We study Fourier integral operators of Hörmander’s type acting on the spaces $ℱL^p{\left({ℝ}^d\right)}_{comp}$, 1 ≤ p ≤ ∞, of compactly supported distributions whose Fourier transform is in $L^p$. We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank r of the Hessian of the phase Φ(x,η) with respect to the space variables x. Indeed, we show that operators of order m = -r|1/2-1/p| are bounded on $ℱL^p{\left({ℝ}^d\right)}_{comp}$ if the mapping $x\mapsto {\nabla }_x\Phi (x,\eta )$ is constant on the fibres, of codimension r, of an affine...

In this paper, the boundedness of the Riesz potential generated by generalized shift operator $I_{B_k}^{\alpha }$ from the spaces $a=(L_{p_m,\nu }\left({ℝ}_n^k\right),a_m)$ to the spaces $a^{\text{'}}=(L_{q_m,\nu }\left({ℝ}_n^k\right),a_m^{\text{'}})$ is examined.

Let $(X,d,\mu )$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $L^2\left(X\right)$. Assume that the semigroup ${\mathrm{e}}^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimate of order $m$ and $L$ satisfies the Plancherel type estimate. Let $H_L^p\left(X\right)$ be the Hardy space associated with $L.$ We show the boundedness of Stein’s square function ${𝒢}_{\delta }\left(L\right)$ arising from Bochner-Riesz means associated to $L$ from Hardy spaces $H_L^p\left(X\right)$ to $L^p\left(X\right)$, and also study the boundedness...

Some boundedness results are established for sublinear operators on the homogeneous Herz spaces. As applications, some new theorems about the boundedness on homogeneous Herz spaces for commutators of singular integral operators are obtained.

The main purpose of this paper is to investigate the behavior of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non-doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on Euclidean space and their known extensions for doubling...

Let T: H → H be an operator in the complex Hilbert space H. Suppose that T is square bounded in average in the sense that there exists a constant M(T) with the property that, for all natural numbers n and for all x ∈ H, the inequality $1/(n+1){\sum }_{j=0}^n\parallel T^{j}x{\parallel }^2\le M{\left(T\right)}^2\parallel x{\parallel }^2$ is satisfied. Also suppose that the adjoint T* of the operator T is square bounded in average with constant M(T*). Then the operator T is power bounded in the sense that $sup\parallel T^{i}n\parallel :n\in \mathbb{N}$ is finite. In fact the following inequality is valid for all n ∈ ℕ: ∥Tn∥ ≤ e M(T)M(T*). Suppose...

Mathematics Subject Classification: 47A56, 47A57,47A63We derive bounds for the norms of the fractional powers of operators with compact Hermitian components, and operators having compact inverses in a separable Hilbert space. Moreover, for these operators, as well as for dissipative operators, the constants in the moment inequalities are established.* This research was supported by the Kamea Fund of Israel.

Let $f$ be a non-zero positive vector of a Banach lattice $L$, and let $T$ be a positive linear operator on $L$ with the spectral radius $r\left(T\right)$. We find some groups of assumptions on $L$, $T$ and $f$ under which the inequalities $$sup\{c\ge 0:Tf\ge cf\}\le r\left(T\right)\le inf\{c\ge 0:Tf\le cf\}$$ hold. An application of our results gives simple upper and lower bounds for the spectral radius of a product of positive operators in terms of positive eigenvectors corresponding to the spectral radii of given operators. We thus extend the matrix result obtained by Johnson and Bru which...

Let $ℒ=$ -div $\left(A\right(x\left)\nabla \right)$ be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ${ℝ}^n$ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed $p\>2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform $\nabla {\left(ℒ\right)}^{-1/2}$ on the $L^p$ space. As an application, for $1\<p\<3+ϵ$, we establish the $L^p$ boundedness of Riesz transforms on Lipschitz domains for operators with $VMO$ coefficients. The range of $p$ is sharp. The closely related boundedness of ...