We prove Lp (and weighted Lp) bounds for singular integrals of the formp.v.  ∫Rn E (A(x) - A(y) / |x - y|) (Ω(x - y) / |x - y|n) f(y) dy,where E(t) = cos t if Ω is odd, and E(t) = sin t if Ω is even, and where ∇ A ∈ BMO. Even in the case that Ω is smooth, the theory of singular integrals with rough kernels plays a key role in the proof. By standard techniques, the trigonometric function E can then be replaced by a large class of smooth functions F. Some related operators are also considered. As...

We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the form (*)     x(t) + a(t)(Tx)(t) = b(t), where $T=M_{n₁,k₁}...M_{n_m,k_m}$ and $M_{n_j,k_j}$ are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, (*) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial $P_T\left(t\right)=t³-t$. By means of the Riemann boundary value...

In this paper the notions of uniformly upper and uniformly lower $\ell$-estimates for Banach function spaces are introduced. Further, the pair $(X,Y)$ of Banach function spaces is characterized, where $X$ and $Y$ satisfy uniformly a lower $\ell$-estimate and uniformly an upper $\ell$-estimate, respectively. The integral operator from $X$ into $Y$ of the form $$Kf\left(x\right)=\varphi \left(x\right){\int }_0^{x}k(x,y)f\left(y\right)\psi \left(y\right)\mathrm{d}y$$ is studied, where $k$, $\varphi$, $\psi$ are prescribed functions under some local integrability conditions, the kernel $k$ is non-negative and is assumed to satisfy certain additional...

We prove the convergence of polynomial collocation method for periodic singular integral, pseudodifferential and the systems of pseudodifferential equations in Sobolev spaces $H^s$ via the equivalence between the collocation and modified Galerkin methods. The boundness of the Lagrange interpolation operator in this spaces when $s>1/2$ allows to obtain the optimal error estimate for the approximate solution i.e. it has the same rate as the best approximation of the exact solution by the polynomials.

Con riferimento all’operatore integrodifferenziale della viscoelasticità lineare nella formulazione creep, si determina la soluzione fondamentale $E$ in corrispondenza di un’arbitraria funzione di memoria. Di conseguenza viene risolto esplicitamente il problema di Cauchy relativo al moto unidimensionale di un sistema viscoelastico $ℬ$, omogeneo ed isotropo, determinato da dati iniziali e storia di stress comunque prefissati. Successivamente, nell’ambito di opportune ipotesi di memoria labile, si dimostrano...

Let $P$ be a selfadjoint classical pseudo-differential operator of order $\>1$ with non-negative principal symbol on a compact manifold. We assume that $P$ is hypoelliptic with loss of one derivative and semibounded from below. Then exp$(-tP)$, $t\ge 0$, is constructed as a non-classical Fourier integral operator and the main contribution to the asymptotic distribution of eigenvalues of $P$ is computed. This paper is a continuation of a series of joint works with A. Menikoff.

We show that the number of derivatives of a non negative 2-order symbol needed to establish the classical Fefferman-Phong inequality is bounded by $\frac{n}{2}+4+ϵ$ improving thus the bound $2n+4+ϵ$ obtained recently by N. Lerner and Y. Morimoto. In the case of symbols of type $S_{0,0}^0$, we show that this number is bounded by $n+4+ϵ$; more precisely, for a non negative symbol $a$, the Fefferman-Phong inequality holds if ${\partial }_x^{\alpha }{\partial }_{\xi }^{\beta }a(x,\xi )$ are bounded for, roughly, $4\le \left|\alpha \right|+\left|\beta \right|\le n+4+ϵ$. To obtain such results and others, we first prove an abstract result which says that...

Characterization of the mapping properties such as boundedness, compactness, measure of non-compactness and estimates of the approximation numbers of Hardy-type integral operators in Banach function spaces are given.