Fractional order derivatives and integrals (differintegrals) are viewed from a frequency-domain perspective using the formalism of Riesz, providing a computational tool as well as a way to interpret the operations in the frequency domain. Differintegrals provide a logical extension of current techniques, generalizing the notion of integral and differential operators and acting as kind of freque...

Consider the second order divergence form elliptic operator L with complex bounded coefficients. In general, the operators related to it (such as Riesz transform or square function) lie beyond the scope of the Calderón-Zygmund theory. They need not be bounded in the classical Hardy, BMO and even some Lp spaces. In this work we generalize the classical approach and develop a theory of Hardy and ...

f(x) det(xx)dx and the Radon transform on the space Mn,m of n × m real matrices x = (xi,j). We present a self-contained proof of the Fourier transform formula for this distribution. Our method differs from that of J. Faraut and A. Korányi [FK] in the part related to justification of the corresponding Bernstein identity. We suggest a new proof of this identity based on explicit representation of...

We are concerned with the following quasilinear Choquard equation: [Formula: see text] where [Formula: see text], [Formula: see text] is the p-Laplacian operator, the potential function [Formula: see text] is continuous and [Formula: see text]. Here, [Formula: see text] is the Riesz potential of order [Formula: see text]. We study the existence of weak solutions for the problem above via the mo...

We prove that the Atiyah-Singer Dirac operator / Dg in L 2 depends Riesz continuously on L∞ perturbations of complete metrics g on a smooth manifold. The Lipschitz bound for the map g → / Dg(1 + / D 2 g) − 12 depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calderón’s first commu...

We obtain a closed form analytic solution for the Dual Phase Lagging equation. This equation is a linear, time-independent partial differential equation modeling the heat distribution in a thin film. The spatial domain is of micrometer and nanometer geometries. We show that the solution is described by a semigroup, and obtain a basis of eigenfunctions. The closure of the set of eigenvalues cont...

The aim of this article is to prove the following theorem. Theorem: Let p be in (1,∞), Hn,m a group of Heisenberg type, R the vector of the Riesz transforms on Hn,m. There exists a constant Cp independent of n and m such that for every f ∈ L (Hn,m) C p e ‖f‖Lp(Hn,m) ≤ ‖|Rf |‖Lp(Hn,m) ≤ Cpe‖f‖Lp(Hn,m). It has been proved by F. Lust-Piquard that the operator norm of R does not depend on n (see [L...

We study the asymptotic behavior, as Planck’s constant ~ → 0, of the number of discrete eigenvalues and the Riesz means of Pauli and Dirac operators with a magnetic field μB(x) and an electric field. The magnetic field strength μ is allowed to tend to infinity as ~ → 0. Two main types of results are established: in the first μ~ ≤ constant as ~ → 0, with magnetic fields of arbitrary direction; t...

We now consider the family of operators Tλ, λ ≥ 0, defined on Rn by T̂λ f (ξ) = mλ f̂ (ξ), where mλ(ξ) = (1− |ξ|)+. These are the Bochner-Riesz multipliers, which can be viewed as an attempt to smooth out the singularity of the disc multiplier to see if we can obtain boundedness on a wider range of Lp spaces. Note that when λ = 0 we obtain S1, and that as λ increases, the multiplier mλ becomes sm...

We prove complex interpolation theorems for analytic families of multilinear operators defined on quasi-Banach spaces, with explicit constants on the intermediate spaces. We obtain analogous results for analytic families of operators defined on spaces generated by the Calderón method applied to couples of quasi-Banach lattices with nontrivial lattice convexity. As an application we derive a mul...