mathbb{R}^{2n})$. A…

Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that operator $L$ generates the analytic semigroup $\{e^{-tL}\}_{t>0}$ whose kernels satisfy the standard Gaussian upper bounds. This paper investigates the sharp weighted…

According to a celebrated result by L\"owner, a real-valued function $f$ is operator monotone if and only if its L\"owner matrix, which is the matrix of divided differences $L_f=(\frac{f(x_i)-f(x_j)}{x_i-x_j})_{i,j=1}^N$, is positive…

We study the regularity of Fourier integral operators, by allowing their symbols to satisfy certain multi-parameter characteristics. As a result, we give an extension of Seeger-Sogge-Stein theorem on product spaces.

The classical Fourier transform on the line sends the operator of multiplication by $x$ to $i\frac{d}{d\xi}$ and the operator of differentiation $\frac{d}{d x}$ to the multiplication by $-i\xi$. For the Fourier transform on the Lobachevsky…

We show how to improve the efficiency of the computation of fast Fourier transforms over F_p where p is a word-sized prime. Our main technique is optimisation of the basic arithmetic, in effect decreasing the total number of reductions…

Let $U(\mathbb T)$ be the space of all continuous functions on the circle $\mathbb T$ whose Fourier series converges uniformly. Salem's well-known example shows that a product of two functions in $U(\mathbb T)$ does not always belongs to…

We consider Fourier transform of vector-valued functions on a locally compact group $G$, which take value in a Banach space $X$, and are square-integrable in Bochner sense. If $G$ is a finite group then Fourier transform is a bounded…

We prove endpoint results for sparse domination of translation invariant multiscale operators. The results are formulated in terms of dilation invariant classes of Fourier multipliers based on natural localized $M^{p\to q}$ norms which…

We study Fourier multiplier operators associated with symbols $\xi\mapsto \exp(i\lambda\phi(\xi/|\xi|))$, where $\lambda$ is a real number and $\phi$ is a real-valued $C^\infty$ function on the standard unit sphere…

In this article, we study $m$-order logarithmic Laplacian $\mathcal{L}_m$, which is a singular integro-differential operator with symbol $\big(2\ln |\cdot|\big)^m$ by the Fourier transform. With help of these logarithmic Laplacians, we…